Trigonometric Graphs and Identities Chapter Overview and Pacing

Transcription

1 Trigonometric Graphs and Identities Chapter verview and Pacing PCING (das) Regular Block Basic/ Basic/ verage dvanced verage dvanced Graphing Trigonometric Functions (pp. 7 7) optional optional Graph trigonometric functions. Find the amplitude and period of variation of the e, coe, and tangent functions. LESSN BJECTIVES Translations of Trigonometric Graphs (pp ) optional optional Graph horizontal translations of trigonometric graphs and find phase shifts. Graph vertical translations of trigonometric graphs. Trigonometric Identities (pp ) optional optional Use identities to find trigonometric values. Use trigonometric identities to simplif epressions. Verifing Trigonometric Identities (pp. 7 7) optional optional Verif trigonometric identities b transforming one side of an equation into the form of the other side. Verif trigonometric identities b transforming each side of the equation into the same form. Sum and Difference of ngles Formulas (pp ) optional optional 0. Find values of e and coe involving sum and difference formulas. Verif identities b ug sum and difference formulas. Double-ngle and Half-ngle Formulas (pp ) optional optional 0. Find values of e and coe involving double-angle formulas. Find values of e and coe involving half-angle formulas. Solving Trigonometric Equations (pp. 79 0) optional optional Preview: Solving Trigonometric Equations (with -7 (with -7 Solve trigonometric equations. Preview) Preview) Use trigonometric equations to solve real-world problems. Stud Guide and Practice Test (pp. 0 09) optional optional 0. Standardized Test Practice (pp. 0 ) Chapter ssessment optional optional TTL Pacing suggestions for the entire ear can be found on pages T0 T. 70 Chapter Trigonometric Graphs and Identities

2 Timesaving Tools Chapter Resource Manager ll-in-ne Planner and Resource Center See pages T T. Stud Guide and Intervention CHPTER RESURCE MSTERS Practice (Skills and verage) Reading to Learn Mathematics Enrichment ssessment Materials graphing calculator, posterboard pplications* -Minute Check Transparencies Interactive Chalkboard lgepss: Tutorial Plus (lessons) 7 9 GCS, graphing calculator, grid paper, string, SC 7 masking tape, rope 9 0 GCS , SC, -7-7 (Preview: graphing calculator) SM 79 9, 9 9 *Ke to bbreviations: GCS Graphing Calculator and Speadsheet Masters, SC School-to-Career Masters, SM Science and Mathematics Lab Manual Chapter Trigonometric Graphs and Identities 70B

3 Mathematical Connections and Background Continuit of Instruction Prior Knowledge In the previous chapter students investigated the si trigonometric functions and worked with angles measured in degrees or radians, as ras in standard position, and as points on the unit circle. lso, the eplored periodicit and inverse trigonometric functions. This Chapter Students use the trigonometric functions to eplore amplitude and period, and the investigate phase shifts and vertical shifts in the graphs of trigonometric functions. The learn how to verif and use trigonometric identities, including sum and difference formulas, double-angle formulas, and half-angle formulas. Finall the solve trigonometric equations ug the ideas of factoring, the zero product propert, trigonometric inverses, and periodic behavior. Future Connections Students will continue to stud amplitude, period, and frequenc for trigonometric and other periodic functions. In later math courses the will use trigonometric formulas and identities, and frequentl when the stud graphs the will analze vertical and horizontal translations of graphs and how those changes are related to changes in the algebraic description of the graph. Graphing Trigonometric Functions This chapter continues the etensive investigation of trigonometric functions from the previous chapter. This lesson focuses on graphs of the trigonometric functions. Each of the e, coe, and tangent functions repeats a pattern of values, or is periodic. For the e and coe functions, the period is radians or 0, while for the tangent function the period is radians or 0. For periodic functions the distance between a horizontal center line and the maimum or minimum value is called the amplitude of the graph. For and cos, the horizontal center line is the -ais and the maimum and minimum values are, so the amplitude is. The lesson also describes these properties algebraicall. For the e function a b and the coe function a cos b, the period is b and the amplitude is a. For a tan b, the tangent function, the period is b. The tangent function has no finite maimum or minimum, so amplitude is not defined for the tangent function. Translations of Trigonometric Graphs In this lesson students eplore the graphs of and cos. More specificall, the use the functions a b( h) k and a cos b( h) k and see how changing each of the values a, b, h, and k affects the graph. lso, the eplore how to sketch a graph for a given set of values of the four variables. ne change in the graph of a periodic function is to move the horizontal center line of the graph. When k 0 the horizontal center line is the -ais and the vertical shift is zero. positive value of k represents a vertical shift upward while a negative value of k represents a downward vertical shift. The amplitude is determined b the value of a, so the maimum values of the function are a units above k and the minimum values of the function are a units below k. third change is the period of the function. The epression for the length of one period has the variable b in the denominator, so as the value of b increases the period of the function decreases. The fourth variable, h, is associated with the phase shift of the function. If h is positive, the entire graph is shifted to the right; if h is negative, the entire graph is shifted to the left. Students also use the equation a tan b( h) k and eplore phase shifts, periods, and vertical shifts for the tangent function. 70C Chapter Trigonometric Graphs and Identities

4 Trigonometric Identities This lesson and the net three deal with trigonometric identities. Students learn the definition of an identit, and the work with arguments that are half of a given angle, twice a given angle, or the sum or difference of two given angles. In this lesson students work with the definitions of the si trigonometric functions in terms of,, and r. B dividing each side of r b r,, or, the results are three identities called the Pthagorean Identities. For other identities, called the Reciprocal Identities, students note that the definitions for e and cosecant, for coe and secant, and for tangent and cotangent are reciprocals. lso, the see that the ratios cos and cos can be simplified to tan and cot, respectivel, resulting in two identities called the Quotient Identities. Students eplore how to use identities to simplif trigonometric epressions, and the use identities to evaluate a complicated trigonometric epression for a given argument. Verifing Trigonometric Identities In this lesson students continue eploring how to identif and use trigonometric identities. For each equation, the goal is to transform each side, replacing epressions with equivalent epressions, until the two sides are identical. There are several approaches for writing equivalent epressions. First, students can make substitutions ug the Pthagorean Identities. Second, the can use the Distributive Propert to factor an epression or to collect like terms. Third, the can transform a term b multipling the term b an epression equivalent to. nd fourth, the can rewrite all the trigonometric functions in terms of or cos b ug the Quotient and Reciprocal Identities. Students also relate trigonometric identities to graphs, ug a graphing calculator to show that the epressions on each side of a trigonometric identit have the same graph. Sum and Difference of ngles Formulas Students derive and then use formulas for rewriting the two-variable functions ( ) and cos ( ) in terms of the one-variable functions,, cos, and cos. The derivation of the difference formula for the coe function begins with the two ordered pairs on the unit circle that correspond to two angles and. The distance d between the two points can be found ug the distance formula, or it can be found as the distance between the point (, 0) and the coordinates of the point on the unit circle associated with angle ( ). fter equating the two epressions for d, algebraic manipulation gives an epression for the two-variable function cos ( ) in terms of one-variable functions. Students use the formulas to find eact values for particular trigonometric epressions. The also use the formulas in problems such as verifing that the equation (0 ) is an identit. Double-ngle and Half-ngle Formulas Students begin with the formulas for ( ) and cos ( ) and replace both and with. The results, called the Double-ngle Formulas, are equations in which each of and cos is epressed in terms of and cos. Then students use an algebraic technique and let represent (so represents ), and derive formulas for and cos in terms of and cos. The two formulas are called the Half-ngle Formulas. Students use the Half-ngle and Double-ngle Formulas, along with other formulas, to find eact values for particular trigonometric epressions. The also substitute the formulas in equations to verif trigonometric identities. Solving Trigonometric Equations In this last lesson of the two-chapter investigation of trigonometric functions, students solve trigonometric equations and review some of the important general ideas of algebra. The first step in solving a trigonometric equation is to use factoring, the zero product propert, and identities to rewrite a complicated equation as a string of simpler trigonometric equations. The second step is to use trigonometric inverses to isolate the variable; that is, to solve an equation such as cos 0. for. The third step is to use ideas of periodicit to include all the occurrences of that value. Students solve trigonometric equations for arguments measured in degrees or in radians, and the use trigonometric equations and their solutions to solve statements of real-world problems. Chapter Trigonometric Graphs and Identities 70D

5 and ssessment Tpe Student Edition Teacher Resources Technolog/Internet SSESSMENT INTERVENTIN ngoing Prerequisite Skills, pp. 7, 7, 77, 7, 7, 790, 797 Practice Quiz, p. 7 Practice Quiz, p. 797 Mied Review Error nalsis Standardized Test Practice pen-ended ssessment Chapter ssessment pp. 7, 77, 7, 7, 790, 797, 0 Find the Error, p. 7 Common Misconceptions, p. 7 pp. 7, 77, 7, 7, 7, 7, 790, 79, 0, 09, 0 Writing in Math, pp. 7, 77, 7, 7, 790, 79, 0 pen Ended, pp. 7, 77, 779, 7, 7, 79, 0 Stud Guide, pp. 0 0 Practice Test, p. 09 -Minute Check Transparencies Quizzes, CRM pp. 9 9 Mid-Chapter Test, CRM p. 9 Stud Guide and Intervention, CRM pp. 7,, 9 0,,, 7, 7 7 Cumulative Review, CRM p. 9 Find the Error, TWE p. 7 Tips for New Teachers, TWE p. 79 TWE p. 7 Standardized Test Practice, CRM pp Modeling: TWE pp. 7, 790 Speaking: TWE pp. 7, 7, 0 Writing: TWE pp. 77, 797 pen-ended ssessment, CRM p. 9 Multiple-Choice Tests (Forms,, B), CRM pp. 79 Free-Response Tests (Forms C, D, ), CRM pp. 90 Vocabular Test/Review, CRM p. 9 lgepss: Tutorial Plus Standardized Test Practice CD-RM standardized_test TestCheck and Worksheet Builder (see below) MindJogger Videoquizzes vocabular_review Ke to bbreviations: TWE = Teacher Wraparound Edition; CRM = Chapter Resource Masters dditional Intervention Resources The Princeton Review s Cracking the ST & PST The Princeton Review s Cracking the CT LEKS TestCheck and Worksheet Builder This networkable software has three modules for intervention and assessment fleibilit: Worksheet Builder to make worksheet and tests Student Module to take tests on screen (optional) Management Sstem to keep student records (optional) Special banks are included for ST, CT, TIMSS, NEP, and End-of-Course tests. 70E Chapter Trigonometric Graphs and Identities

6 Intervention Technolog lgepss: Tutorial Plus CD-RM offers a complete, self-paced algebra curriculum. lgebra Lesson lgepss Lesson - 7 Graphing Trigonometric Functions - Trigonometric Identities LEKS is an online mathematics learning sstem that adapts assessment and tutoring to the student s needs. Subscribe at Intervention at Home Log on for student stud help. For each lesson in the Student Edition, there are Etra Eamples and Self-Check Quizzes. For chapter review, there is vocabular review, test practice, and standardized test practice For more information on Intervention and ssessment, see pp. T T. Reading and Writing in Mathematics Glencoe lgebra provides numerous opportunities to incorporate reading and writing into the mathematics classroom. Student Edition Foldables Stud rganizer, p. 7 Concept Check questions require students to verbalize and write about what the have learned in the lesson. (pp. 7, 77, 779, 7, 7, 79, 0, 0) Writing in Math questions in ever lesson, pp. 7, 77, 7, 7, 790, 79, 0 Reading Stud Tip, pp. 7, 7 WebQuest, pp. 77, 0 Teacher Wraparound Edition Foldables Stud rganizer, pp. 7, 0 Stud Notebook suggestions, pp. 7, 77, 779, 7, 7, 79, 0 Modeling activities, pp. 7, 790 Speaking activities, pp. 7, 7, 0 Writing activities, pp. 77, 797 ELL Resources, pp. 70, 77, 77, 70, 7, 79, 79, 0, 0 dditional Resources Vocabular Builder worksheets require students to define and give eamples for ke vocabular terms as the progress through the chapter. (Chapter Resource Masters, pp. vii-viii) Reading to Learn Mathematics master for each lesson (Chapter Resource Masters, pp., 7,, 9,, 7, 77) Vocabular PuzzleMaker software creates crossword, jumble, and word search puzzles ug vocabular lists that ou can customize. Teaching Mathematics with Foldables provides suggestions for promoting cognition and language. Reading and Writing in the Mathematics Classroom WebQuest and Project Resources For more information on Reading and Writing in Mathematics, see pp. T T7. Chapter Trigonometric Graphs and Identities 70F

7 Notes Have students read over the list of objectives and make a list of an words with which the are not familiar. Point out to students that this is onl one of man reasons wh each objective is important. thers are provided in the introduction to each lesson. Trigonometric Graphs and Identities Lessons - and - Graph trigonometric functions and determine period, amplitude, phase shifts, and vertical shifts. Lessons - and - Use and verif trigonometric identities. Lessons - and - Use sum and difference formulas and double- and half-angle formulas. Lesson -7 Solve trigonometric equations. Some equations contain one or more trigonometric functions. It is important to know how to simplif trigonometric epressions to solve these equations. Trigonometric functions can be used to model man real-world applications, such as music. You will learn how a trigonometric function can be used to describe music in Lesson -. Ke Vocabular amplitude (p. 7) phase shift (p. 79) vertical shift (p. 77) trigonometric identit (p. 777) trigonometric equation (p. 799) NCTM Local Lesson Standards bjectives -,,, 7,, 9, 0 -,,,,,, 9, 0 -,,,, 9, 0 -, -,,, 7,, 9, 0 -,, 7,, 9, 0-7, 0 Preview -7,,, 7,, 9, 0 Ke to NCTM Standards: =Number & perations, =lgebra, =Geometr, =Measurement, =Data nalsis & Probabilit, =Problem Solving, 7=Reasoning & Proof, =Communication, 9=Connections, 0=Representation 70 Chapter Trigonometric Graphs and Identities Vocabular Builder ELL The Ke Vocabular list introduces students to some of the main vocabular terms included in this chapter. For a more thorough vocabular list with pronunciations of new words, give students the Vocabular Builder worksheets found on pages vii and viii of the Chapter Resource Masters. Encourage them to complete the definition of each term as the progress through the chapter. You ma suggest that the add these sheets to their stud notebooks for future reference when studing for the Chapter test. 70 Chapter Trigonometric Graphs and Identities

8 Prerequisite Skills To be successful in this chapter, ou'll need to master these skills and be able to appl them in problem-solving situations. Review these skills before beginning Chapter. For Lessons - and - Trigonometric Values Find the eact value of each trigonometric function. (For review, see Lesson -.).. tan. cos tan. 7. cos 7.. tan not defined For Lessons -, -, and - Circular Functions Find the eact value of each trigonometric function. (For review, see Lesson -.) 9. cos (0 ) cot 9. sec. tan. csc (70 ). cos 7. tan not defined not defined For Lesson - Factor Polnomials Factor completel. If the polnomial is not factorable, write prime. (For review, see Lesson -.) 7. ( ). ( ) 9. prime 0... ( )( ) ( )( ) ( )( ) For Lesson -7 Solve Quadratic Equations Solve each equation b factoring. (For review, see Lesson -.). 0,. 0,. 0 0,. 0 0, ,, This section provides a review of the basic concepts needed before beginning Chapter. Page references are included for additional student help. Prerequisite Skills in the Getting Read for the Net Lesson section at the end of each eercise set review a skill needed in the net lesson. For Prerequisite Lesson Skill - Graphs of Quadratic Functions (p. 7) - Reference ngles (p. 77) - Properties of Equalit (p. 7) - Simplifing Radical Epressions (p. 7) - Solving Equations Ug the Square Root Propert (p. 790) -7 Solving Equations Ug the Zero Product Propert (p. 797) Make this Foldable to help ou organize information about trigonometric graphs and identities. Begin with eight sheets of grid paper. Staple Cut and Label Staple the stack of grid paper along the top to form a booklet. Cut seven lines from the bottom of the top sheet, si lines from the second sheet, and so on. Label with lesson numbers as shown. Trigonometric Graphs & Identities Reading and Writing s ou read and stud the chapter, use each page to write notes and to graph eamples for each lesson. Chapter Trigonometric Graphs and Identities 7 For more information about Foldables, see Teaching Mathematics with Foldables. TM Writing Instructions and Sequencing Data fter students make their Foldable, have them label each tab to correspond to a lesson in this chapter. Students use their Foldable to take notes, define terms, record concepts, and write eamples. fter each lesson, ask students to write a set of instructions on how to do something presented in the lesson. For eample, a student might write instructions for graphing trigonometric functions. Have students follow their own instructions to check them for accurac. Use their notes and tetbook to make needed revisions. Chapter Trigonometric Graphs and Identities 7

9 Lesson Notes Graphing Trigonometric Functions Focus -Minute Check Transparenc - Use as a quiz or review of Chapter. Mathematical Background notes are available for this lesson on p. 70C. Vocabular amplitude Graph trigonometric functions. Find the amplitude and period of variation of the e, coe, and tangent functions. can ou predict the behavior of tides? The rise and fall of tides can have great impact on the communities and ecosstems that depend upon them. ne tpe of tide is a semidiurnal tide. This means that bodies of water, like the tlantic cean, have two high tides and two low tides a da. Because tides are periodic, the behave the same wa each da. High Tide Still Water Level Period Low Tide Tidal Range Building on Prior Knowledge In Chapter, students learned the e, coe, and tangent of 0,, and 0 angles. In this lesson, students will learn the e, coe, and tangent of angles with other measures and use all of these values to create graphs of the si trigonometric functions. can ou predict the behavior of tides? sk students: Wh would ou need to know the times for the high and low tides? Sample answer: roads might be flooded at high tide How is the period of a tide defined? Sample answer: the length of time between two high tides (or two low tides) What is a tidal range? Sample answer: the difference in water level between low tide and high tide Stud Tip Look Back To review period and periodic functions, see Lesson -. GRPH TRIGNMETRIC FUNCTINS The diagram below illustrates the water level as a function of time for a bod of water with semidiurnal tides. Water Level High Tide 0 0 Time Low Tide In each ccle of high and low tides, the pattern repeats itself. Recall that a function whose graph repeats a basic pattern is called a periodic function. To find the period, start from an point on the graph and proceed to the right until the pattern begins to repeat. The simplest approach is to begin at the origin. Notice that after about hours the graph begins to repeat. Thus, the period of the function is about hours. To graph the periodic functions, cos, or tan, use values of epressed either in degrees or radians. rdered pairs for points on these graphs are of the form (, ), (, cos ), and (, tan ), respectivel nearest tenth cos 0 0 nearest tenth tan 0 nd 0 nd 0 nearest nd nd tenth nd not defined 7 Chapter Trigonometric Graphs and Identities Resource Manager Workbook and Reproducible Masters Chapter Resource Masters Stud Guide and Intervention, pp. 7 Skills Practice, p. 9 Practice, p. 0 Reading to Learn Mathematics, p. Enrichment, p. Transparencies -Minute Check Transparenc - Real-World Transparenc nswer Ke Transparencies Technolog Interactive Chalkboard

