Chapter 14 Resource Masters

Transcription

1 hapter Resource Masters

2 onsumable Workbooks Man of the worksheets contained in the hapter Resource Masters booklets are available as consumable workbooks. Stud Guide and Intervention Workbook X Skills Practice Workbook Practice Workbook NSWERS FR WRKKS The answers for hapter of these workbooks can be found in the back of this hapter Resource Masters booklet. opright b The McGraw-Hill ompanies, Inc. ll rights reserved. Printed in the United States of merica. Permission is granted to reproduce the material contained herein on the condition that such material be reproduced onl for classroom use; be provided to students, teacher, and families without charge; and be used solel in conjunction with Glencoe s lgebra. n other reproduction, for use or sale, is prohibited without prior written permission of the publisher. Send all inquiries to: The McGraw-Hill ompanies rion Place olumbus, H 0-0 ISN: lgebra hapter Resource Masters

3 ontents Vocabular uilder vii Lesson - Stud Guide and Intervention Skills Practice Practice Reading to Learn Mathematics Enrichment Lesson - Stud Guide and Intervention Skills Practice Practice Reading to Learn Mathematics Enrichment Lesson - Stud Guide and Intervention Skills Practice Practice Reading to Learn Mathematics Enrichment Lesson - Stud Guide and Intervention Skills Practice Practice Reading to Learn Mathematics Enrichment Lesson - Stud Guide and Intervention Skills Practice Practice Reading to Learn Mathematics Enrichment Lesson - Stud Guide and Intervention Skills Practice Practice Reading to Learn Mathematics Enrichment Lesson - Stud Guide and Intervention Skills Practice Practice Reading to Learn Mathematics Enrichment hapter ssessment hapter Test, Form hapter Test, Form hapter Test, Form hapter Test, Form hapter Test, Form D hapter Test, Form hapter pen-ended ssessment hapter Vocabular Test/Review hapter Quizzes & hapter Quizzes & hapter Mid-hapter Test hapter umulative Review hapter Standardized Test Practice. 9 9 Unit Test/Review (h. ) Second Semester Test (h. ) Final Test (h. ) Standardized Test Practice Student Recording Sheet NSWERS Glencoe/McGraw-Hill iii Glencoe lgebra

4 Teacher s Guide to Using the hapter Resource Masters The Fast File hapter Resource sstem allows ou to convenientl file the resources ou use most often. The hapter Resource Masters includes the core materials needed for hapter. These materials include worksheets, extensions, and assessment options. The answers for these pages appear at the back of this booklet. ll of the materials found in this booklet are included for viewing and printing in the lgebra TeacherWorks D-RM. Vocabular uilder Pages vii viii include a student stud tool that presents up to twent of the ke vocabular terms from the chapter. Students are to record definitions and/or examples for each term. You ma suggest that students highlight or star the terms with which the are not familiar. WHEN T USE Give these pages to students before beginning Lesson -. Encourage them to add these pages to their lgebra Stud Notebook. Remind them to add definitions and examples as the complete each lesson. Stud Guide and Intervention Each lesson in lgebra addresses two objectives. There is one Stud Guide and Intervention master for each objective. WHEN T USE Use these masters as reteaching activities for students who need additional reinforcement. These pages can also be used in conjunction with the Student Edition as an instructional tool for students who have been absent. Skills Practice There is one master for each lesson. These provide computational practice at a basic level. WHEN T USE These masters can be used with students who have weaker mathematics backgrounds or need additional reinforcement. Practice There is one master for each lesson. These problems more closel follow the structure of the Practice and ppl section of the Student Edition exercises. These exercises are of average difficult. WHEN T USE These provide additional practice options or ma be used as homework for second da teaching of the lesson. Reading to Learn Mathematics ne master is included for each lesson. The first section of each master asks questions about the opening paragraph of the lesson in the Student Edition. dditional questions ask students to interpret the context of and relationships among terms in the lesson. Finall, students are asked to summarize what the have learned using various representation techniques. WHEN T USE This master can be used as a stud tool when presenting the lesson or as an informal reading assessment after presenting the lesson. It is also a helpful tool for ELL (English Language Learner) students. Enrichment There is one extension master for each lesson. These activities ma extend the concepts in the lesson, offer an historical or multicultural look at the concepts, or widen students perspectives on the mathematics the are learning. These are not written exclusivel for honors students, but are accessible for use with all levels of students. WHEN T USE These ma be used as extra credit, short-term projects, or as activities for das when class periods are shortened. Glencoe/McGraw-Hill iv Glencoe lgebra

5 ssessment ptions The assessment masters in the hapter Resource Masters offer a wide range of assessment tools for intermediate and final assessment. The following lists describe each assessment master and its intended use. hapter ssessment HPTER TESTS Form contains multiple-choice questions and is intended for use with basic level students. Forms and contain multiple-choice questions aimed at the average level student. These tests are similar in format to offer comparable testing situations. Forms and D are composed of freeresponse questions aimed at the average level student. These tests are similar in format to offer comparable testing situations. Grids with axes are provided for questions assessing graphing skills. Form is an advanced level test with free-response questions. Grids without axes are provided for questions assessing graphing skills. ll of the above tests include a freeresponse onus question. The pen-ended ssessment includes performance assessment tasks that are suitable for all students. scoring rubric is included for evaluation guidelines. Sample answers are provided for assessment. Vocabular Test, suitable for all students, includes a list of the vocabular words in the chapter and ten questions assessing students knowledge of those terms. This can also be used in conjunction with one of the chapter tests or as a review worksheet. Intermediate ssessment Four free-response quizzes are included to offer assessment at appropriate intervals in the chapter. Mid-hapter Test provides an option to assess the first half of the chapter. It is composed of both multiple-choice and free-response questions. ontinuing ssessment The umulative Review provides students an opportunit to reinforce and retain skills as the proceed through their stud of lgebra. It can also be used as a test. This master includes free-response questions. The Standardized Test Practice offers continuing review of algebra concepts in various formats, which ma appear on the standardized tests that the ma encounter. This practice includes multiplechoice, grid-in, and quantitativecomparison questions. ubble-in and grid-in answer sections are provided on the master. nswers Page is an answer sheet for the Standardized Test Practice questions that appear in the Student Edition on pages 0. This improves students familiarit with the answer formats the ma encounter in test taking. The answers for the lesson-b-lesson masters are provided as reduced pages with answers appearing in red. Full-size answer kes are provided for the assessment masters in this booklet. Glencoe/McGraw-Hill v Glencoe lgebra