10 fter plotting several points, complete the graphs of and cos b connecting the points with a smooth, continuous curve. Recall from Chapter that each of these functions has a period of 0 or radians. That is, the graph of each function repeats itself ever 0 or radians. (90, ) (, 0.7) (, 0.7) 90 Notice that both the e and coe have a maimum value of and a minimum value of. The amplitude of the graph of a periodic function is the absolute value of half the difference between its maimum value and its minimum value. So, for both the e and coe functions, the amplitude of their graphs is ( ) or. The graph of the tangent function can also be drawn b plotting points. B eamining the values for tan in the table, ou can see that the tangent function is not defined for 90, 70,, 90 k 0, where k is an integer. The graph is separated b vertical asmptotes whose -intercepts are the values for which tan is not defined (, 0.7) (, 0.7) (70, ) tan (0, ) (, 0.7) (, 0.7) 90 cos (, 0.7) (, 0.7) (0, ) Teach GRPH TRIGNMETRIC FUNCTINS Teaching Tip Have students brainstorm to make a list of real-world phenomena that fluctuate in a regular periodic pattern. (Eamples: temperatures, the number of people in a mall over the course of a week) Teaching Tip You might wish to have students draw their own graphs of the e, coe, and tangent functions ug the data from the table on p. 7. Their graphs can be compared to those shown on p. 7. This immediate feedback can be beneficial in helping students acquire the skills for graphing trigonometric functions, which are more complicated than most of the graphing students have done to this point. Teaching Tip Review how the e and coe functions are related in a right triangle. The period of the tangent function is 0 or radians. Since the tangent function has no maimum or minimum value, it has no amplitude. The graphs of the secant, cosecant, and cotangent functions are shown below. Compare them to the graphs of the coe, e, and tangent functions, which are shown in red. sec csc cot cos tan Notice that the period of the secant and cosecant functions is 0 or radians. The period of the cotangent is 0 or radians. Since none of these functions have a maimum or minimum value, the have no amplitude. Lesson - Graphing Trigonometric Functions 7 Differentiated Instruction Visual/Spatial Have groups of students make posters showing sketches of the graphs of the si trigonometric functions. Encourage students to color-code the ke features of all the graphs, such as period, amplitude, asmptotes, and so on. Lesson - Graphing Trigonometric Functions 7

11 VRITINS F TRIGNMETRIC FUNCTINS VRITINS F TRIGNMETRIC FUNCTINS Just as with other functions, a trigonometric function can be used to form a famil of graphs b changing the period and amplitude. Concept Check fter discusg the Ke Concept bo about amplitudes and periods, ask: What is an equation involving the e function with a period of 90 and an amplitude of? ne of the following:, (),, or () Stud Tip mplitude and Period Note that the amplitude affects the graph along the vertical ais and the period affects it along the horizontal ais. 7. When a is positive, the amplitude is a. When a is negative, then amplitude is a. a has no effect on the period. Period and mplitude n a TI- Plus graphing calculator, set the MDE to degrees. Think and Discuss. Graph and. What is the maimum value of each function?. How man times does each function reach a maimum value?,. Graph. What is the maimum value of this function? How man times does this function reach its maimum value? ;. Use the equations b and cos b. Repeat Eercises for maimum values and the other values of b. What conjecture can ou make about the effect of b on the maimum values and the periods of these functions? The greater the value of b, the smaller the period. b has no effect on the maimum value.. Graph and. What is the maimum value of each function? What is the period of each function?, ; 0. Graph. What is the maimum value of this function? What is the period of this function? ; 0 [0, 70] scl: b [.,.] scl: 0. [0, 70] scl: b [.,.] scl: Use the equations a and a cos. Repeat Eercises and for other values of a. What conjecture can ou make about the effect of a on the amplitudes and periods of a and a cos? The results of the investigation suggest the following generalization. mplitudes and Periods Words For functions of the form a b and a cos b, the amplitude is a, and the period is 0 or. b b For functions of the form a tan b, the amplitude is not defined, and the period is 0 or. b b Eamples amplitude and period 0 or 90 cos amplitude or and period tan no amplitude and period 7 Chapter Trigonometric Graphs and Identities Graphing Trigonometric Functions To set the calculator for degrees, press MDE and move the cursor to highlight DEGREE and press ENTER. lso, be sure to have students clear the Y= lists before beginning Eercise. 7 Chapter Trigonometric Graphs and Identities

12 You can use the amplitude and period of a trigonometric function to help ou graph the function. Eample Graph Trigonometric Functions Find the amplitude and period of each function. Then graph the function. a. cos First, find the amplitude. a The coefficient of cos is. Net, find the period. 0 0 b b 0 Use the amplitude and period to graph the function. In-Class Eample Power Point Find the amplitude and period of each function. Then graph the function. a. ampl: ; period: cos b. cos ampl: ; period: 0 cos b. mplitude: Period: a 0 0 b c. ampl: ; period: 0 c. mplitude: a 0. ( ) Period: b 0. Figure Lesson - Graphing Trigonometric Functions 7 Teacher to Teacher Berchie Hollida uthor, Cincinnati, H I have m students construct a unit circle on a coordinate plane with a toothpick length radius. The mark ever. Students form right triangles inside the circle and break toothpicks to match the lengths of each vertical leg. (See Figure at the right.) The transfer each leg to its appropriate degree mark on a second -ais and place a dot at the top of each toothpick. (See Figure.) Finall, students connect the dots with a smooth curve. Figure Lesson - Graphing Trigonometric Functions 7

13 In-Class Eample Practice/ppl Stud Notebook Power Point CENGRPHY Refer to the application at the beginning of the lesson. The tidal range in the Ba of Fund in Canada measures 0 feet. a. Write a function to represent the height h of the tide. ssume that the tide is at equilibrium at t 0 and that the high tide is beginning. t b. Graph the tide function. 0 0 h t Have students add the definitions/eamples of the vocabular terms to their Vocabular Builder worksheets for Chapter. include an other item(s) that the find helpful in mastering the skills in this lesson. ceanograph Lake Superior has one of the smallest tidal ranges. It can be measured in inches, while the tidal range in the Ba of Funda in Canada measures up to 0 feet. Source: ffice of Naval Research Concept Check. Sample answer: The graph repeats itself ever 0.. Jamile; the amplitude is, and the period is. You can use trigonometric functions to describe real-world situations. Eample Use Trigonometric Functions CENGRPHY Refer to the application at the beginning of the lesson. Suppose the tidal range of a cit on the tlantic coast is feet. tide is at equilibrium when it is at its normal level, halfwa between its highest and lowest points. a. Write a function to represent the height h of the tide. ssume that the tide is at equilibrium at t 0 and that the high tide is beginning. Since the height of the tide is 0 at t 0, use the e function h a bt, where a is the amplitude of the tide and t is the time in hours. Find the amplitude. The difference between high tide and low tide is the tidal range or feet. a or 9 Find the value of b. Each tide ccle lasts about hours. period b b b or Solve for b. Thus, an equation to represent the height of the tide is h 9 t. b. Graph the tide function. h t 0. PEN ENDED Eplain wh tan has no amplitude. See margin.. Eplain what it means to sa that the period of a function is 0.. FIND THE ERRR Dante and Jamile graphed cos. Dante 0 Jamile 0 FIND THE ERRR Students should quickl notice that Dante must be incorrect because his graph does not have an amplitude of. sk students if the can identif the mistake that Dante made. 7 Chapter Trigonometric Graphs and Identities nswer Who is correct? Eplain our reasoning.. Sample answer: mplitude is half the difference between the maimum and minimum values of a graph; tan has no maimum or minimum value. 7 Chapter Trigonometric Graphs and Identities

14 Guided Practice GUIDED PRCTICE KEY Eercises Eamples, indicates increased difficult Practice and ppl Homework Help For Eercises See Eamples Etra Practice See page 9. pplication. months; Sample answer: The pattern in the population will repeat itself ever months. Find the amplitude, if it eists, and period of each function. Then graph each function.. See pp. N.... cos 7. tan. csc cos. sec. cos BILGY For Eercises and, use the following information. In a certain wildlife refuge, the population of field mice can be modeled b t, where represents the number of mice and t represents the number of months past March of a given ear.. Determine the period of the function. What does this period represent?. What is the maimum number of mice and when does this occur? 0; June Find the amplitude, if it eists, and period of each function. Then graph each function.. See pp. N... cos 7. csc. tan sec... sec. cot. tan. cot Stud Guide and Intervention, p. 7 (shown) and p. NME DTE PERID - Stud Guide and Intervention Graphing Trigonometric Functions Graph Trigonometric Functions To graph a trigonometric function, make a table of values for known degree measures (0, 0,, 0, 90, and so on). Round function values to the nearest tenth, and plot the points. Then connect the points with a smooth, continuous curve. The period of the e, coe, secant, and cosecant functions is 0 or radians. mplitude of a Function The amplitude of the graph of a periodic function is the absolute value of half the difference between its maimum and minimum values. Eample Graph for 0 0. First make a table of values Graph the following functions for the given domain.. cos, 0 0. tan, 0 cos tan 0 70 Eercises 70 What is the amplitude of each function? NME 7 DTE PERID Practice (verage) Graphing Trigonometric Functions Find the amplitude, if it eists, and period of each function. Then graph each function.. cos.. cot ; 0 no amplitude; 0 ; 7 Skills Practice, p. 9 and Practice, p. 0 (shown) Lesson cos 9. csc Medicine The tuning fork was invented in 7 b English trumpeter John Shore. Source: 0. t, 0. t, 0. 0t. f() cos and f() sec ; See pp. N for graphs cot. tan.. Draw a graph of a e function with an amplitude and a period of 90. Then write an equation for the function. See pp. N for graph;.. Draw a graph of a coe function with an amplitude of 7 and a period of. Then write an equation for the function. See pp. N for graph; 7 cos.. CMMUNICTINS The carrier wave for a certain FM radio station can be modeled b the equation (0 7 t), where is the amplitude of the wave and t is the time in seconds. Determine the period of the carrier wave. 0 7 MEDICINE For Eercises and 7, use the following information. Doctors ma use a tuning fork that resonates at a given frequenc as an aid to diagnose hearing problems. The sound wave produced b a tuning fork can be modeled ug a e function.. If the amplitude of the e function is 0., write the equations for tuning forks that resonate with a frequenc of,, and Hertz. 7. How do the periods of the tuning forks compare? See margin.. CRITICL THINKING function is called even if the graphs of ƒ() and ƒ() are eactl the same. Which of the si trigonometric functions are even? Justif our answer with a graph of each function. Lesson - Graphing Trigonometric Functions 77. csc. tan. ;0 no amplitude; 0 no amplitude; FRCE For Eercises 7 and, use the following information. n anchoring cable eerts a force of 00 Newtons on a pole. The force has the horizontal and vertical components F and F. ( force of one Newton (N), is the force that gives an acceleration of m/sec to a mass of kg.) 7. The function F 00 cos describes the relationship between the angle and the horizontal force. What are the amplitude and period of this function? 00; 0 0. The function F 00 describes the relationship between the angle and the vertical force. What are the amplitude and period of this function? 00; 0 WETHER For Eercises 9 and 0, use the following information. The function 0 t, where t is in months and t 0 corresponds to pril, models the average high temperature in degrees Fahrenheit in Centerville. 9. Determine the period of this function. What does this period represent? ; a calendar ear 0. What is the maimum high temperature and when does this occur? F; Jul Reading to Learn Mathematics, p. NME 0 DTE PERID Reading to Learn Mathematics Graphing Trigonometric Functions Pre-ctivit Wh can ou predict the behavior of tides? Read the introduction to Lesson - at the top of page 7 in our tetbook. Consider the tides of the tlantic cean as a function of time. pproimatel what is the period of this function? hours Reading the Lesson. Determine whether each statement is true or false. a. The period of a function is the distance between the maimum and minimum points. false b. The amplitude of a function is the difference between its maimum and minimum values. false c. The amplitude of the function is. false d. The function cot has no amplitude. true e. The period of the function sec is. false f. The amplitude of the function cos is. false N 70 F ELL 0 F Lesson - nswer 7. Sample answer: The amplitudes are the same. s the frequenc increases, the period decreases. - Enrichment Blueprints NME DTE PERID Enrichment, p. Interpreting blueprints requires the abilit to select and use trigonometric functions and geometric properties. The figure below represents a plan for an improvement to a roof. The metal fitting shown makes a 0 angle with the horizontal. The vertices of the geometric shapes are not labeled in these plans. Relevant information must be selected and the appropriate function used to find the unknown measures. Eample Find the unknown measures in the figure at the right. The measures and are the legs of a right triangle. The measure of the hpotenuse is in. in. or 0 in. " 0 " Roofing Improvement 0 " top view side view " metal fitting 0.09" g. The function has a period of. true h. The period of the function cot is. true i. The amplitude of the function is. false j. The period of the function csc is. false k. The graph of the function has no asmptotes. true l. The graph of the function tan has an asmptote at 0. false m. When 0, the values of cos and sec are equal. true n. When 70, cot is undefined. false o. When 0, csc is undefined. true Helping You Remember. What is an eas wa to remember the periods of a b and a cos b? Sample answer: The period of the functions and cos is 0 or. Divide 0 or b the absolute value of the coefficient of, depending on whether ou want to find the period in degrees or in radians. Lesson - Graphing Trigonometric Functions 77

15 bout the Eercises rganization b bjective Graph Trigonometric Functions: Variations of Trigonometric Functions: dd/even ssignments Eercises are structured so that students practice the same concepts whether the are assigned odd or even problems. ssignment Guide Basic: odd, verage: odd, dvanced: even, (optional: ) ssess pen-ended ssessment Modeling Provide students with a coordinate grid and a length of string. Give students one of the si basic trigonometric functions and have them model the graph ug the string. Have students check their model with a graphing calculator. Getting Read for Lesson - PREREQUISITE SKILL Students will graph horizontal and vertical translations of trigonometric functions in Lesson -. Students will appl what the learned about families of quadratic functions. Use Eercises to determine our students familiarit with the graphs of families of functions. nswers t t Standardized Test Practice BTING For Eercises 9, use the following information. marker buo off the coast of Gulfport, Mississippi, bobs up and down with the waves. The distance between the highest and lowest point is feet. The buo moves from its highest point to its lowest point and back to its highest point ever 0 seconds. 9. Write an equation for the motion of the buo. ssume that it is at equilibrium at t 0 and that it is on the wa up from the normal water level. t 0. Draw a graph showing the height of the buo as a function of time. See margin.. What is the height of the buo after seconds? about.9 ft. WRITING IN MTH nswer the question that was posed at the beginning of the lesson. See margin. Wh can ou predict the behavior of tides? Include the following in our answer: an eplanation of wh certain tidal characteristics follow the patterns seen in the graph of the e function, and a description of how to determine the amplitude of a function ug the maimum and minimum values.. What is the period of ƒ() cos? 0 B 0 C 0 D 70. Identif the equation of the graphed function. C B C D Maintain Your Skills Mied Review Solve each equation. (Lesson -7) Getting Read for the Net Lesson. See pp. N. 7 Chapter Trigonometric Graphs and Identities. Sample answer: Tides displa periodic behavior. This means that their pattern repeats at regular intervals. nswers should include the following information. Tides rise and fall in a periodic manner, similar to the e function. In f() a b, the amplitude is the absolute value of a Sin 90. rc () rccos Find the eact value of each function. (Lesson -) ( ) 0. cos 0. PRBBILITY There are girls and bos on the Facult dvisor Board. Three are juniors. Find the probabilit of selecting a bo or a girl from the committee who is not a junior. (Lesson -). Find the first five terms of the sequence in which a, a n a n. (Lesson -),, 7, 9, PREREQUISITE SKILL Graph each pair of functions on the same set of aes. (To review graphs of quadratic functions, see Lesson -.).,.,., ( )., ( ) 7 Chapter Trigonometric Graphs and Identities

16 Translations of Trigonometric Graphs Lesson Notes Vocabular phase shift vertical shift midline TECHING TIP For Eercise, point out that ce the calculator has no preprogrammed button for the secant function, the will need to graph. co s Graph horizontal translations of trigonometric graphs and find phase shifts. Graph vertical translations of trigonometric graphs. can translations of trigonometric graphs be used to show animal populations? In predator-pre ecosstems, the number of predators and the number of pre tend to var in a periodic manner. In a certain region with cootes as predators and rabbits as pre, the rabbit population R can be modeled b the equation R 00 0 t, where t is the time in ears ce Januar, 00. Horizontal Translations n a TI- Plus, set the MDE to degrees. Think and Discuss. See margin. ( 0). Graph and ( 0). ( 0) How do the two graphs compare?. Graph ( 0). How does this graph compare to the other two?. What conjecture can ou make about the effect of h in the function [0, 70] scl: b [.,.] scl: 0. ( h)?. Test our conjecture on the following pairs of graphs. cos and cos ( 0) See pp. N for graphs; tan and tan ( ) the conjecture holds. sec and sec ( 7) Notice that when a constant is added to an angle measure in a trigonometric function, the graph is shifted to the left or to the right. If (, ) are coordinates of, then ( h, ) are coordinates of ( h). horizontal translation of a trigonometric function is called a phase shift. Number of Rabbits R Years Since 00 HRIZNTL TRNSLTINS Recall that a translation is a tpe of transformation in which the image is identical to the preimage in all aspects ecept its location on the coordinate plane. horizontal translation shifts to the left or right, and not upward or downward. Lesson - Translations of Trigonometric Graphs 79 t Focus -Minute Check Transparenc - Use as a quiz or review of Lesson -. Mathematical Background notes are available for this lesson on page 70C. can translations of trigonometric graphs be used to show animal populations? sk students: Wh might the two animal populations var? Sample answer: s the number of predators increases, more pre are eaten and there are fewer pre left. Lower numbers of pre means increased competition for food b the predator species, so the number of predators decreases as their food suppl diminishes. What are the minimum and maimum rabbit populations shown b the graph? 90 rabbits, 0 rabbits How could ou find the range of the population without calculating the maimum and minimum values or graphing the function? The range, 00, is twice the amplitude, 0, of the graph. nswers See p. 770 for Graphing Calculator answers. Resource Manager Workbook and Reproducible Masters Chapter Resource Masters Stud Guide and Intervention, pp. Skills Practice, p. Practice, p. Reading to Learn Mathematics, p. 7 Enrichment, p. ssessment, p. 9 Graphing Calculator and Spreadsheet Masters, p. School-to-Career Masters, p. 7 Transparencies -Minute Check Transparenc - nswer Ke Transparencies Technolog lgepss: Tutorial Plus, Lesson 7 Interactive Chalkboard Lesson - Lesson Title 79

17 Teach Building on Prior Knowledge In Lesson -, students learned about the translations of graphs of quadratic functions. In this lesson, students will use similar techniques to stud the translations of graphs of the trigonometric functions. HRIZNTL TRNSLTINS In-Class Eample Power Point State the amplitude, period, and phase shift for each function. Then graph the function. a. ( 0) amplitude: ; period: 0; phase shift: 0 left b. cos amplitude: ; period: ; phase shift: right cos nswers (p. 79) 90 ( 0) 0 Graphing Calculator Investigation. The graph of ( 0) is shifted 0 to the right of the graph of.. The graph of ( 0) is shifted 0 to the left of the graph of.. Sample answer: When h is positive the graph shifts right h units. When h is negative the graph shifts left h units Words: Models: Stud Tip The phase shift of the functions a b ( h), a cos b ( h), and a tan b ( h) is h, where b 0. If h 0, the shift is to the right. If h 0, the shift is to the left. Sine Coe Tangent a b a b( h); h 0 a b( h); h 0 Verifing a Graph fter drawing the graph of a trigonometric function, select value of and evaluate them in the equation to verif our graph. 770 Chapter Trigonometric Graphs and Identities Eample Graph Horizontal Translations State the amplitude, period, and phase shift for each function. Then graph the function. a. cos ( 0 ) Since a and b, the amplitude and period of the function are the same as cos. However, h 0, so the phase shift is 0. Because h 0, the parent graph is shifted to the right. To graph cos ( 0 ), consider the graph of cos. Graph this function and then shift the graph 0 to the right. The graph cos ( 0 ) is the graph of cos shifted to the right. b. mplitude: a or Period: or b a cos b a cos b( h); h 0 a cos b( h); h 0 The secant, cosecant, and cotangent can be graphed ug the same rules. Phase Shift: h The phase shift is to the left ce 0. a tan b( h); h 0 a tan b Phase Shift a tan b( h); h 0 Graphing the Secant Function The graph of the secant function should look like a pattern of U shapes, alternatel opening upward and downward. Students might want to etend the activit b investigating the graphs of cosecant (csc) and cotangent (cot). The graphing calculator does not have a button for graphing either of these functions so students should enter Y= / to graph the cosecant function, and the should enter Y= /tan to graph the cotangent function. cos ( 0 ) cos ( ) 770 Chapter Trigonometric Graphs and Identities

18 Stud Tip Look Back Pa close attention to trigonometric functions for the placement of parentheses. Note that ( ). The epression on the left represents a phase shift while the epression on the right represents a vertical shift. VERTICL TRNSLTINS In Chapter, ou learned that the graph of is a vertical translation of the parent graph of. Similarl, graphs of trigonometric functions can be translated verticall through a vertical shift. When a constant is added to a trigonometric function, the graph is shifted upward or downward. If (, ) are coordinates of, then (, k) are coordinates of k. new horizontal ais called the midline becomes the reference line about which the graph oscillates. For the graph of k, midline the midline is the graph of k k Words The vertical shift of the functions a b( h) k, a cos b( h) k, and a tan b( h) k is k. If k 0, the shift is up. If k 0, the shift is down. The midline is k. Models: Sine Coe Tangent a b( h) k; k 0 a b( h) k; k 0 cos a cos b( h) k; k 0 a cos b( h) k; k 0 k tan The secant, cosecant, and cotangent can be graphed ug the same rules. Eample Graph Vertical Translations Vertical Shift a tan b( h) k; k 0 a tan b( h) k; k 0 State the vertical shift, equation of the midline, amplitude, and period for each function. Then graph the function. a. tan Since tan tan (), k, and the vertical shift is. Draw the midline,. The tangent function has no amplitude and the period is the same as that of tan. Draw the graph of the function relative to the midline. VERTICL TRNSLTINS Teaching Tip Make sure students are clear about the difference between the terms vertical and horizontal. Students can use the words vertigo (a sense of dizziness some people eperience when looking down from a great height) and horizon to help link vertical and horizontal to their relative direction. In-Class Eample Power Point State the vertical shift, equation of the midline, amplitude, and period for each function. Then graph the function. a. vertical shift: ; midline: ; amplitude: ; period: b. cos vertical shift: ; midline: ; amplitude: ; period: cos tan Lesson - Translations of Trigonometric Graphs 77 Interactive Chalkboard PowerPoint Presentations This CD-RM is a customizable Microsoft PowerPoint presentation that includes: Step-b-step, dnamic solutions of each In-Class Eample from the Teacher Wraparound Edition dditional, Your Turn eercises for each eample The -Minute Check Transparencies Hot links to Glencoe nline Stud Tools Lesson - Translations of Trigonometric Graphs 77

19 Teaching Tip Suggest that students write the general form of the e, coe, and tangent functions given in the Ke Concept bo on p. 77 at the top of separate inde cards. Ug the steps listed in the Concept Summar on p. 77, have students identif the parts of the equation used in each step to find the information listed in each step. The can then use their inde cards to work Eample. Stud Tip Look Back It ma be helpful to first graph the parent graph in one color. Then appl the vertical shift and graph the function in another color. Then appl the change in amplitude and graph the function in the final color. b. Vertical shift: k, so the midline is the graph of. mplitude: a or Period: b Since the amplitude of the function is, draw dashed lines parallel to the midline that are unit above and below the midline. Then draw the e curve. In-Class Eample Power Point State the vertical shift, amplitude, period, and phase shift of. Then graph the function. The vertical shift is. The amplitude is. The period is. The phase shift is to the right. In general, use the following steps to graph an trigonometric function. Step Step Step Step Eample Determine the vertical shift, and graph the midline. Graphing Trigonometric Functions Determine the amplitude, if it eists. Use dashed lines to indicate the maimum and minimum values of the function. Determine the period of the function and graph the appropriate function. Determine the phase shift and translate the graph accordingl. Graph Transformations State the vertical shift, amplitude, period, and phase shift of cos. Then graph the function. The function is written in the form a cos [b( h)] k. Identif the values of k, a, b, and h. k, so the vertical shift is. a, so the amplitude is or. b, so the period is or. h, so the phase shift is to the right. Then graph the function. Step Step Step Step The vertical shift is. Graph the midline. The amplitude is. Draw dashed lines units above and below the midline at and 0. The period is, so the graph will be stretched. Graph cos ug the midline as a reference. Shift the graph to the right. cos cos ( ) 0 77 Chapter Trigonometric Graphs and Identities 77 Chapter Trigonometric Graphs and Identities

20 Health Blood preasure can change from minute to minute and can be affected b the slightest of movements, such as tapping our fingers or crosg our arms. Source: merican Heart ssociation You can use information about amplitude, period, and translations of trigonometric functions to model real-world applications. Eample Use Translations to Solve a Problem HELTH Suppose a person s resting blood pressure is 0 over 0. This means that the blood pressure oscillates between a maimum of 0 and a minimum of 0. If this person s resting heart rate is 0 beats per minute, write a e function that represents the blood pressure for t seconds. Then graph the function. Eplore Plan Solve You know that the function is periodic and can be modeled ug e. Let P represent blood pressure and let t represent time in seconds. Use the equation P a [b(t h)] k. Write the equation for the midline. Since the maimum is 0 and the minimum is 0, the midline lies halfwa between these values. P 0 0 or 00 Determine the amplitude b finding the difference between the midline value and the maimum and minimum values. a 0 00 a or 0 0 or 0 Thus, a 0. Determine the period of the function and solve for b. Recall that the period of a function can be found ug the epression. Since the b heart rate is 0 beats per minute, there is one heartbeat, or ccle, per second. So, the period is second. Write an equation. b b Multipl each side b b. b Solve. For this eample, let b. The use of the positive or negative value depends upon whether ou begin a ccle with a maimum value (positive) or a minimum value (negative). There is no phase shift, so h 0. So, the equation is P 0 t 00. Graph the function. In-Class Eample Power Point HELTH Refer to Eample in the Student Edition. Write a e function that represents the blood pressure for t seconds of a person with a resting blood pressure of 0 over 90 and a heart rate of 7 beats per minute. Then graph the function. P t P 0 P t 0 Eamine Step Step Step Step Draw the midline P 00. Draw maimum and minimum reference lines. Use the period to draw the graph of the function. There is no phase shift. P P 0 t 00 0 Notice that each ccle begins at the midline, rises to 0, drops to 0, and then returns to the midline. This represents the blood pressure of 0 over 0 for one heartbeat. Since each ccle lasts second, there will be 0 ccles, or heartbeats, in minute. Therefore, the graph accuratel represents the information. t Lesson - Translations of Trigonometric Graphs 77 Differentiated Instruction Kinesthetic Make coordinate aes with masking tape on the classroom floor. Give students at least feet of rope and have them stand along the -ais, positioning the rope to model the graph of. s ou call out equations of functions whose graphs are horizontal phase shifts of the graph of, students can step left or right to model the translated graph. Similarl, call out functions whose graphs are vertical shifts of the graph of. Lesson - Translations of Trigonometric Graphs 77

21 Practice/ppl Stud Notebook Have students add the definitions/eamples of the vocabular terms to their Vocabular Builder worksheets for Chapter. record the Ke Concepts bo on p. 77 and the Concept Summar on p. 77. include an other item(s) that the find helpful in mastering the skills in this lesson. bout the Eercises rganization b bjective Horizontal Translations: 9, Vertical Translations: dd/even ssignments Eercises 9 are structured so that students practice the same concepts whether the are assigned odd or even problems. ssignment Guide Basic: 9 9 odd, odd, 7 verage: 9 odd, 7 dvanced: 0 even,, (optional: 7) Concept Check. See margin. Guided Practice GUIDED PRCTICE KEY Eercises Eamples 7 indicates increased difficult Practice and ppl 9. See pp. N for graphs. pplication 77 Chapter Trigonometric Graphs and Identities. Identif the vertical shift, amplitude, period, and phase shift of the graph of cos ( 90 ).. Define the midline of a trigonometric graph.. PEN ENDED Write the equation of a trigonometric function with a phase shift of. State the amplitude, period, and phase shift for each function. Then graph the function.. See pp. N for graphs. no amplitude; 0 ; 0. ; ;. tan ( 0 ). cos ( ) ; 0 ; 7. sec State the vertical shift, equation of the midline, amplitude, and period for each function. Then graph the function.. cos ; ; ; ; ; 0 9. sec no amplitude; 0 0. tan. 0. ; ; no amplitude; 0 0.; 0.; ; 0 State the vertical shift, amplitude, period, and phase shift of each function. Then graph the function.. 0; ; 0 ; 0. ; no amplitude; 0 ;. [( 0 )] 0. cot ( ). sec. cos ; no amplitude; ; PHYSICS For Eercises, use the following information. ceiling weight is attached to a spring and suspended from the ceiling. t equilibrium, the weight is located feet above the floor. The weight is pulled down foot and released. ft. Determine the vertical shift, amplitude, and ft period of a function that represents the height floor of the weight above the floor if the weight returns to its lowest position ever seconds. ; ; s 7. Write the equation for the height h of the weight above the floor as a function of time t seconds. h cos t or h cos 90 t. Draw a graph of the function ou wrote in Eercise 7. See pp. N. State the amplitude, period, and phase shift for each function. Then graph the function. 0. no amplitude; 0 ; 0 9. cos ( 90 ) ; 0 ; cot ( 0 ) no amplitude; ;. ; ;. cos ; ; ; ; ;. tan (. ). ( 7 ) ; 0 ; 7 no amplitude; 0 ;. nswers. vertical shift: ; amplitude: ; period: 0; phase shift:. The midline of a trigonometric function is the line about which the graph of the function oscillates after a vertical shift.. Sample answer: ( ).. 0 tan ( ) cos ( 0 ) 77 Chapter Trigonometric Graphs and Identities

22 Homework Help For Eercises See Eamples 9 0,, 7 0 Etra Practice See page 9.. ; ; no amplitude; 0 Translations of trigonometric graphs can be used to describe temperature trends. Visit webquest to continue work on our WebQuest project. 7. h 9 9 (t.) State the vertical shift, equation of the midline, amplitude, and period for each function. Then graph the function. 0. See pp. N for graphs.. ; ; ; 0. sec ; ; no amplitude; 0 7. cos ; ; ; 0. csc 9. ; ; ; 0 0. cos..;.; ; 0. Graph tan. Describe the transformation to the parent graph tan. See margin for graph; translation units left and units up.. Draw a graph of the function cos ( 0 ). How does this graph compare to the graph of cos? See margin for graph; translation 0 right and units up with an amplitude of unit. State the vertical shift, amplitude, period, and phase shift of each function. Then graph the function.. See pp. N.. [( )]. cos [( 0 )]. csc ( 0 ).. cot ( 90 ) cos ( 0 ). tan ( ) sec. Graph cos and cos ( ). How do the graphs compare?. Compare the graphs of and cos.. MUSIC When represented on an oscilloscope, the note above middle C has period of. Which of the following can be an equation for an oscilloscope 0 graph of this note? The amplitude of the graph is K. c a. K 0t b. K 0t c. K 0t ZLGY For Eercises, use the following information. The population of predators and pre in a closed ecological sstem tends to var periodicall over time. In a certain sstem, the population of owls can be represented b t where t is the time in ears ce Januar, 00. In that same sstem, the population of mice M can be represented b M t ; r. Find the maimum number of owls. fter how man ears does this occur?. What is the minimum number of mice? How long does it take for the population of mice to reach this level? 00;. r. Wh would the maimum owl population follow behind the population of mice? See margin. 7. TIDES The height of the water in a harbor rose to a maimum height of feet at :00 P.M. and then dropped to a minimum level of feet b :00.M. ssume that the water level can be modeled b the e function. Write an equation that represents the height h of the water t hours after noon on the first da. Stud Guide and Intervention, p. (shown) and p. NME DTE PERID - Stud Guide and Intervention Translations of Trigonometric Graphs Horizontal Translations When a constant is subtracted from the angle measure in a trigonometric function, a phase shift of the graph results. Phase Shift The horizontal phase shift of the graphs of the functions a b( h), a cos b( h), and a tan b( h) is h, where b 0. If h 0, the shift is to the right. If h 0, the shift is to the left. Eample State the amplitude, period, and phase shift for cos. Then graph the function. mplitude: a or Period: or b Phase Shift: h The phase shift is to the right ce 0. Eercises State the amplitude, period, and phase shift for each function. Then graph the function.. ( 0). tan ; 0; 0 to the left no amplitude; ; to the right. cos ( ). ; to the right ; 0; to the right ; NME DTE PERID Practice (verage) Skills Practice, p. and Practice, p. (shown) State the vertical shift, amplitude, period, and phase shift for each function. Then graph the function.. tan. cos ( 0). csc ( 0). no vertical shift; no ; ; 0; 0.; no amplitude; amplitude; ; 0; 0 Translations of Trigonometric Graphs ECLGY For Eercises, use the following information. The population of an insect species in a stand of trees follows the growth ccle of a particular tree species. The insect population can be modeled b the function 0 0 t, where t is the number of ears ce the stand was first cut in November, 90.. How often does the insect population reach its maimum level? ever 0 r. When did the population last reach its maimum? What condition in the stand do ou think corresponds with a minimum insect population? Sample answer: The species on which the insect feeds has been cut. BLD PRESSURE For Eercises 7 9, use the following information. Jason s blood pressure is 0 over 70, meaning that the pressure oscillates between a maimum of 0 and a minimum of 70. Jason s heart rate is beats per minute. The function that represents Jason s blood pressure P can be modeled ug a e function with no phase shift. 7. Find the amplitude, midline, and period in seconds of the function. 0; P 90; s. Write a function that represents Jason s blood pressure P after t seconds. P 0 70t 90 Jason s Blood Pressure P 9. Graph the function Reading to Learn Mathematics, p. 7 NME DTE PERID Reading to Learn Mathematics 70 Translations of Trigonometric Graphs Pressure Time ELL Pre-ctivit How can translations of trigonometric graphs be used to show animal populations? Read the introduction to Lesson - at the top of page 79 in our tetbook. ccording to the model given in our tetbook, what would be the estimated rabbit population for Januar, 00? 00 t Lesson - nswer nline Research Data Update Use the Internet or another resource to find tide data for a location of our choice. Write a e function to represent our data. Then graph the function. Visit to learn more.. Sample answer: When the pre (mouse) population is at its greatest the predator will consume more and the predator population will grow while the pre population falls. Lesson - Translations of Trigonometric Graphs 77 NME DTE PERID Enrichment, p. - Enrichment Translating Graphs of Trigonometric Functions Three graphs are shown at the right: ( 0) Replacing with ( 0) translates the graph to the right. Replacing with translates the graph units down. Eample Graph one ccle of cos ( 0). Step Transform the equation into Step the form k a cos b( h). cos ( ) cos. = = ( 0 ) + = = cos Reading the Lesson. Determine whether the graph of each function represents a shift of the parent function to the left, to the right, upward, or downward. (Do not actuall graph the functions.) a. ( 90) to the left b. upward c. cos to the right d. tan downward. Determine whether the graph of each function has an amplitude change, period change, phase shift, or vertical shift compared to the graph of the parent function. (More than one of these ma appl to each function. Do not actuall graph the functions.) a. amplitude change and phase shift b. cos ( 70) period change and phase shift c. cos amplitude change and period change d. sec period change and vertical shift e. tan phase shift and vertical shift f. amplitude change, period change, phase shift, and vertical shift Helping You Remember. Man students have trouble remembering which of the functions ( ) and ( ) represents a shift to the left and which represents a shift to the right. Ug, eplain a good wa to remember which is which. Sample answer: lthough e curves are infinitel repeating periodic graphs, think of starting a period or ccle at (0, 0). Then ( ) starts earl at (), a shift of to the left, while ( ) starts late at, a shift of to the right. Lesson - Translations of Trigonometric Graphs 77 Lesson -

23 ssess pen-ended ssessment Writing Have students write a summar eplaining how to use the equation for a trigonometric function to identif how its graph is shifted verticall and/or horizontall from its parent graph. Getting Read for Lesson - PREREQUISITE SKILL Students will find the value of a trigonometric function in Lesson -. Students should be familiar with values of the e, coe, and tangent functions for various reference angles in order to determine the sign of the value. Use Eercises 7 to determine our students familiarit with reference angles. Standardized Test Practice. CRITICL THINKING The graph of cot is a transformation of the graph of tan. Determine a, b, and h so that cot a tan [b( h)] for all values of for which each function is defined. a, b, h 9. WRITING IN MTH nswer the question that was posed at the beginning of the lesson. See pp. N. How can translations of trigonometric graphs be used to show animal populations? Include the following in our answer: a description of what each number in the equation R 00 0 t represents, and a comparison of the graphs of a cos b, a cos b k, and a cos [b( h)]. 0. Which equation is represented b the graph? B B C D cot ( ) cot ( ) tan ( ) tan ( ). Identif the equation for a e function of period 90, after a phase shift 0 to the left. D [0.( 0 )] B [( 0 )] C [0.( 0 )] D [( 0 )] ssessment ptions Quiz (Lessons - and -) is available on p. 9 of the Chapter Resource Masters. nswers. amplitude: does not eist; period: 0 or csc. amplitude: ; period: 70 or Maintain Your Skills Mied Review a. (a ) (a ) 0. ( )( ) Getting Read for the Net Lesson 77 Chapter Trigonometric Graphs and Identities. amplitude: does not eist; period: 70 or Find the amplitude, if it eists, and period of each function. Then graph each function. (Lesson -). See margin.. csc.. tan Find each value. (Lesson -7). Cos. cos Cos 7 7. Sin. cos Tan GEMETRY Find the total number of diagonals that can be drawn in a decagon. (Lesson -) Solve each equation. Round to the nearest hundredth. (Lesson 0-) Simplif each epression. (Lesson 9-) w w.. a a w w. 0 PREREQUISITE SKILL Find the value of each function. (To review reference angles, see Lesson -.). cos () 0 7. tan 7. cos 0 7. tan. 9. cos tan 0 7. tan 0 77 Chapter Trigonometric Graphs and Identities

24 Trigonometric Identities Lesson Notes Vocabular trigonometric identit Use identities to find trigonometric values. Use trigonometric identities to simplif epressions. can trigonometr be used to model the path of a baseball? model for the height of a baseball after it is hit as a function of time can be determined ug trigonometr. If the ball is hit with an initial velocit of v feet per second at an angle of from the horizontal, then the height h of the ball after t seconds can be represented b h v cos t c os t h 0, where h 0 is the height of the ball in feet the moment it is hit. FIND TRIGNMETRIC VLUES In the equation above, the second term in c so s t can also be written as (tan )t. c os t (tan )t is an eample of a trigonometric identit. trigonometric identit is an equation involving trigonometric functions that is true for all values for which ever epression in the equation is defined. The identit tan is true ecept for angle measures such as 90, 70, c os 0,, 90 0 k. The coe of each of these angle measures is 0, so none of the epressions tan 90, tan 70, tan 0, and so on, are defined. n identit similar to this is cot c os. These identities are sometimes called quotient identities. These and other basic trigonometric identities are listed below. Quotient Identities tan c os Basic Trigonometric Identities cot c os Reciprocal Identities csc sec cot co s ta n h 0 h t Focus -Minute Check Transparenc - Use as a quiz or review of Lesson -. Mathematical Background notes are available for this lesson on page 70D. can trigonometr be used to model the path of a baseball? sk students: What assumptions are made when ug this function as a model for the height of the ball above the ground? Sample answers: There is no wind; there is no friction as the ball passes through the air; the ball does not hit an obstruction. To hit the ball the farthest horizontal distance, is it better for the angle to be a large angle or a small angle? nswers will var. Ignoring friction, an angle measure of will result in the farthest horizontal distance. What might be a reasonable value for h 0? Sample answer: ft Pthagorean Identities cos tan sec cot csc You can use trigonometric identities to find values of trigonometric functions. Lesson - Trigonometric Identities 777 Resource Manager Workbook and Reproducible Masters Chapter Resource Masters Stud Guide and Intervention, pp. 9 0 Skills Practice, p. Practice, p. Reading to Learn Mathematics, p. Enrichment, p. Graphing Calculator and Spreadsheet Masters, p. Transparencies -Minute Check Transparenc - nswer Ke Transparencies Technolog Interactive Chalkboard Lesson - Lesson Title 777

25 Teach FIND TRIGNMETRIC VLUES Teaching Tip You can use the familiar definitions of e, coe, and tangent as the ratios of the opposite side, adjacent side, and hpotenuse of a right triangle to show wh c os tan. In-Class Eample In-Class Eample Power Point a. Find tan if sec and tan b. Find if cos and SIMPLIFY EXPRESSINS Power Point Simplif (csc ). cos Eample Find a Value of a Trigonometric Function a. Find cos if and cos Trigonometric identit cos Subtract from each side. cos Substitute for. cos 9 Square. cos Subtract. cos Take the square root of each side. Since is in the second quadrant, cos is negative. Thus, cos. b. Find csc if cot and cot csc Trigonometric identit csc Substitute for cot. csc Square. 7 csc dd. 7 csc Take the square root of each side. Since is in the fourth quadrant, csc is negative. Thus, csc 7. SIMPLIFY EXPRESSINS Trigonometric identities can also be used to simplif epressions containing trigonometric functions. Simplifing an epression that contains trigonometric functions means that the epression is written as a numerical value or in terms of a gle trigonometric function, if possible. Eample Simplif an Epression Simplif csc cot. cos csc cot c os csc cos, cot c os cos cos cos s in cos dd. cos co s s in sec sec co s 77 Chapter Trigonometric Graphs and Identities Differentiated Instruction Logical Have students work in groups of three. sk each group to choose one of the identities in the Ke Concept bo on p. 777 and work together to demonstrate that it is true. Students should verif their results ug the definitions of e, coe, and tangent in terms of the sides of a right triangle. 77 Chapter Trigonometric Graphs and Identities

26 Concept Check. See margin. Guided Practice GUIDED PRCTICE KEY Eercises Eamples 7 9 pplication Practice and ppl Practice and ppl.. nswers Eample Simplif and Use an Epression BSEBLL Refer to the application at the beginning of the lesson. Rewrite the equation in terms of tan. h v cos t c os t h 0 riginal equation v cos t c os t h 0 v cos t (tan )t h 0 Factor. tan c os (sec v )t (tan )t h 0 Since sec, co s cos sec. ( v tan )t (tan )t h 0 sec tan Thus, v cos t c os t h 0 ( v tan )t (tan )t h 0.. Describe how ou can determine the quadrant in which the terminal side of angle lies if.. Eplain wh the Pthagorean identities are so named.. PEN ENDED Eplain what it means to simplif a trigonometric epression. Find the value of each epression.. tan, if ; csc, if cos ; cos, if ; sec, if tan ; 70 0 Simplif each epression.. csc cos tan 9. sec tan 0. t an sec. ( cot ) csc. PHYSICL SCIENCE When a person moves along a circular path, the bod leans awa from a vertical position. The nonnegative acute angle that the bod makes with the vertical is called the angle of inclination and is represented b the v equation tan, where R is the radius of the circular path, v is the speed g R of the person in meters per second, and g is the acceleration due to gravit, 9. meters per second squared. Write an equivalent epression ug and cos. cos g v R Find the value of each epression.. tan, if cot ; 0 90., if cos ; sec, if tan ; tan, if sec ; Sample answer: The e function is negative in the third and fourth quadrants. Therefore, the terminal side of the angle must lie in one of those two quadrants.. Sample answer: Pthagorean identities are derived b appling the Pthagorean Theorem to trigonometric concepts. Lesson - Trigonometric Identities 779. Sample answer: Simplifing a trigonometric epression means writing the epression as a numerical value or in terms of a gle trigonometric function, if possible. In-Class Eample Practice/ppl Stud Notebook Power Point BSEBLL Refer to the application at the beginning of the lesson. Rewrite the equation in terms of sec. h v (sec )t (sec )t h 0 Have students add the definitions/eamples of the vocabular terms to their Vocabular Builder worksheets for Chapter. record the basic trigonometric identities from the Ke Concept bo on p include an other item(s) that the find helpful in mastering the skills in this lesson. bout the Eercises rganization b bjective Find Trigonometric Values: Simplif Epressions: dd/even ssignments Eercises are structured so that students practice the same concepts whether the are assigned odd or even problems. ssignment Guide Basic: odd, 7, verage: odd, 7, dvanced: even, 7 (optional: ) ll: Practice Quiz ( ) Lesson - Trigonometric Identities 779

27 Stud Guide and Intervention, p. 9 (shown) and p. 0 NME DTE PERID - Stud Guide and Intervention Trigonometric Identities Find Trigonometric Values trigonometric identit is an equation involving trigonometric functions that is true for all values for which ever epression in the equation is defined. Quotient Identities tan cot c os c os Basic Trigonometric Reciprocal Identities csc sec cot co s ta n Identities Pthagorean Identities cos tan sec cot csc Eample Find the value of cot if csc ; cot csc Trigonometric identit cot Substitute for csc. cot Square. cot 9 Subtract from each side. cot Take the square root of each side. Since is in the third quadrant, cot is positive, Thus cot. Eercises Find the value of each epression.. tan, if cot ; csc, if cos ; cos, if ;0 90. sec, if ;0 90. cos, if tan ;90 0. tan, if 7 ; sec, if cos 7 ; , if cos 7 ; cot, if csc ; , if csc 9 ; NME 9 DTE PERID Practice (verage) Skills Practice, p. and Practice, p. (shown) Trigonometric Identities Find the value of each epression.., if cos and sec, if and cot, if cos 0 and 70 0., if cot and cot, if csc and sec, if csc and sec, if tan and 0 70., if tan and Lesson - Homework Help For See Eercises Eamples 7 Etra Practice See page Find the value of each epression. 7. csc, if cos ; cos, if sec ; cos, if ; csc, if cos ; tan, if cos ; cos, if csc ; sec, if ; 90 0., if tan ; Simplif each epression cos csc cot. tan cot 7. cot cos. cos tan 9. (csc cot ) 0. (tan sec ). cos csc cot tan. cos. cos csc. csc tan cot. cot. ta n tan MUSEMENT PRKS For Eercises 7 9, use the following information. Suppose a child is riding on a merr-go-round and is seated on an outside horse. The diameter of the merr-go-round is meters. 7. If the e of the angle of inclination of the child is, what is the angle of inclination made b the child? Refer to Eercise for information on the angle of inclination. about.. What is the velocit of the merr-go-round? about m/s 9. If the speed of the merr-go-round is. meters per second, what is the value of the angle of inclination of a rider? about cot, if tan and cot, if cos and 70 0 Simplif each epression.. csc tan sec. cos. cot cos tan csc cot. cot csc. csc. csc cot cos cos cos cos 7. cos cot. 9. sec cos tan csc tan sec 0. ERIL PHTGRPHY The illustration shows a plane taking an aerial photograph of point. Because the point is directl below the plane, there is no distortion in the image. For an point B not directl below the plane, however, the increase in distance creates distortion in the photograph. This is because as the distance from the camera to the point being photographed increases, the eposure of the film reduces b ( )(csc ). Epress ( )(csc ) in terms of cos onl. cos. TSUNMIS The equation a t represents the height of the waves pasg a buo at a time t in seconds. Epress a in terms of csc t. a csc t Reading to Learn Mathematics, p. - NME DTE PERID Reading to Learn Mathematics Trigonometric Identities Pre-ctivit How can trigonometr be used to model the path of a baseball? Read the introduction to Lesson - at the top of page 777 in our tetbook. Suppose that a baseball is hit from home plate with an initial velocit of feet per second at an angle of with the horizontal from an initial height of feet. Show the equation that ou would use to find the height of the ball 0 seconds after the ball is hit. (Show the formula with the appropriate numbers substituted, but do not do an calculations.) h 0 0 cos cos Reading the Lesson. Match each epression from the list on the left with an epression from the list on the right that is equal to it for all values for which each epression is defined. (Some of the epressions from the list on the right ma be used more than once or not at all.) a. sec tan iii i. b. cot v ii. tan ELL B musement Parks The oldest operational carousel in the United States is the Fling Horse Carousel at Martha s Vineard, Massachusetts. Source: Martha s Vineard Preservation Trust LIGHTING For Eercises 0 and, use the following information. The amount of light that a source provides to a surface is called the illuminance. The illuminance E in foot candles on a surface is related to the distance R in feet from I the light source. The formula sec ER, where I is the intensit of the light source measured in candles and is the angle between the light beam and a line perpendicular to the surface, can be used in situations in which lighting is important. 0. Solve the formula in terms of E. E I c os R. Is the equation in Eercise 0 equivalent to R I tan cos? Eplain. E No; R I tan cos simplifies to E I s in. E R ELECTRNICS For Eercises and, use the following information. When an alternating current of frequenc ƒ and a peak current I pass through a resistance R, then the power delivered to the resistance at time t seconds is P I R I R cos (ft).. Write an epression for the power in terms of ft. P I R ft. Write an epression for the power in terms of tan ft. P I I R R t an ft. CRITICL THINKING If tan, find sec 9. cot 70 Chapter Trigonometric Graphs and Identities c. c ii iii. os d. cos iii iv. sec e. csc i v. csc f. co iv vi. cot s g. c os vi. Write an identit that ou could use to find each of the indicated trigonometric values and tell whether that value is positive or negative. (Do not actuall find the values.) a. tan, if and 0 70 tan ; positive cos b. sec, if tan and 90 0 tan sec ; negative Helping You Remember NME DTE PERID - Enrichment, p. Planetar rbits The orbit of a planet around the sun is an ellipse with the sun at one focus. Let the pole of a polar coordinate sstem be that focus and the polar ais be toward the other focus. The polar equation of an ellipse is ep r. Since p b and b e cos c a c, p a c a c c c. a Because e c, a r Polar is. good wa to remember something new is to relate it to something ou alread know. How can ou use the unit circle definitions of the e and coe that ou learned in Chapter to help ou remember the Pthagorean identit cos? Sample answer: n a unit circle, cos and. The equation of the unit circle is, so this is equivalent to the equation cos. 70 Chapter Trigonometric Graphs and Identities p a a c c a a e ( e ). Therefore ep a( e ). Substituting into the polar equation of an ellipse ields an equation that is useful for finding distances from the planet to the sun. a( e r ) e cos ccentricit of d a

28 Standardized Test Practice Maintain Your Skills P Mied Review 9. See margin for graphs. Getting Read for the Net Lesson ractice Quiz WRITING IN MTH nswer the question that was posed at the beginning of the lesson. See margin. How can trigonometr be used to model the path of a baseball? Include the following in our answer: an eplanation of wh the equation at the beginning of the lesson is the same as sec v (tan ) h 0, and eamples of how ou might use this equation for other situations.. If m and 0 90, then tan B m m.. m B C. D. m mm m 7. sec B sec C csc D csc State the vertical shift, equation of the midline, amplitude, and period for each function. Then graph the function. (Lesson -) ; ;. ; ; ; 0 9. tan no amplitude; 0 Find the amplitude, if it eists, and period of each function. Then graph each function. (Lesson -) 0. See pp. N. 0. csc. cos. cot. Find the sum of a geometric series for which a, a n, and r. (Lesson -) 9. Write an equation of a parabola with focus at (, ) and whose directri is. (Lesson -) ( ) PREREQUISITE SKILL Name the propert illustrated b each statement. (To review properties of equalit, see Lesson -.). Smmetric (). If, then.. If 7 s, then s. Subt. () 7. If, then.. If q ( ), then q. Multiplication () Substitution (). Find the amplitude and period of. Then graph the function.. (Lesson -), 70 or. State the vertical shift, amplitude, period, and phase shift for cos. Then graph the function. Find the value of each epression. (Lesson -). cos, if ; csc, if cot ; sec, if tan ; 0 90 (Lesson -),,, Lessons - through - See pp. N for graphs. Lesson - Trigonometric Identities 7 ssess pen-ended ssessment Speaking Have students eplain how to rewrite and simplif epressions ug trigonometric identities, ug an eample the worked during the lesson. Getting Read for Lesson - PREREQUISITE SKILL Students will verif trigonometric identities in Lesson - b rewriting them ug the properties of equalit. Use Eercises to determine our students familiarit with the properties of equalit. ssessment ptions Practice Quiz The quiz provides students with a brief review of the concepts and skills in Lessons - through -. Lesson numbers are given to the right of the eercises or instruction lines so students can review concepts not et mastered. nswers nswer. Sample answer: You can use equations to find the height and the horizontal distance of a baseball after it has been hit. The equations involve ug the initial angle the ball makes with the ground with the e function. nswers should include the following information. Both equations are quadratic in nature with a leading negative coefficient. Thus, both are inverted parabolas which model the path of a baseball. model rockets, hitting a golf ball, kicking a rock tan Lesson - Trigonometric Identities 7

29 Lesson Notes Verifing Trigonometric Identities Focus -Minute Check Transparenc - Use as a quiz or review of Lesson -. Mathematical Background notes are available for this lesson on p. 70D. can ou verif trigonometric identities? sk students: Can ou give an eample of two functions in whose values are equal for some values of but not all? Sample answer: and are the same for 0 but not for 0. Wh is it not sufficient to show that two functions have equal values for specific values of when tring to justif that two functions are equivalent? The functions might have equal values for some values of but not others. Stud Tip Common Misconception You cannot perform operations to the quantities from each side of an unverified identit as ou do with equations. Until an identit is verified it is not considered an equation, so the properties of equalit do not appl. Verif trigonometric identities b transforming one side of an equation into the form of the other side. Verif trigonometric identities b transforming each side of the equation into the same form. can ou verif trigonometric identities? Eamine the graphs of tan and tan. Recall that when the graphs of two functions coincide, the functions are equivalent. However, the graphs onl show a limited range of solutions. It is not sufficient to show some values of and conclude that the statement is true for all values of. In order to show that the equation tan tan for all values of, ou must consider the general case. TRNSFRM NE SIDE F N EQUTIN You can use the basic trigonometric identities along with the definitions of the trigonometric functions to verif identities. For eample, if ou wish to show that tan tan is an identit, ou need to show that it is true for all values of. Verifing an identit is like checking the solution of an equation. You must simplif one or both sides of an equation separatel until the are the same. In man cases, it is easier to work with onl one side of an equation. You ma choose either side, but it is often easier to begin with the more complicated side of the equation. Transform that epression into the form of the simpler side. Eample Transform ne Side of an Equation Verif that tan tan is an identit. Transform the left side. tan tan riginal equation c os tan tan c os c os cos cos tan Rewrite ug the LCD, cos. cos cos ( cos ) cos tan tan co s tan c os tan tan tan Subtract. Factor. cos a b a c c b c os tan tan (tan )( ) 7 Chapter Trigonometric Graphs and Identities Resource Manager Workbook and Reproducible Masters Chapter Resource Masters Stud Guide and Intervention, pp. Skills Practice, p. 7 Practice, p. Reading to Learn Mathematics, p. 9 Enrichment, p. 0 ssessment, pp. 9, 9 Transparencies -Minute Check Transparenc - nswer Ke Transparencies Technolog lgepss: Tutorial Plus, Lesson Interactive Chalkboard

30 Standardized Test Practice Test-Taking Tip Verif our answer b choog values for. Then evaluate the original epression and compare to our answer choice. Eample Multiple-Choice Test Item TRNSFRM BTH SIDES F N EQUTIN Sometimes it is easier to transform both sides of an equation separatel into a common form. The following suggestions ma be helpful as ou verif trigonometric identities. Substitute one or more basic trigonometric identities to simplif an epression. Factor or multipl to simplif an epression. Multipl both the numerator and denominator b the same trigonometric epression. Write both sides of the identit in terms of e and coe onl. Then simplif each side as much as possible. Find an Equivalent Epression c os cot cos B C cos D Read the Test Item Find an epression that is equal to the given epression. Solve the Test Item Write a trigonometric identit b ug the basic trigonometric identities and the definitions of trigonometric functions to transform the given epression to match one of the choices. c os cos cot cot c s cos so in cos cos Simplif. cos Simplif. Since c os cot cos, the answer is C. Eample Distributive propert. cos Verif b Transforming Both Sides Verif that sec tan tan cot is an identit. sec tan tan cot riginal equation cos c os c os c os Epress all terms ug e and coe. cos Standardized Test Practice c os cos Subtract on the left. Multipl on the right. cos Simplif the left side. Eample Urge students to read slowl and carefull so the do not mistake cot for cos or csc for cos. In addition, students ma find the correct answer, cos, but choose choice because the did not notice the difference between cos and cos. Lesson - Verifing Trigonometric Identities 7 Teach TRNSFRM NE SIDE F N EQUTIN In-Class Eamples In-Class Eample Practice/ppl Stud Notebook Power Point Verif that csc cos tan is an identit. csc cos tan cos c os csc c os tan cot B cos C 0 D cos TRNSFRM BTH SIDES F N EQUTIN Power Point Verif that csc sec cot is an identit. cos csc sec cot cos c os co s cos co c os s cos cos co s cos si n cos s in cos Have students add the definitions/eamples of the vocabular terms to their Vocabular Builder worksheets for Chapter. include an other item(s) that the find helpful in mastering the skills in this lesson. Lesson - Verifing Trigonometric Identities 7

31 bout the Eercises rganization b bjective Transform ne Side of an Equation:, 0 Transform Both Sides of an Equation: dd/even ssignments Eercises 0 are structured so that students practice the same concepts whether the are assigned odd or even problems. lert! Eercises 7 require a graphing calculator. ssignment Guide Basic: odd,, verage: odd,, (optional: 7) dvanced: 0 even, 0 (optional: ) ssess pen-ended ssessment Speaking Have students eplain some of the techniques the have seen or used to verif trigonometric identities in this lesson. Getting Read for Lesson - PREREQUISITE SKILL Students will simplif radical epressions in the process of ug the Sum and Difference of ngles Formulas in Lesson -. Use Eercises to determine our students familiarit with simplifing radical epressions. ssessment ptions Quiz (Lessons - and -) is available on p. 9 of the Chapter Resource Masters. Mid-Chapter Test (Lessons - through -) is available on p. 9 of the Chapter Resource Masters. Concept Check. See pp. N. Guided Practice GUIDED PRCTICE KEY Eercises Eamples, 0, Standardized Test Practice Practice and ppl Homework Help For See Eercises Eamples,, Etra Practice See page 0. 7 Chapter Trigonometric Graphs and Identities. Eplain the steps used to verif the identit tan sec cos.. Describe the various methods ou can use to show that two trigonometric epressions form an identit.. PEN ENDED Write a trigonometric equation that is not an identit. Eplain how ou know it is not an identit. Verif that each of the following is an identit. 9. See pp. N.. tan (cot tan ) sec. tan cos cos cos. 7. tan csc tan. 9. sec tan s ec tan cot tan sec 0. Which epression is equivalent to se c csc? D tan B cos C tan D csc Verif that each of the following is an identit.. See pp. N.. cos tan cos. cot (cot tan ) csc. sec sec. sec cot. cos (csc cot ) cos. cos tan cot cos cot csc 7. cot csc. cos tan s in tan sec 9. s ec cot 0. cos csc c os cos. cot. tan cs c cot c os tan. sec csc. co s se c. tan sec sec. cos cos 7. cos cos cos. sec cos 9. Verif that tan cos csc is an identit. See pp. N. 0. Show that cos and form an identit. See pp. N. cos PHYSICS For Eercises and, use the following information. If an object is propelled from ground level, the maimum height that it reaches is given b h v, where is the angle between the ground and the initial g path of the object, v is the object s initial velocit, and g is the acceleration due to gravit, 9. meters per second squared.. Verif the identit v v tan. g g sec See pp. N.. model rocket is launched with an initial velocit of 0 meters per second at an angle of 0 with the ground. Find the maimum height of the rocket. 9.7 m Differentiated Instruction Interpersonal Have groups or pairs of students work together to verif some of the identities in Eercises 9. Have students record the techniques the found helpful. sk students to compare their list of techniques to the list of suggestions given above Eample on p Chapter Trigonometric Graphs and Identities

32 Standardized Test Practice Graphing Calculator 7. See pp. N for graphs. Maintain Your Skills Mied Review ; ;.. Getting Read for the Net Lesson. CRITICL THINKING Present a logical argument for wh the identit cos is true when 0. See margin.. WRITING IN MTH nswer the question that was posed at the beginning of the lesson. See pp. N. How can ou verif trigonometric identities? Include the following in our answer: an eplanation of wh ou cannot perform operations to each side of an unverified identit, an eplanation of how ou can tell if two epressions are equivalent, and an eplanation of wh ou cannot use the graphs of two equations to verif an identit.. Which of the following is not equivalent to cos? D cos cos B C cot D tan csc cos. Which of the following is equivalent to cot cos? B cos cos B C D VERIFYING TRIGNMETRIC IDENTITIES You can determine whether or not an equation ma be a trigonometric identit b graphing the epressions on either side of the equals sign as two separate functions. If the graphs do not match, then the equation is not an identit. If the two graphs do coincide, the equation might be an identit. The equation has to be verified algebraicall to ensure that it is an identit. Determine whether each of the following ma be or is not an identit. 7. cot tan csc cot is not. sec sec ma be 9. ( )( ) cos ma be 0. csc sec ma be tan. s ec sec csc ma be. tan se c cs is not c Find the value of each epression. (Lesson -). sec, if tan ; cos, if ; csc, if cot ; 90 0., if cos ; 70 0 State the amplitude, period, and phase shift of each function. Then graph each function. (Lesson -) 7 9. See pp. N for graphs. 7. cos ( 0 ). ( ) 9. cos ; 0 ; 0 ; 0 ; 0. What is the probabilit that an event occurs if the odds of the event occurring are :? (Lesson -) PREREQUISITE SKILL Simplif each epression. (To review simplifing radical epressions, see Lesson -.).... Stud Guide and Intervention, p. (shown) and p. NME DTE PERID - Stud Guide and Intervention Verifing Trigonometric Identities Transform ne Side of an Equation Use the basic trigonometric identities along with the definitions of the trigonometric functions to verif trigonometric identities. ften it is easier to begin with the more complicated side of the equation and transform that epression into the form of the simpler side. Eample Verif that each of the following is an identit. tan a. sec cos b. cos sec cot csc Transform the left side. Transform the left side. sec cos tan cot cos sec csc cos c os cos c os cos sec cos cos cos cos sec cos cos cos cos cos sec cos cos cos sec cos cos cos sec sec Eercises Verif that each of the following is an identit.. csc cos csc cot cos. cos cos cos csc ( cos ) c os ( cos ) cos csc ( cos )( cos ) cos c os c os csc cos csc csc cos cos cos cos cos ( ) cos (cos ) - cos cos cos cos cos cos cos NME DTE PERID Practice (verage) Skills Practice, p. 7 and Practice, p. (shown) Verifing Trigonometric Identities Verif that each of the following is an identit.. cos sec cos os cos c sec cos sec sec sec. ( )( ) cos ( )( ) cos cos cos cos. cos cot cot cos cos cot cot cos cos cot c os cos cos cot cos cos cos cot (cos )( ) cos cot cos c os cos cot cos cot cos. cos c os cos. tan tan sec tan tan sec (tan ) sec (sec ) sec sec sec. ( )(csc sec ) sec ( )(csc sec ) sec ( ) cos sec c os sec tan sec sec sec 7. PRJECTILES The square of the initial velocit of an object launched from the ground is v gh, where is the angle between the ground and the initial path, h is the maimum height reached, and g is the acceleration due to gravit. Verif the identit gh gh sec. sec gh gh gh gh s in cos gh sec sec se c sec sec. LIGHT The intensit of a light source measured in candles is given b I ER sec, where E is the illuminance in foot candles on a surface, R is the distance in feet from the light source, and is the angle between the light beam and a line perpendicular to the surface. Verif the identit ER ( tan ) cos ER sec. ER ( tan ) cos ER sec cos ER sec ER se c sec Reading to Learn Mathematics, p. 9 - NME DTE PERID Reading to Learn Mathematics Verifing Trigonometric Identities Pre-ctivit How can ou verif trigonometric identities? Read the introduction to Lesson - at the top of page 7 in our tetbook. For, 0, or,. Does this mean that is an identit? Eplain our reasoning. Sample answer: No; an identit is an equation that is true for all values of a variable for which the functions involved are defined, not just some values. If,, and. Reading the Lesson. Determine whether each equation is an identit or not an identit. a. tan identit ELL Lesson - Lesson - Verifing Trigonometric Identities 7 cos b. not an identit tan c. c os cos c not an identit os d. cos (tan ) identit nswer. Sample answer: Consider a right triangle BC with right angle at C. If an angle, sa, has a e of, then angle B must have a coe of. Since and B are both in a right triangle and neither is the right angle, their sum must be. NME DTE PERID - Enrichment, p. 0 Heron s Formula Heron s formula can be used to find the area of a triangle if ou know the lengths of the three sides. Consider an triangle BC. Let K represent the area of BC. Then K bc B K b c Square both sides. c a b c ( cos ) b b c ( cos )( cos ) b c b c a b c a bc bc Use the law of coes. b c a b c a a b c a b c Simplif. C e. c os csc sec identit f. cos not an identit g. tan cos identit csc h. s ec ta n co not an identit t. Which of the following is not permitted when verifing an identit? B. simplifing one side of the identit to match the other side B. cross multipling if the identit is a proportion C. simplifing each side of the identit separatel to get the same epression on both sides Helping You Remember. Man students have trouble knowing where to start in verifing a trigonometric identit. What is a simple rule that ou can remember that ou can alwas use if ou don t see a quicker approach? Sample answer: Write both sides in terms of es and coes. Then simplif each side as much as possible. Lesson - a hen s a b b c. Lesson - Verifing Trigonometric Identities 7

33 7 GHI 0 per BC JKL TUV MN WXYZ DEF Lesson Notes Sum and Difference of ngles Formulas Focus -Minute Check Transparenc - Use as a quiz or review of Lesson -. Mathematical Background notes are available for this lesson on p. 70D. are the sum and difference formulas used to describe communication interference? sk students: Where else have ou heard the term interference? Sample answer: television and radio t its peak, how does the amplitude of the combined wave compare to the amplitude of the initial two waves? The amplitude of the combined wave is the sum of the amplitudes of the two initial waves. Wh does the combined wave cross the -ais at a point where neither of the two initial waves are crosg the ais? The combined wave is the sum of the other two waves. It crosses the -ais at points where one of the initial waves is above the -ais and the other wave is an equal distance below the -ais. Stud Tips Reading Math The Greek letter beta,, can be used to denote the measure of an angle. It is important to realize that ( ) is not the same as. Find values of e and coe involving sum and difference formulas. Verif identities b ug sum and difference formulas. are the sum and difference formulas used to describe communication interference? Have ou ever been talking on a cell phone and temporaril lost the signal? Radio waves that pass through the same place at the same time cause interference. Constructive interference occurs when two waves combine to have a greater amplitude than either of the component waves. Destructive interference occurs when the component waves combine to have a smaller amplitude. ( ) [ ( ) ] SUM ND DIFFERENCE FRMULS Notice that the third equation shown above involves the sum of and. It is often helpful to use formulas for the trigonometric values of the difference or sum of two angles. For eample, ou could find b evaluating (0 ). Formulas can be developed that can be used to evaluate epressions like ( ) or cos ( ). The figure at the right shows two angles (cos, ) and in standard position on the unit circle. d Use the Distance Formula to find d, where (cos, ) (, ) (cos, ) and (, ) (cos, ). d (cos cos ) ( ) d (cos cos ) ( ) d (cos cos cos cos ) ( ) d cos cos cos cos d cos cos + cos = and d cos cos + cos = Now find the value of d when the angle having [cos ( ), ( )] measure is in standard position on the unit d circle, as shown in the figure at the left. d [cos ( ) ] [) ( ] 0 (, 0) d [cos ( ) ] [ ( ) 0] 9 PQRS * # SEPT. 7 FRI. :0 PM S S [cos ( ) cos ( ) ] ( ) cos ( ) ( ) cos ( ) cos ( ) cos ( ) 7 Chapter Trigonometric Graphs and Identities Resource Manager Workbook and Reproducible Masters Chapter Resource Masters Stud Guide and Intervention, pp. Skills Practice, p. Practice, p. Reading to Learn Mathematics, p. Enrichment, p. Transparencies -Minute Check Transparenc - nswer Ke Transparencies Technolog Interactive Chalkboard

34 B equating the two epressions for d, ou can find a formula for cos ( ). d d cos ( ) cos cos cos ( ) cos cos Divide each side b. cos ( ) cos cos Sum and Difference of ngles Formulas The following identities hold true for all values of and. cos ( ) cos cos ( ) cos cos dd to each side. Use the formula for cos ( ) to find a formula for cos ( ). cos ( ) cos [ ()] cos cos () () cos cos cos () cos ; () You can use a similar method to find formulas for ( ) and ( ). Notice the smbol in the formula for cos ( ). It means minus or plus. In the coe formula, when the sign on the left side of the equation is plus, the sign on the right side is minus; when the sign on the left side is minus, the sign on the right side is plus. The signs match each other in the e formula. Eample Use Sum and Difference of ngles Formulas Find the eact value of each epression. a. cos 7 Use the formula cos ( ) cos cos. cos 7 cos (0 ) 0, cos 0 cos 0 Evaluate each epression. Multipl. Simplif. b. (0 ) Use the formula ( ) cos cos. (0 ) (0 70 ) = 0, = 70 0 cos 70 cos 0 70 (0) () Evaluate each epression. 0 Multipl. Simplif. Lesson - Sum and Difference of ngles Formulas 77 Differentiated Instruction Naturalist Have students appl what the know about geograph, or have them conduct research, to find out whether the light energ per square foot would be increag or decreag as ou travel toward the equator. Have students research the latitude of our cit, or a cit that interests them, and repeat Eample for that cit. Teach Building on Prior Knowledge The Distance Formula was first discussed and used back in Lesson -. In this lesson, students will learn how to appl the Distance Formula to derive the Sum and Difference of ngles Formulas. SUM ND DIFFERENCE FRMULS Teaching Tip Eplain that can be found b evaluating (0 ) because the eact values of 0 and are known. Stress that ug a difference such as (90 7) is ineffective because 7 is not easil computed or remembered. Reading Tip Have the entire class read aloud the paragraph directl below the Ke Concept bo, about the minus and plus signs. Stress that its use in the formula for the coe of a sum or difference indicates that the sign on the right side of the identit is the opposite of the sign on the left side. In-Class Eamples Power Point Find the eact value of each epression. a. 7 b. cos (7) Refer to Eample in the Student Edition. Use the difference of angles formula to determine the amount of light energ in Raleigh, North Carolina, located at a latitude of. N. The maimum light energ per square foot is E. Lesson - Sum and Difference of ngles Formulas 77

35 VERIFY IDENTITIES In-Class Eample Verif that each of the following is an identit. a. cos (90 ) cos (90) cos 90 cos 90 0 cos b. cos (0 ) cos cos (0 ) cos cos 0 cos 0 cos cos 0 cos cos cos Practice/ppl Stud Notebook Power Point Have students add the definitions/eamples of the vocabular terms to their Vocabular Builder worksheets for Chapter. record the sum and difference formulas. include an other item(s) that the find helpful in mastering the skills in this lesson. bout the Eercises rganization b bjective Sum and Difference Formulas: 7 Verif Identities: 9 dd/even ssignments Eercises 9 are structured so that students practice the same concepts whether the are assigned odd or even problems. ssignment Guide Basic: odd, 9 7 odd, 0, odd, 7 verage: 9 odd, 0, odd, 7 dvanced: 0 even, (optional: 7 7) Stud Tip Reading Math The smbol is the lowercase Greek letter phi. Concept Check. See margin.. Sometimes; sample answer: The coe function can equal. Guided Practice GUIDED PRCTICE KEY Eercises Eamples 9, 0 Eample VERIFY IDENTITIES verif identities. 7 Chapter Trigonometric Graphs and Identities nswers Use Sum and Difference Formulas to Solve a Problem PHYSICS n June, the maimum amount of light energ falling on a square foot of ground at a location in the northern hemisphere is given b E (. ), where is the latitude of the location and E is the amount of light energ when the Sun is directl overhead. Use the difference of angles formula to determine the amount of light energ in Rochester, New York, located at a latitude of. N. Use the difference formula for e. (. ). cos cos.. cos. cos (0.97) In Rochester, New York, the maimum light energ per square foot is 0.90E. Eample Verif Identities You can also use the sum and difference formulas to Verif that each of the following is an identit. a. (0 ) (0 ) riginal equation 0 cos cos 0 Sum of angles formula 0 cos () Evaluate each epression. Simplif. b. cos (0 ) cos cos (0 ) cos cos 0 cos 0 cos () cos 0 cos cos cos riginal equation Sum of angles formula Evaluate each epression. Simplif.. Determine whether ( ) is an identit.. Describe a method for finding the eact value of 0. Then find the value.. PEN ENDED Determine whether cos ( ) is sometimes, alwas, or never true. Eplain our reasoning. Find the eact value of each epression cos 7. cos (0 ). (0 ) 9. cos (0 ) Verif that each of the following is an identit. 0. See pp N. 0. cos (70 ). cos. ( 0 ) cos ( 0 ) cos. ( ) cos cos. Use the formula ( ) cos cos. Since 0 (0), replace with 0 and with to get 0 cos cos 0. B finding the sum of the products of the values, the result is or about Chapter Trigonometric Graphs and Identities

36 pplication indicates increased difficult Practice and ppl Homework Help For Eercises See Eamples 7 9 0,, Etra Practice See page Phsics In the northern hemisphere, the da with the least number of hours of dalight is December or, the first da of winter. Source: GEMETRY Determine the eact value of tan. Find the eact value of each epression... cos cos. cos cos.. ( ). cos ( ). cos (0 ). ( ). What is the eact value of 7? 7. Find the eact value of cos 0 cos. Verif that each of the following is an identit. 9. See pp. N.. (70 ) cos 9. cos (90 ) 0. cos (90 ). (90 ) cos. cos. cos ( ) cos. cos ( ) cos. ( ). (0 ) (0 ) cos 7. cos. ( ) ( ) 9. cos ( ) tan tan sec sec CMMUNICTIN For Eercises 0 and, use the following information. radio transmitter sends out two signals, one for voice communication and another for data. Suppose the equation of the voice wave is v 0 (t 0 ) and the equation of the data wave is d 0 cos (t 0 ). 0. Draw a graph of the waves when the are combined. See margin.. Refer to the application at the beginning of the lesson. What tpe of interference results? Eplain. Destructive; the resulting graph has a smaller amplitude than the two initial graphs. PHYSICS For Eercises, use the following information. n December, the maimum amount of light energ that falls on a square foot of ground at a certain location is given b E (. ), where is the latitude of the location. Use the sum of angles formula to find the amount of light energ, in terms of E, for each location.. 0. E E. Salem, R (Latitude:.9 N). Chicago, IL (Latitude:. N). Charleston, SC (Latitude:. N). San Diego, C (Latitude.7 N) 0.7 E 0. E. CRITICL THINKING Use the sum and difference formulas for e and coe to derive formulas for tan ( ) and tan ( ). See pp. N. 0 Lesson - Sum and Difference of ngles Formulas 79 0 Stud Guide and Intervention, p. (shown) and p. NME DTE PERID - Stud Guide and Intervention Sum and Difference of ngles Formulas Sum and Difference Formulas The following formulas are useful for evaluating an epression like from the known values of e and coe of 0 and. Sum and Difference of ngles The following identities hold true for all values of and. cos ( ) cos cos ( ) cos cos Eample Find the eact value of each epression. a. cos cos cos (00 ) cos 00 cos 00 b. (0) (0) ( 0) cos 0 cos 0 Eercises Find the eact value of each epression.. 0. cos. cos (7). cos (). 9. cos 0 7. (7). cos 9. cos () 0.. cos (0). 9 - NME DTE PERID Practice (verage) Skills Practice, p. and Practice, p. (shown) Sum and Difference of ngles Formulas Find the eact value of each epression.. cos 7. cos 7. (). (0). 0. cos (7) 9. 9 Verif that each of the following is an identit. 0. cos (0 ) cos cos (0 ) cos cos 0 cos 0 cos cos 0 cos cos cos. (0 ) (0 ) 0 cos cos 0 0 cos. ( ) ( ) ( ) ( ) cos cos ( cos cos ) cos. cos cos cos cos cos cos cos cos. SLR ENERGY n March, the maimum amount of solar energ that falls on a square foot of ground at a certain location is given b E (90 ), where is the latitude of the location and E is a constant. Use the difference of angles formula to find the amount of solar energ, in terms of cos, for a location that has a latitude of. E cos ELECTRICITY In Eercises and, use the following information. In a certain circuit carring alternating current, the formula i (0t) can be used to find the current i in amperes after t seconds. Sample answer:. Rewrite the formula ug the sum of two angles. i (90t 0t). Use the sum of angles formula to find the eact current at t second. amperes Reading to Learn Mathematics, p. - NME DTE PERID Reading to Learn Mathematics Sum and Difference of ngles Formulas Pre-ctivit How are the sum and difference formulas used to describe communication interference? Read the introduction to Lesson - at the top of page 7 in our tetbook. Consider the functions and. Do the graphs of these two functions have constructive interference or destructive interference? constructive Reading the Lesson ELL. Match each epression from the list on the left with an epression from the list on the right that is equal to it for all values of the variables. (Some of the epressions from the list on the right ma be used more than once or not at all.) a. ( ) v i. b. cos ( ) vi ii. cos cos c. (0 ) vii iii. cos Lesson - d. (0 ) i iv. cos cos 0. NME DTE PERID Enrichment, p. - Enrichment e. cos (0 ) iii v. cos cos f. ( ) ii vi. cos cos g. cos (90 ) i vii. 0 (t 0 ) 0 cos (t 0 ) Identities for the Products of Sines and Coes B adding the identities for the es of the sum and difference of the measures of two angles, a new identit is obtained. h. cos ( ) iv viii. cos. Which epressions are equal to? (There ma be more than one correct choice.). cos 0 cos 0 B. cos 0 cos 0 B and C t ( ) cos cos ( ) cos cos (i) ( ) ( ) cos This new identit is useful for epresg certain products as sums. Eample Write cos as a sum. In the identit let and so that cos ( ) ( ). Thus, cos. B subtracting the identities for ( ) and ( ), a similar identit for epresg a product as a difference is obtained. C. 0 cos cos 0 D. cos 0 cos 0 Helping You Remember. Some students have trouble remembering which signs to use on the right-hand sides of the sum and difference of angle formulas. What is an eas wa to remember this? Sample answer: In the e identities, the signs are the same on both sides. In the coe identities, the signs are opposite on the two sides. Lesson - Lesson - Sum and Difference of ngles Formulas 79

37 ssess pen-ended ssessment Modeling Show students a graph of the function. Have students point out on the graph which values would be most useful to use with the sum and difference of angles formulas, and ask them to eplain their reasoning. Getting Read for Lesson - PREREQUISITE SKILL Students will find values ug half-angle formulas in Lesson -. The halfangle formulas include epressions within square root smbols, so students must be comfortable evaluating square roots. Use Eercises 7 7 to determine our students familiarit with the Square Root Propert. nswers 7. Sample answer: To determine communication interference, ou need to determine the e or coe of the sum or difference of two angles. nswers should include the following information. Interference occurs when waves pass through the same space at the same time. When the combined waves have a greater amplitude, constructive interference results and when the combined waves have a smaller amplitude, destructive interference results.., cos, tan, csc, sec, cot Standardized Test Practice Maintain Your Skills Mied Review 0. See pp. N.. about mi Getting Read for the Net Lesson WRITING IN MTH nswer the question that was posed at the beginning of the lesson. See margin. How are the sum and difference formulas used to describe communication interference? Include the following in our answer: an eplanation of the difference between constructive and destructive interference, and a description of how ou would eplain wave interference to a friend.. Find the eact value of. 790 Chapter Trigonometric Graphs and Identities C B D 9. Find the eact value of cos (0 ). C 0. B C Verif that each of the following is an identit. (Lesson -) 0. cot sec co s cos D tan ( cos ) s ec csc. ( csc ) cos sec. csc t an Simplif each epression. (Lesson -). tan csc. sec sec c os. (cot tan ) sec 7. csc tan sec sec Find the eact values of the si trigonometric functions of if the terminal side of in standard position contains the given point. (Lesson -) 0. See margin. (, ) 9. (, ) 0. (0, ) Evaluate each epression. (Lesson -). P(, ) 0. P(, 7),99,0. C(, ). C(0, ) 0. VITIN pilot is fling from Chicago to Columbus, a distance of 00 miles. In order to avoid an area of thunderstorms, she alters her initial course b and flies on this course for 7 miles. How far is she from Columbus? (Lesson -). Write 0 in standard form. (Lesson -) PREREQUISITE SKILL Solve each equation. (To review solving equations ug the Square Root Propert, see Lesson -.) , cos, tan, csc, sec, cot 0., cos 0, tan undefined, csc, sec undefined, cot Chapter Trigonometric Graphs and Identities

38 Double-ngle and Half-ngle Formulas Lesson Notes Find values of e and coe involving double-angle formulas. Find values of e and coe involving half-angle formulas. can trigonometric functions be used to describe music? Stringed instruments such as a piano, guitar, or violin rel on waves to produce the tones we hear. When the strings are struck or plucked, the vibrate. If the motion of the strings were observed in slow motion, ou could see that there are places on the string, called nodes, that do not move under the vibration. Halfwa between each pair of consecutive nodes are antinodes that undergo the maimum vibration. The nodes and antinodes form harmonics. These harmonics can be represented ug variations of the equations and. DUBLE-NGLE FRMULS You can use the formula for ( ) to find the e of twice an angle,, and the formula for cos ( ) to find the coe of twice an angle, cos. ( ) cos cos ( ) cos cos cos cos cos cos You can find alternate forms for cos b making substitutions into the epression cos. cos ( ) Substitute for cos. cos cos ( cos ) cos Simplif. Substitute cos for. Simplif. These formulas are called the double-angle formulas. Double-ngle Formulas The following identities hold true for all values of. cos cos cos cos cos cos Fundamental, first harmonic N N Second harmonic N Third harmonic N Fourth harmonic N N N N Fifth harmonic N N n n n n n Focus -Minute Check Transparenc - Use as a quiz or review of Lesson -. Mathematical Background notes are available for this lesson on page 70D. can trigonometric functions be used to describe music? sk students: What do the values of n represent on the right side of the diagram? the number of antinodes What is occurring at an antinode? maimum vibration t what point do ou think a guitar string would be more likel to break, a node or an antinode? at an antinode Teach DUBLE-NGLE FRMULS Teaching Tip Point out that the double-angle formulas are derived ug the sum and difference of angles formulas presented in Lesson -. Lesson - Double-ngle and Half-ngle Formulas 79 Resource Manager Workbook and Reproducible Masters Chapter Resource Masters Stud Guide and Intervention, pp. 7 Skills Practice, p. 9 Practice, p. 70 Reading to Learn Mathematics, p. 7 Enrichment, p. 7 ssessment, p. 9 Transparencies -Minute Check Transparenc - nswer Ke Transparencies Technolog Interactive Chalkboard Lesson - Lesson Title 79

39 Teaching Tip Remind students of the identit cos, and its two variations that occur b subtracting either or cos from both sides. Eplain that there are three versions of the formula for cos because of the three variations of the identit cos. In-Class Eample Power Point Find the eact value of each epression if and is between 0 and 90. a. 7 b. cos Eample Double-ngle Formulas Find the eact value of each epression if and is between 90 and 0. a. Use the identit cos. First, find the value of cos. cos cos cos cos 9 cos Subtract. Find the square root of each side. Since is in the second quadrant, coe is negative. Thus, cos. Now find. cos Double-angle formula, cos The value of is. b. cos Use the identit cos. cos Double-angle formula 7 7 The value of cos is. HLF-NGLE FRMULS You can derive formulas for the e and coe of half a given angle ug the double-angle formulas. Find. Find cos. cos Double-angle formula cos Substitute for and for. cos Solve for. co s Take the square root of each side. cos cos Double-angle formula cos cos Substitute for and for. cos cos cos co s Solve for cos. Take the square root of each side. 79 Chapter Trigonometric Graphs and Identities 79 Chapter Trigonometric Graphs and Identities

40 Stud Tip Choog the Sign You ma want to determine the quadrant in which the terminal side of will lie in the first step of the solution. Then ou can use the correct sign from the beginning. These are called the half-angle formulas. The signs are determined b the function of. The following identities hold true for all values of. cos Eample Half-ngle Formulas cos c Find cos if and is in the third quadrant. Since cos co s, we must find cos first. cos cos cos cos 7 cos 7 Simplif. Take the square root of each side. Since is in the third quadrant, cos 7. cos co s Half-angle formula 7 cos Simplif the radicand. Rationalize. 7 Half-ngle Formulas os Multipl. Since is between 0 and 70, is between 90 and. Thus, cos is 7 negative and equals. HLF-NGLE FRMULS Intervention Stress the Stud Tip provided in the margin net to Eample. Determining the proper sign for the answer at the beginning of the computation will help some students avoid forgetting this step at the end of their computations. In-Class Eample New Power Point Find cos if and is in the second quadrant. Eample Evaluate Ug Half-ngle Formulas Find the eact value of each epression b ug the half-angle formulas. a cos 0 c os (continued on the net page) Lesson - Double-ngle and Half-ngle Formulas 79 Differentiated Instruction uditor/musical If possible, ask a music teacher at our school to talk to students about harmonics. Students plaing stringed instruments ma also be willing to share what the have learned about harmonics and waves. If a music teacher is not available, a phsics teacher ma also be able to demonstrate harmonics or bring a device that creates standing waves to class. Lesson - Double-ngle and Half-ngle Formulas 79

41 In-Class Eamples Practice/ppl Stud Notebook Power Point Find the eact value of each epression b ug the halfangle formulas. a. b. cos 9 Verif that (cos cos ) is an identit. (cos cos ) [cos (cos )] (cos cos ) ( ) Have students add the definitions/eamples of the vocabular terms to their Vocabular Builder worksheets for Chapter. record the double-angle and half-angle formulas. include an other item(s) that the find helpful in mastering the skills in this lesson. nswers. Sample answer: If is in the third quadrant, then is between 90 and. Use the half-angle formula for coe knowing that the value is negative.. Sample answer: ; cos () cos 90 or 0, cos or. Sample answer: The identit used for cos depends on whether ou know the value of, cos, or both values.. 7,,, Concept Check. See margin. Guided Practice GUIDED PRCTICE KEY Eercises Eamples cos 0 = b. cos cos cos cos cos = Recall that ou can use the sum and difference formulas to verif identities. Double- and half-angle formulas can also be used to verif identities.. Eplain how to find cos if is in the third quadrant.. Find a countereample to show that cos cos is not an identit.. PEN ENDED Describe the conditions under which ou would use each of the three identities for cos. Find the eact values of, cos,, and cos for each of the following.. cos ; cos ; ; ; 70 0, 9, 7. See margin. 0, Find the eact value of each epression b ug the half-angle formulas cos 9 79 Chapter Trigonometric Graphs and Identities., 9 9, 0,.,,, 7. 7,, Simplif the radicand. Simplif the denominator. = cos Simplif the radicand. Eample Verif Identities Simplif the denominator. Verif that ( cos ) is an identit. ( cos ) riginal equation cos cos 7, 7 Multipl. cos + cos = Double-angle formula 79 Chapter Trigonometric Graphs and Identities

42 Practice and ppl Homework Help For Eercises See Eamples,,, 9 0, 7 Etra Practice See page. pplication. See margin. Verif that each of the following is an identit. 0. See margin. 0. cot. cos cos cos. VITIN When a jet travels at speeds greater than the speed of sound, a sonic boom is created b the sound waves forming a cone behind the jet. If is the measure of the angle at the verte of the cone, then the Mach number M can be determined ug the formula. Find the Mach number M of a jet if a sonic boom is created b a cone with a verte angle of 7.. Find the eact values of, cos,, and cos for each of the following.. ; cos ; cos ; ; ; cos ; cos ; cos ; ; ; 0 70 bout the Eercises rganization b bjective Double ngle Formulas:, Half ngle Formulas: 0,, dd/even ssignments Eercises are structured so that students practice the same concepts whether the are assigned odd or even problems. ssignment Guide Basic: 7 odd,, 9, verage: 7 odd,, 9, dvanced: even, 9 9 (optional: 0 ) ll: Practice Quiz ( 0). cos ; ; 90 0 ptics rainbow appears when the sun shines through water droplets that act as a prism. nswers Find the eact value of each epression b ug the half-angle formulas.. cos. 7. cos cos Verif that each of the following is an identit.. See pp. N.. cot. cos cos. cos. ( cos ). tan cos. c os tan cos cos 7. PTICS If a glass prism has an ape angle of measure and an angle of deviation of measure, then the inde of refraction n of 0. cot cos ( cos ) s in cos co s s in cot ( ) the prism is given b n. What is the angle of deviation of a prism with an ape angle of 0 and an inde of refraction of?. Lesson - Double-ngle and Half-ngle Formulas 79. cos cos cos. 0, 9,, 9 9.,, 0, nswers., 7 9 9,,. 7,, 0, ,,,., 7, 0, 9., 7,, 0. 0, 9,, 9 9., 7 9 9,,., 7,,., 9 9,, 0., 7 0,, Lesson - Double-ngle and Half-ngle Formulas 79

43 Stud Guide and Intervention, p. 7 (shown) and p. NME DTE PERID - Stud Guide and Intervention Double-ngle and Half-ngle Formulas Double-ngle Formulas The following identities hold true for all values of. Double-ngle cos cos cos Formulas cos cos cos Eample Find the eact values of and cos if 9 and First, find the value of cos. cos cos cos cos cos 0 9 Since is in the third quadrant, cos is negative. Thus cos 0. To find, use the identit cos. cos The value of is 0. To find cos, use the identit cos. cos The value of cos is. 0 Eercises Find the eact values of and cos for each of the following..,0 90, 7., ,. cos, 0 70, 7. cos,90 0 7,., cos,90 0, 7, NME 7 DTE PERID Practice (verage) Skills Practice, p. 9 and Practice, p. 70 (shown) Double-ngle and Half-ngle Formulas Find the eact values of, cos,, and cos for each of the following.. cos, 0 90., , 9,, 0,, 7, cos, 70 0., 0 70, 7,, 0, 9 9,, Find the eact value of each epression b ug the half-angle formulas.. tan 0. tan 7. cos 7.. Verif that each of the following is an identit. 9. tan tan cos tan ; tan t an cos tan t an ; cos cos t an tan 0. cos cos cos cos () cos cos cos cos cos ( cos )(cos ) cos cos cos cos cos cos. ERIL PHTGRPHY In aerial photograph, there is a reduction in film eposure for an point X not directl below the camera. The reduction E is given b E E 0 cos, where is the angle between the perpendicular line from the camera to the ground and the line from the camera to point X, and E 0 is the eposure for the point directl below the. camera. Ug the identit cos, verif that E 0 cos E 0 cos E 0 cos E 0 (cos ) E 0 ( ) E 0 si n E 0 c os E0 cos. IMGING scanner takes thermal images from altitudes of 00 to,000 meters. The width W of the swath covered b the image is given b W H tan, where H is the height and is half the scanner s field of view. Verif that H H tan. cos H H cos H cos cos cos H Htan ( cos ) cos Reading to Learn NME 70 DTE PERID - Reading to Learn Mathematics Mathematics, p. 7 Double-ngle and Half-ngle Formulas Pre-ctivit How can trigonometric functions be used to describe music? Read the introduction to Lesson - at the top of page 79 in our tetbook. Suppose that the equation for the second harmonic is a. Then what would be the equations for the fundamental tone (first harmonic), third harmonic, fourth harmonic, and fifth harmonic? 0.a;.a; a;.a Reading the Lesson ELL. Match each epression from the list on the left with all epressions from the list on the right that are equal to it for all values of. a. v i. cos Lesson -. c os L cos L c os L cos L Standardized Test Practice GEGRPHY For Ecercises and 9, use the following information. Mercator projection map uses a flat projection of Earth in which the distance between the lines of latitude increases with their distance from the equator. The calculation of the location of a point on this projection uses the epression tan L, where L is the latitude of the point.. Write this epression in terms of a trigonometric function of L. 0 N 0 9. Find the eact value of the epression if L 0. PHYSICS For Eercises 0 and, use the following information. n object is propelled from ground level with an initial velocit of v at an angle of elevation. 0. See pp. N. 0. The horizontal distance d it will travel can be determined ug d v, g where g is acceleration due to gravit. Verif that this epression is the same as g v (tan tan ).. The maimum height h the object will reach can be determined ug the formula h v. Find the ratio of the maimum height attained g to the horizontal distance traveled. tan CRITICL THINKING For Eercises, use the following information. Consider the functions f(), g(), h() cos, and k() cos.. See pp. N.. Draw the graphs of g(), h(), and k() on the same coordinate plane on the interval from to. What do ou notice about the graphs?. Where do the maima and minima of g, h, and k occur?. Draw the graph of f() on a separate coordinate plane.. What is the behavior of the graph of f() at the locations found in Eercise?. Use what ou know about transformations to determine c and d so that g() h() c k() d. 7. WRITING IN MTH nswer the question that was posed at the beginning of the lesson. See margin. How can trigonometric functions be used to describe music? Include the following in our answer: a description of what happens to the graph of the function of a vibrating string as it moves from one harmonic to the net, and an eplanation of what happens to the period of the function as ou move from the nth harmonic to the (n )th harmonic.. Find the eact value of cos if and D 0 B C D Find the eact value of if cos and B B C D 0 N 0 N 0 N 0 N b. cos ii and iii ii. c. cos iv iii. cos 79 Chapter Trigonometric Graphs and Identities d. i iv. v. cos cos. Determine whether ou would use the positive or negative square root in the half-angle identities for and cos in each of the following situations. (Do not actuall calculate and cos.) a., if cos and is in Quadrant I positive b. cos, if cos 0.9 and is in Quadrant II positive c. cos, if 0.7 and is in Quadrant III negative d., if 0. and is in Quadrant IV positive Helping You Remember. Man students find it difficult to remember a large number of identities. How can ou obtain all three of the identities for cos b remembering onl one of them and ug a Pthagorean identit? Sample answer: Just remember the identit cos cos. Ug the Pthagorean identit cos, ou can substitute either for cos or cos for to get the other two identities for cos. 79 Chapter Trigonometric Graphs and Identities NME DTE PERID - Enrichment, p. 7 lternating Current The figure at the right represents an alternating current generator. rectangular coil of wire is suspended between the poles of a magnet. s the coil of wire is rotated, it passes through the magnetic field and generates current. s point X on the coil passes through the points and C, its motion is along the direction of the magnetic field between the poles. Therefore, no current is generated. However, through points Band D, the motion of X is perpendicular to the magnetic field. This induces maimum current in the coil. Between and B, B and C, C and D, and D and, the current in the coil will have an intermediate value. Thus, the graph of the current of an alternating current generator is closel related to the e curve. The actual current, i, in a household current is given b i I M 0t ) where I M is the maimum t is the el The maimum current ma have a positive or negative value. i(amperes) B X B C C D t(seconds)

44 Maintain Your Skills.. P Mied Review Getting Read for the Net Lesson ractice Quiz Find the eact value of each epression. (Lesson -) 0. cos.. ( ). cos cos (00 ) Verif that each of the following is an identit. (Lesson -). cot cos csc See pp. N. csc 7. cos (cos cot ) cot cos ( ) See pp. N. ERTHQUKE For Eercises and 9, use the following information. The magnitude of an earthquake M measured on the Richter scale is given b M log 0, where represents the amplitude of the seismic wave caug ground motion. (Lesson 0-). 0 or 0. How man times as great was the 90 Chile earthquake as the 9 Indonesia earthquake? 9. The largest aftershock of the 9 laskan earthquake was.7 on the Richter scale. How man times as great was the main earthquake as this aftershock? 0. or about times as great US TDY Snapshots Strongest quakes in 0th centur Chile, 90 laska, 9 Russia, 9 Ecuador, 90 laska, 97 Kuril Islands, 9 laska, 9 India, 90 Chile, 9 Indonesia, 9 Source: U.S. Geological Surve Magnitude B William Risser and Marc E. Mullins, US TDY PREREQUISITE SKILL Solve each equation. (To review solving equations ug the Zero Product Propert, see Lesson -.) 0. ( )( ) 0,. ( )( ) 0,. ( ) 0 0,. ( )( ) 0,. ( )( ) 0,. ( ) 0 0, Lessons - through - Verif that each of the following is an identit. (Lessons -). See pp. N.. sec tan. sec cos tan. tan ( cos ) cos Verif that each of the following is an identit. (Lessons - and -). See pp. N.. (90 ) cos. cos. ( 0 ) cos ( 0 ) cos Find the eact value of each epression b ug the double-angle or half-angle formulas. (Lesson -) 7. if cos ; cos if 9 ; cos nline Lesson Plans Lesson - Double-ngle and Half-ngle Formulas 797 US TDY Education s nline site offers resources and interactive features connected to each da s newspaper. Eperience TDY, US TDY s dail lesson plan, is available on the site and delivered dail to subscribers. This plan provides instruction for integrating US TDY graphics and ke editorial features into our mathematics classroom. Log on to ssess pen-ended ssessment Writing Have students write their own problems like Eamples and, and have them write an eplanation of how to use a double-angle or half-angle formula to solve their eamples. Getting Read for Lesson -7 PREREQUISITE SKILL In Lesson -7, students will solve trigonometric equations ug the Zero Product Propert. Use Eercises 0 to determine our students familiarit with the Zero Product Propert. ssessment ptions Practice Quiz The quiz provides students with a brief review of the concepts and skills in Lessons - through -. Lesson numbers are given to the right of the eercises or instruction lines so students can review concepts not et mastered. Quiz (Lessons - and -) is available on p. 9 of the Chapter Resource Masters. nswer (p. 79) 7. Sample answer: The sound waves associated with music can be modeled ug trigonometric functions. nswers should include the following information. In moving from one harmonic to the net, the number of vibrations that appear as e waves increases b. The period of the function as ou move from the nth harmonic to the (n )th harmonic decreases from to n. n Lesson - Double-ngle and Half-ngle Formulas 797

45 Graphing Calculator Investigation Preview of Lesson -7 Getting Started pproimate Solutions In Eample, approimate solutions can also be found b ug the Trace feature. In most situations however, the Intersect feature will give more accurate solutions. Teach You might wish to demonstrate the technique shown in Eample b first ug a simple quadratic equation like 9, whose solutions students will readil know. Graph and 9 to see that the intersect at two points, where and, just as students will epect. Remind students that the solutions of the equation are the values of the points of intersection, not the values. If the epression on the right side of an equation is just 0 (as in Eercises and ), ou can graph the function for the left side of the equation and then just use the Zero feature on the CLC menu to find approimate solutions. ssess In Eercises, check to see that students can eplain wh there are no real solutions. In Eercise, make sure students found all four possible values. Students who found fewer than four are not ug the correct domain for. Solving Trigonometric Equations The graph of a trigonometric function is made up of points that represent all values that satisf the function. To solve a trigonometric equation, ou need to find all values of the variable that satisf the equation. You can use a TI- Plus to solve trigonometric equations b graphing each side of the equation as a function and then locating the points of intersection. Eample Use a graphing calculator to solve 0. if 0 0. Rewrite the equation as two functions, and 0.. Then graph the two functions. Look for the point of intersection. Make sure that our calculator is in degree mode to get the correct viewing window. KEYSTRKES: MDE ENTER WINDW 0 ENTER 0 ENTER 90 ENTER ENTER ENTER ENTER SIN X,T,,n ENTER 0. ENTER Based on the graph, ou can see that there are two points of intersection in the interval 0 0. Use or nd [CLC] to approimate the solutions. The approimate solutions are. and.. Eample GRPH Use a graphing calculator to solve tan cos cos 0 if 0 0. Because the tangent function is not continuous, place the calculator in Dot mode. The related functions to be graphed are tan cos cos and 0. These two functions do not intersect. Therefore, the equation tan cos cos 0 has no real solutions. 79 Investigating Slope-Intercept Form 79 Chapter Trigonometric Graphs and Identities Preview of Lesson -7. See pp. N for graphs. Eercises..,.9. no real solution. no real solution Use a graphing calculator to solve each equation for the values of indicated.. 0. if 0 0. tan if 0 0 0, 0. cos 0 if cos. if if if 0 0 0, 0, 00 0, 0, 0, 0 [0, 0] scl: 90 b [, ] scl: Like other equations ou have studied, some trigonometric equations have no real solutions. Carefull eamine the graphs over their respective periods for points of intersection. If there are no points of intersection, then the trigonometric equation has no real solutions. [0, 0] scl: 90 b [, ] scl: 79 Chapter Trigonometric Graphs and Identities

46 Solving Trigonometric Equations Lesson Notes Vocabular trigonometric equation Solve trigonometric equations. Use trigonometric equations to solve real-world problems. SLVE TRIGNMETRIC EQUTINS You have seen that trigonometric identities are true for all values of the variable for which the equation is defined. However, most trigonometric equations, like some algebraic equations, are true for some but not all values of the variable. Eample can trigonometric equations be used to predict temperature? The average dail high temperature for a region can be described b a trigonometric function. For eample, the average dail high temperature for each month in rlando, Florida, can be modeled b the function T. (0..) 0.9, where T represents the average dail high temperature in degrees Fahrenheit and represents the month of the ear. This equation can be used to predict the months in which the average temperature in rlando will be at or above a desired temperature. Solve Equations for a Given Interval Find all solutions of each equation for the given interval. a. cos ; 0 0 cos riginal equation cos 0 Solve for 0. (cos )(cos ) 0 Factor. Now use the Zero Product Propert. cos 0 or cos 0 cos cos 0 0 The solutions are 0 and 0. b. cos ; 0 cos riginal equation cos cos = cos cos cos 0 Solve for 0. cos ( ) 0 Factor. (continued on the net page) Lesson -7 Solving Trigonometric Equations 799 Focus -Minute Check Transparenc -7 Use as a quiz or review of Lesson -. Mathematical Background notes are available for this lesson on page 70D. can trigonometric equations be used to predict temperature? sk students: Wh would temperature not be modeled b a quadratic function? Sample answer: Temperature varies periodicall and does not continue upward or downward to infinit. So temperature should be modeled b a trigonometric, not a quadratic, function. What can ou tell about the range of temperatures b studing the function T? The temperature varies. degrees (the amplitude of the function) above and below a temperature of 0.9 F (the midline of the function). What can ou tell about the maimum temperature b studing the function? The maimum temperature is. 0.9, or 9. F. Resource Manager Workbook and Reproducible Masters Chapter Resource Masters Stud Guide and Intervention, pp. 7 7 Skills Practice, p. 7 Practice, p. 7 Reading to Learn Mathematics, p. 77 Enrichment, p. 7 ssessment, p. 9 School-to-Career Masters, p. Science and Mathematics Lab Manual, pp. Teaching lgebra With Manipulatives Masters, p. 0 Transparencies -Minute Check Transparenc -7 nswer Ke Transparencies Technolog Interactive Chalkboard Lesson - Lesson Title 799

47 Teach Building on Prior Knowledge In Lesson -, students learned to use the Zero Product Propert to solve equations. In this lesson, students will use the Zero Product Propert to solve trigonometric equations. SLVE TRIGNMETRIC EQUTINS In-Class Eamples Power Point Find all solutions of each equation for the given interval. a. cos ; 0 0 0, 0, 70 b. ; 0,,, Teaching Tip In Eample, show students how to look for patterns in the solution of part a. Students should look for pairs of solutions that differ b eactl or. a. Solve cos cos for all values of if is measured in radians. k, k, k, k, where k is an integer b. Solve cos cos for all values of if is measured in degrees. 0 k 0, 0 k 0, 00 k 0, where k is an integer Stud Tip Epresg Solutions as Multiples The epression 90 k 0 includes 70 and its multiples, so it is not necessar to list them separatel. Use the Zero Product Propert. cos 0 or 0 cos 0 Trigonometric equations are usuall solved for values of the variable between 0 and 0 or 0 radians and radians. There are solutions outside that interval. These other solutions differ b integral multiples of the period of the function. Eample or The solutions are and. Solve Trigonometric Equations a. Solve for all values of if is measured in radians. riginal equation Divide each side b. Look at the graph of to find solutions of. The solutions are 7,, 9,, and so on, and 7,, 9,, and so on. The onl solutions in the interval 0 to are 7 and. The period of the e function is radians. So the solutions can be written as 7 k and k, where k is an integer. b. Solve cos cos 0 for all values of if is measured in degrees. cos cos 0 riginal equation cos cos 0 cos cos cos cos 0 Simplif. cos ( cos ) 0 Factor. Solve for in the interval 0 to 0. cos 0 or cos 0 90 or 70 cos cos 0 or 0 The solutions are 90 k 0, 0 k 0, and 0 k 0. If an equation cannot be solved easil b factoring, tr rewriting the epression ug trigonometric identities. However, ug identities and some algebraic operations, such as squaring, ma result in etraneous solutions. So, it is necessar to check our solutions ug the original equation. 00 Chapter Trigonometric Graphs and Identities 00 Chapter Trigonometric Graphs and Identities

48 Eample Solve Trigonometric Equations Ug Identities Solve cos tan 0. cos tan 0 riginal equation cos cos 0 0 ( ) 0 tan = c os Multipl. Factor. 0 or 0 0, 0, or 0 90 In-Class Eamples Power Point Solve cot cos. 90 k 0, where k is an integer Solve cos. 0 k 0, 00 k 0, where k is an integer CHECK cos tan 0 cos tan 0 cos 0 tan = 0 cos 0 tan = cos tan 0 cos tan 0 cos 0 tan = 0 cos 90 tan = tan 90 is undefined. Thus, 90 is not a solution. The solution is 0 k 0. Some trigonometric equations have no solution. For eample, the equation cos has no solution ce all values of cos are between and, inclusive. Thus, the solution set for cos is empt. Eample Determine Whether a Solution Eists Solve cos cos. cos cos riginal equation ( cos ) cos cos cos cos cos Multipl. cos cos 0 Subtract from each side. ( cos )( cos ) 0 Factor. cos 0 or cos 0 cos cos cos cos Not possible ce 0 or 0 cos cannot be greater than. Thus, the solutions are 0 k 0 and 0 k 0. Lesson -7 Solving Trigonometric Equations 0 Differentiated Instruction Interpersonal s students work through this lesson, have them create a class list on the chalkboard that identifies common errors the made. Encourage students to add suggestions for how to avoid their errors. For eample, one common error is having one s calculator set to degrees when it needs to be set to radians for a problem, and vice versa. Lesson -7 Solving Trigonometric Equations 0

49 USE TRIGNMETRIC EQUTINS In-Class Eample Practice/ppl Stud Notebook Power Point Teaching Tip s students work Eample, some of them ma calculate 0.9 as.079. Point out that this result occurs when their calculator is set to degrees instead of radians. GRDENING Refer to Eample in the Student Edition. If Rhonda decides to wait until there are hours of dalight, on what da can she plant her flowers? around the th da of the ear, or around pril Have students complete the definitions/eamples for the remaining terms on their Vocabular Builder worksheets for Chapter. summarize some techniques the used for solving trigonometric equations. include an other item(s) that the find helpful in mastering the skills in this lesson. bout the Eercises rganization b bjective Solve Trigonometric Equations: 0 Use Trigonometric Equations: dd/even ssignments Eercises 0 are structured so that students practice the same concepts whether the are assigned odd or even problems. ssignment Guide Basic: 9 odd, verage: 9 odd, dvanced: 0 even, Concept Check. See margin. Guided Practice GUIDED PRCTICE KEY Eercises Eamples 7,. 7 k, k or 0 k 0, 0 k 0 pplication. k, k, k or 0 k 0, 0 k 0, 90 k 0 0 Chapter Trigonometric Graphs and Identities nswers USE TRIGNMETRIC EQUTINS used to solve real-world situations. Eample Use a Trigonometric Equation Trigonometric equations are often GRDENING Rhonda wants to wait to plant her flowers until there are at least hours of dalight. The number of hours of dalight H in her town can be represented b H.. (0.0d.), where d is the da of the ear and angle measures are in radians. n what da is it safe for Rhonda to plant her flowers? H.. (0.0d.) riginal equation.. (0.0d.) H =.. (0.0d.) Subtract. from each side. 0.9 (0.0d.) Divide each side b d. Sin d dd. to each side. 0. d Divide each side b 0.0. Rhonda can safel plant her flowers around the 0th da of the ear, or around pril.. Tell wh the equation sec 0 has no solutions.. Eplain wh the number of solutions to the equation is infinite.. PEN ENDED Write an eample of a trigonometric equation that has no solution. Find all solutions of each equation for the given interval..,,,. cos ; ; , 0, 0, 00,. cos ; 0 7. cos 0; 0 Solve each equation for all values of if is measured in radians.. cos cos 0 k 9. cos 0 0 k Solve each equation for all values of if is measured in degrees. 0. cos. cos cos 90 k 0, 0 k 0 0 k 0, 00 k 0 Solve each equation for all values of.. 0. cos 0. PHYSICS ccording to Snell s law, the angle at which light enters water is related to the angle at which light travels in water b the equation.. t what angle does a beam of light enter the water if the beam travels at an angle of through the water?.. Sample answer: If sec 0 then 0. Since co s no value of makes 0, there are no solutions. co s. Sample answer: The function is periodic with two solutions in each of its infinite number of periods.. Sample answer:. k, k. k, k, k k, k. 0 k 7. k, k. 0 k, k, k 0 Chapter Trigonometric Graphs and Identities

50 Practice and ppl Homework Help For Eercises See Eamples 0, Etra Practice See page.. 0, 00. 0, , 0. 0, 0, 0, 0 9.,, 0.. 7,.,,, Waves In the oceans, the height and period of water waves are determined b wind velocit, the duration of the wind, and the distance the wind has blown across the water. Source: Find all solutions of each equation for the given interval.. cos 0; 0 0. ; ; 0 0. cos ; cos ; 0 0. cos ; 0. 0;. cos cos ; 0 Solve each equation for all values of if is measured in radians.. cos cos 0. cos 0. cos cos 0. cos cos 7. cos cos 0. cos. See margin. Solve each equation for all values of if is measured in degrees. 9. cos 0. tan tan tan 0 9. See margin.. cos 7 cos 0 Solve each equation for all values of. 0. See pp. N.. cos cos cos. cos cos 0 9. cos 0. cos. S or S cot t an LIGHT For Eercises and, use the information shown.. The length of the shadow S of the International Peace Memorial at Put-In-Ba, hio, depends upon the angle of inclination of the Sun,. Epress S as a function of. ft. Find the angle of inclination that will produce a shadow 0 feet long. about WVES For Eercises and, use the following information. For a short time after a wave is created b a boat, the height of the wave can be modeled ug h h t, where h is the maimum height of the wave in P feet, P is the period in seconds, and t is the propagation of the wave in seconds.. If h and P seconds, write the equation for the wave and draw its graph over a 0-second interval. See pp. N.. How man times over the first 0 seconds does the graph predict the wave to be one foot high? 0 Lesson -7 Solving Trigonometric Equations 0 Stud Guide and Intervention, p. 7 (shown) and p. 7 NME DTE PERID -7 Stud Guide and Intervention Solving Trigonometric Equations Solve Trigonometric Equations You can use trigonometric identities to solve trigonometric equations, which are true for onl certain values of the variable. Eample Find all solutions of Eample Solve cos 0 0 for the interval for all values of. Give our answer in 0 0. both radians and degrees. 0 cos 0 cos cos 0 cos ( ) 0 cos 0 or 0 0, 0, 0, 0 90 k 0; 0 k 0, k 0 k 0; 7 k, k Eercises Find all solutions of each equation for the given interval.. cos cos, 0. cos 0, 0,, 0,,,. cos,0 0. 0, 0,, 9,, Solve each equation for all values of if is measured in radians.. 0. cos cos 0 k, k k, k, 7 k, k Solve each equation for all values of if is measured in degrees. 7. cos. tan k k 0, 7. k 0-7 NME 7 DTE PERID Practice (verage) Skills Practice, p. 7 and Practice, p. 7 (shown) Find all solutions of each equation for the given interval.. cos, cos,0 0 90, 0, 90,, 70. cos cos, 0 0. cos cos (90 ) 0, 0 0, 0, 00 7,. cos,,. tan sec, Solve each equation for all values of if is measured in radians. 7. cos. cot cot k k and k cos k, k, and k k. cos. sec k Solving Trigonometric Equations k Solve each equation for all values of if is measured in degrees.. cos cos. csc csc 0 90 k 0 0 k 0, 90k 0, and 0 k 0. ( cos ) cos 0 k 0 and 0 k 0. cos cos 90 k 0 and 0 k 0 Solve each equation for all values of. 7. k and k,. 0 k and k, or 0 k 0 and 0 k 0 or 0 k 0 and 0 k 0 9. k, or 0 k 0 0. cos 0 k and k, or k 0 and 0 k 0. WVES Waves are caug a buo to float in a regular pattern in the water. The vertical position of the buo can be described b the equation h. Write an epression that describes the position of the buo when its height is at its midline. k or k 0. ELECTRICITY The electric current in a certain circuit with an alternating current can be described b the formula i 0t, where i is the current in amperes and t is the time in seconds. Write an epression that describes the times at which there is no current. 0.7kt Reading to Learn Mathematics, p NME 7 DTE PERID Reading to Learn Mathematics Solving Trigonometric Equations Pre-ctivit How can trigonometric equations be used to predict temperature? Read the introduction to Lesson -7 at the top of page 799 in our tetbook. Describe how ou could use a graphing calculator to determine the months in which the average dail high temperature is above 0F. (ssume that represents Januar.) Specif the graphing window that ou would use. Sample answer: Graph the functions. (0..) 0.9 (ug radian mode) and 0 on the same screen. Use the window [, ] b [0, 00] with Xscl and Yscl. Note the values for which the curve is above the horizontal line. Reading the Lesson. Identif which equations have no solution. C, E, and G. B. tan 0.00 C. sec ELL D. csc E. cos.0 F. cot 000 G. cos H. sec. 0 I Lesson -7 Lesson -7 nswers 9. k0 0. 0k0. 70k0. 0k0, 0k0. 0k0, 0k0. 0k0, 0k0-7 Enrichment, p. 7 Families of Curves Use these graphs for the problems below NME DTE PERID The Famil n n = 0 n = n = n = n = n = n = m = m = The Famil e m m = m = m = m = m = m = m = m = m = 0. Use a trigonometric identit to write the first step in the solution of each trigonometric equation. (Do not complete the solution.) a. tan cos,0 tan b. 0, 0 0 ( ) 0 c. cos, 0 0 d. cos, 0 cos cos e. cos cos, 0 0 ( cos ) cos f. tan tan 0 ( tan )(tan ) 0 Helping You Remember. good wa to remember something is to eplain it to someone else. How would ou eplain to a friend the difference between verifing a trigonometric identit and solving a trigonometric equation. Sample answer: Verifing a trigonometric identit means showing that the two sides are equal for all values of the variable for which the functions involved are defined. This is done b transforming one or both sides until the same epression is obtained on both sides. Solving a trigonometric equation means finding the values of the variable for which both sides are equal. This process ma require simplifing trigonometric epressions, but it also requires finding the angles for which a trigonometric function has a particular value. l ft Lesson -7 Solving Trigonometric Equations 0

51 ssess pen-ended ssessment Speaking Have students show ou a trigonometric equation the solved and eplain step b step how the performed each step of their computation. ssessment ptions Quiz (Lesson -7) is available on p. 9 of the Chapter Resource Masters. nswer. Sample answer: Temperatures are cclic and can be modeled b trigonometric functions. nswers should include the following information. temperature could occur twice in a given period such as when the temperature rises in the spring and falls in autumn. Standardized Test Practice Maintain Your Skills Mied Review 9., 7, 0, ,,,. 7,,,., 7,,. CRITICL THINKING Computer games often use transformations to distort images on the screen. In one such transformation, an image is rotated counterclockwise ug the equations ' cos and ' cos. If the coordinates of an image point are (, ) after a 0 rotation, what are the coordinates of the preimage point? (.9, 0.9). WRITING IN MTH nswer the question that was posed at the beginning of the lesson. See margin. How can trigonometric equations be used to predict temperature? Include the following in our answer: an eplanation of wh the e function can be used to model the average dail temperature, and an eplanation of wh, during one period, ou might find a specific average temperature twice. 7. Which of the following is not a possible solution of 0 cos tan? D 7 B C D. The graph of the equation cos is shown. Which is a solution for cos? B B C 0 D Find the eact value of, cos,, and cos for each of the following. (Lesson -) 9. ; cos ; cos ; ; 0 90 Find the eact value of each epression. (Lesson -). 0. cos. Solve BC. Round measures of sides and angles to the nearest tenth. (Lesson -) b.0, c., mc 7 0 C b c B Trig Class ngles for Lessons in Lit It is time to complete our project. Use the information and data ou have gathered about the applications of trigonometr to prepare a poster, report, or Web page. Be sure to include graphs, tables, or diagrams in the presentation. 0 Chapter Trigonometric Graphs and Identities 0 Chapter Trigonometric Graphs and Identities

52 Stud Guide and Review Vocabular and Concept Check amplitude (p. 7) double-angle formula (p. 79) half-angle formula (p. 79) midline (p. 77) phase shift (p. 79) trigonometric equation (p. 799) Choose the correct letter that best matches each phrase.. horizontal translation of a trigonometric function h. a reference line about which a graph oscillates b. vertical translation of a trigonometric function d. the formula used to find cos f. cos e. a measure of how long it takes for a graph to repeat itself c 7. cos ( ) cos cos g. the absolute value of half the difference between the maimum and minimum values of a periodic function a - See pages 7 7. Eample Graphing Trigonometric Functions trigonometric identit (p. 777) vertical shift (p. 77) a. amplitude b. midline c. period d. vertical shift e. double-angle formula f. half-angle formula g. difference of angles formula h. phase shift Concept Summar For trigonometric functions of the form a b and a cos b, the amplitude is a, and the period is 0 or. b b The period of a tan b is 0 or b b. Find the amplitude and period of = cos. Then graph the function. The amplitude is or. The period is 0 or 90. Use the amplitude and period to graph the function cos Vocabular and Concept Check This alphabetical list of vocabular terms in Chapter includes a page reference where each term was introduced. ssessment vocabular test/review for Chapter is available on p. 9 of the Chapter Resource Masters. Lesson-b-Lesson Review For each lesson, the main ideas are summarized, additional eamples review concepts, and practice eercises are provided. Vocabular PuzzleMaker ELL The Vocabular PuzzleMaker software improves students mathematics vocabular ug four puzzle formats crossword, scramble, word search ug a word list, and word search ug clues. Students can work on a computer screen or from a printed handout. Eercises Find the amplitude, if it eists, and period of each function. Then graph each function. See Eample on page See pp. N. 9. cos 0... sec. csc. tan For more information about Foldables, see Teaching Mathematics with Foldables. TM Chapter Stud Guide and Review 0 Have students look through the chapter to make sure the have included notes and eamples of graphs for each lesson in this chapter in their Foldable. Encourage students to refer to their Foldables while completing the Stud Guide and Review and to use them in preparing for the Chapter Test. MindJogger Videoquizzes ELL MindJogger Videoquizzes provide an alternative review of concepts presented in this chapter. Students work in teams in a game show format to gain points for correct answers. The questions are presented in three rounds. Round Concepts ( questions) Round Skills ( questions) Round Problem Solving ( questions) Chapter Stud Guide and Review 0

53 Stud Guide and Review Chapter Stud Guide and Review nswers.,, 0, , does not eist, 70, , does not eist,,.,,, tan [ ( 90 )] sec [ ( )] [( 0 )] cos [ ( )] - See pages Eample - See pages Eample 0 Chapter Trigonometric Graphs and Identities Translations of Trigonometric Graphs Concept Summar For trigonometric functions of the form a b( h), a cos ( h), and a tan ( h), the phase shift is to the right when h 0 and to the left when h 0. For trigonometric functions of the form a b( h) k, a cos ( h) k, and a tan ( h) k, the vertical shift is up when k 0 and down when k 0. State the vertical shift, amplitude, period, and phase shift of. Then graph the function. Identif the values of k, a, b, and h. k, so the vertical shift is. a, so the amplitude is. b, so the period is or. h, so the phase shift is to the right. Eercises State the vertical shift, amplitude, period, and phase shift of each function. Then graph the function. See Eample on page 77.. See margin.. [( 0 )]. tan ( 90 ) 7. sec. cos Trigonometric Identities Concept Summar Quotient Identities: Reciprocal Identities: csc Pthagorean Identities: Simplif cot cos. cot cos c os co s cos tan, cot c os c os, sec, cot co s tan cos, tan sec, cot sec cot = c os Multipl. Eercises Find the value of each epression. See Eample on page cot, if csc ; sec, if ; 0 90 Simplif each epression. See Eample on page cot. sec. csc cos. cos sec csc. cos tan [( )] 0 Chapter Trigonometric Graphs and Identities

54 Chapter Stud Guide and Review Stud Guide and Review - See pages 7 7. Eample - See pages Eample Verifing Trigonometric Identities Concept Summar Use the basic trigonometric identities to transform one or both sides of a trigonometric equation into the same form. Verif that tan cot sec csc. tan cot sec csc riginal equation c os sec csc tan c os c os cos sec csc cos, cot c os Rewrite ug the LCD, cos. sec csc cos cos sec csc co s Rewrite as the product of two epressions. sec csc sec csc sec, csc co s Eercises Verif that each of the following is an identit. See Eamples on pages See margin.. c os cos. csc cot t an cot cos. cot sec cot 7. sec ( sec cos ) tan Sum and Difference of ngles Formulas Concept Summar For all values of and : cos () cos cos ( ± ) cos cos Find the eact value of 9. 9 (0 ) cos cos 0 0, Evaluate each epression. Simplif. nswers. csc cot cos c os cos cos cos co cos s cos cos cos cos ( cos ) cos ( cos ) cos cos. cot sec cot c os cos cot cot csc cot cot cot 7. sec (sec cos ) tan co s co s cos tan cos tan sec tan tan tan. 7. See pp. N. Eercises Find the eact value of each epression. See Eample on page 77.. cos 9. cos cos (0 ). (0 ) Verif that each of the following is an identit. See Eample on page 7.. cos (90 ). (0 ) cos (0 ). ( ) 7. cos cos ( ) Chapter Stud Guide and Review 07 nswer. c os cos t an cot cos cos c os s c os in c os cos cos c os cos cos Chapter Stud Guide and Review 07

55 Stud Guide and Review Etra Practice, see pages 9. Mied Problem Solving, see page 7. nswers., 7,, 9. 0, 9, 9 9, ,,, , 9, 9 9, nswers (p. 09).,, 0, no phase shift ,, 70, cos [ ( 0 )] - See pages Eample -7 See pages Eample Double-ngle and Half-ngle Formulas Concept Summar Double-angle formulas: Half-angle formulas: sec Verif that csc is an identit. sec csc riginal equation co s csc, sec co s Simplif the comple fraction. cos cos, cos cos, cos, cos cos co s co s cos, cos Eercises Find the eact values of, cos,, and cos for each of the following. See Eamples and on pages 79 and 79.. See margin.. ; ; cos 7 ; cos ; 70 0 Solving Trigonometric Equations Concept Summar Solve trigonometric equations b factoring or b ug trigonometric identities. Solve 0 if riginal equation cos 0 = cos ( cos ) 0 Factor. 0 or cos 0 0 or 0 0 or 0 Eercises Find all solutions of each equation for the interval 0 0. See Eample on page 799.., 7, 9,. cos cos 0 Solve each equation for all values of if is measured in radians. See Eample on page k, k. k, k. 0. cos 0 Chapter Trigonometic Graphs and Identities 0 Chapter Trigonometric Graphs and Identities

56 Practice Test Vocabular and Concepts Choose the correct term to complete each sentence.. The ( period, phase shift) of ( 0 ) is 0.. midline is used with a (phase shift, vertical shift ) of a trigonometric function.. The amplitude of cos [( )] is,.. The (coe, cosecant ) has no amplitude. Skills and pplications State the vertical shift, amplitude, period, and phase shift of each function. Then graph the function.. See margin... cos ( 0 ) Find the value of each epression. 7. tan, if ; sec, if cot ; 0 70 Verif that each of the following is an identit. 9. See pp. N. 9. ( cos ) cos 0. sec. s ec cot. tan c os cos sec Find the eact value of each epression..... cos.. ( ). cos 0 7. cos Solve each equation for all values of if is measured in degrees. 0 k 0, 0 k 0, 9. sec tan 0 k 0 0. cos cos 0 k 0. cos 0 k 0, 0 k 0,. tan 0 k 0 0 k 0 GLF For Eercises and, use the following information. golf ball is hit with an initial velocit of 00 feet per second. The distance the ball travels is found b the formula d v 0, where v g 0 is the initial velocit, g is the acceleration due to gravit, feet per second squared, and is the measurement of the angle that the path of the ball makes with the ground.. Find the distance that the ball travels if the angle between the path of the ball and the ground measures ft. If a ball travels. feet, what was the angle the path of the ball made with the ground to the nearest degree?. STNDRDIZED TEST PRCTICE Identif the equation of the graphed function. B C cos B cos cos D cos Portfolio Suggestion Chapter Practice Test 09 Introduction In mathematics, trigonometric functions can be used to model real-world problems. sk Students Find a real-world problem modeled in this chapter that interests ou and show how ou solved it. Eplain how the function models the real-world problem and what could be gained b understanding the real-world problem better. Place our work in our portfolio. ssessment ptions Vocabular Test vocabular test/review for Chapter can be found on p. 9 of the Chapter Resource Masters. Chapter Tests There are si Chapter Tests and an pen-ended ssessment task available in the Chapter Resource Masters. Chapter Tests Form Tpe Level Pages MC basic 79 0 MC average B MC average C FR average D FR average 7 FR advanced 9 90 MC = multiple-choice questions FR = free-response questions pen-ended ssessment Performance tasks for Chapter can be found on p. 9 of the Chapter Resource Masters. sample scoring rubric for these tasks appears on p.. Unit Test unit test/review can be found on pp of the Chapter Resource Masters. End-of-Year Tests Second Semester Test for Chapters and a Final Test for Chapters can be found on pp of the Chapter Resource Masters. TestCheck and Worksheet Builder This networkable software has three modules for assessment. Worksheet Builder to make worksheets and tests. Student Module to take tests on-screen. Management Sstem to keep student records. Chapter Practice Test 09

57 Standardized Test Practice These two pages contain practice questions in the various formats that can be found on the most frequentl given standardized tests. practice answer sheet for these two pages can be found on p. of the Chapter Resource Masters. Standardized Test Practice NME DTE PERID Standardized Test Practice Student Recording Sheet (Use with pages Sheet, 0 ofp. the Student Edition.) Part Multiple Choice Select the best answer from the choices given and fill in the corresponding oval. B C D B C D 7 B C D 9 B C D B C D B C D B C D 0 B C D Part Record our answers on the answer sheet provided b our teacher or on a sheet of paper.. Which of the following is not equal to. 0? D C Multiple Choice B D (0.)(0.007) 00. The sum of five consecutive odd integers is. What is the sum of the greatest and least of these integers? B. In right triangle QRS, what is the value of tan R? D C 7 7 B D 7 7. What is the value of cos? B R 7 Q B S B C D B C Part Short Response/Grid In D B C D 0 C D Solve the problem and write our answer in the blank. For Questions 9, also enter our answer b writing each number or smbol in a bo. Then fill in the corresponding oval for that number or smbol. / / / / / / / / / / Part Quantitative Comparison / / / / Select the best answer from the choices given and fill in the corresponding oval. 0 B C D B C D B C D B C D B C D dditional Practice See pp in the Chapter Resource Masters for additional standardized test practice. nswers. If bananas cost a cents and oranges cost b cents, what is the cost of bananas and oranges in terms of a and b? D C ab a b a b a b D. bag contains peppermint candies, 0 butterscotch candies, and cherr candies. Emma chooses one piece at random, puts it in her pocket, and then repeats the process. If she has chosen peppermint candies, butterscotch candies, and cherr cand, what is the probabilit that the net piece of cand she chooses will be cherr? C 7 B C D. What is the value of? B cos B. What is the least positive value for where reaches its minimum? C C D 9. Which of the following is equivalent to os sec c? C cos B tan D 0. If cos and is in Quadrant II, what is the value of? D B B B C D C D 0 Chapter Trigonometric Graphs and Identities Log n for Test Practice The Princeton Review offers additional test-taking tips and practice problems at their web site. Visit or TestCheck and Worksheet Builder Special banks of standardized test questions similar to those on the ST, CT, TIMSS, NEP, and lgebra End-of-Course tests can be found on this CD-RM. 0 Chapter Trigonometric Graphs and Identities

58 ligned and verified b Part Short Response/Grid In Record our answers on the answer sheet provided b our teacher or on a sheet of paper.. If k is a positive integer, and 7k equals a prime number that is less than 0, then what is one possible value of 7k? 7 or. It costs $ to make a book. The selling price will include an additional 00%. What will be the selling price? $. The mean of seven numbers is 0. The sum of three of the numbers is 9. What is the sum of the remaining four numbers? 9. If a b 0 and c 9b, what is the ratio of a to c? /. What is the value of if?. The ages of children at a part are, 7,,, 7, 7,,, 7,, 9, 7, and 7. Let N represent the median of their ages and m represent the mode. What is N m? 0 7. In the figure below, CEFG is a square, BD is a right triangle, D is the midpoint of side CE, H is the midpoint of side CG, and C is the midpoint of side BD. BCDE is a line segment, and HD is a line segment. If the measure of the area of square CEFG is, what is the measure of the area of quadrilateral BCH?. line with a slope of passes through points (, n) and (0, n). What is the value of n? / 9. If 0, what is the value of 0 cos 0? Part Compare the quantit in Column and the quantit in Column B. Then determine whether: B C D 0. the quantit in Column is greater, the quantit in Column B is greater, the two quantities are equal, or the relationship cannot be determined from the information given.. c c c B Quantitive Comparison Column the length of a diagonal of a square whose area is 00. w, w Column B the length of a diagonal of a rectangle B C D E C w H. (a b) a b G F C (a b) a b Test-Taking Tip lwas write down our calculations on scrap paper or in the test booklet, even if ou think ou can do the calculations in our head. Writing down our calculations will help ou avoid making simple mistakes.. B tan Chapters Standardized Test Practice Chapter Standardized Test Practice