6 NME DTE PERID Reading to Learn Mathematics Vocabular uilder This is an alphabetical list of the ke vocabular terms ou will learn in hapter. s ou stud the chapter, complete each term s definition or description. Remember to add the page number where ou found the term. dd these pages to our lgebra Stud Notebook to review vocabular at the end of the chapter. amplitude Vocabular Term Found on Page Definition/Description/Example Vocabular uilder M pluh TD double-angle formula half-angle formula midline phase shift FYZ (continued on the next page) Glencoe/McGraw-Hill vii Glencoe lgebra

7 NME DTE PERID Reading to Learn Mathematics Vocabular uilder (continued) Vocabular Term trigonometric equation Found on Page Definition/Description/Example trigonometric identit vertical shift Glencoe/McGraw-Hill viii Glencoe lgebra

8 - 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 sine, cosine, 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 maximum and minimum values. Example Graph sin for 0 0. First make a table of values sin sin 0 Lesson - sin Exercises Graph the following functions for the given domain.. cos, 0 0. tan, What is the amplitude of each function?.. x x Glencoe/McGraw-Hill Glencoe lgebra

9 - NME DTE PERID Stud Guide and Intervention (continued) Graphing Trigonometric Functions Variations of Trigonometric Functions For functions of the form a sin b and a cos b, the amplitude is a, 0 mplitudes and the period is or. b b and Periods For functions of the form a tan b, the amplitude is not defined, 0 and the period is or. b b Example a. cos Find the amplitude and period of each function. Then graph the function. First, find the amplitude. a, so the amplitude is. Next find the period Use the amplitude and period to help graph the function. cos b. tan The amplitude is not defined, and the period is Exercises Find the amplitude, if it exists, and period of each function. Then graph each function.. sin. tan Glencoe/McGraw-Hill Glencoe lgebra

10 - NME DTE PERID Skills Practice Graphing Trigonometric Functions Find the amplitude, if it exists, and period of each function. Then graph each function.. cos. sin. sec Lesson -. tan. sin. csc tan. cos 9. sin Glencoe/McGraw-Hill 9 Glencoe lgebra

11 NME DTE PERID - Practice Graphing Trigonometric Functions Find the amplitude, if it exists, and period of each function. Then graph each function.. sin. cot. cos csc. tan. sin FRE For Exercises and, use the following information. n anchoring cable exerts a force of 00 Newtons on a pole. The force has the horizontal and vertical components F x and F. ( force of one Newton (N), is the force that gives an acceleration of m/sec to a mass of kg.) 00 N F. The function F x 00 cos describes the relationship between the angle and the horizontal force. What are the amplitude and period of this function? F x. The function F 00 sin describes the relationship between the angle and the vertical force. What are the amplitude and period of this function? WETHER For Exercises 9 and 0, use the following information. The function 0 sin t, where t is in months and t 0 corresponds to pril, models the average high temperature in degrees Fahrenheit in enterville. 9. Determine the period of this function. What does this period represent? 0. What is the maximum high temperature and when does this occur? Glencoe/McGraw-Hill 0 Glencoe lgebra

12 - NME 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 in our textbook. onsider the tides of the tlantic cean as a function of time. pproximatel what is the period of this function? Reading the Lesson. Determine whether each statement is true or false. a. The period of a function is the distance between the maximum and minimum points. b. The amplitude of a function is the difference between its maximum and minimum values. Lesson - c. The amplitude of the function sin is. d. The function cot has no amplitude. e. The period of the function sec is. f. The amplitude of the function cos is. g. The function sin has a period of. h. The period of the function cot is. i. The amplitude of the function sin is. j. The period of the function csc is. k. The graph of the function sin has no asmptotes. l. The graph of the function tan has an asmptote at 0. m. When 0, the values of cos and sec are equal. n. When 0, cot is undefined. o. When 0, csc is undefined. Helping You Remember. What is an eas wa to remember the periods of a sin b and a cos b? Glencoe/McGraw-Hill Glencoe lgebra

13 - NME DTE PERID Enrichment lueprints 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. Example Find the unknown measures in the figure at the right. Roofing Improvement The measures x and are the legs of a right triangle. The measure of the hpotenuse " top view metal fitting is in. in. or 0 in. cos in. x sin 0 0 x 0. in. x " 0 " side view " 0.09" Find the unknown measures of each of the following.. himne on roof. ir vent. Elbow joint ' ' x ' 9 ' D ' ' 0 r ' t 0 ' Glencoe/McGraw-Hill Glencoe lgebra

14 - 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 sin 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. Example State the amplitude, period, and phase shift for cos. Then graph the function. mplitude: a or Period: or b.0 Phase Shift: h The phase shift is to the right since 0. Exercises State the amplitude, period, and phase shift for each function. Then graph the function.. sin ( 0). tan Lesson cos ( ). sin Glencoe/McGraw-Hill Glencoe lgebra

15 - NME DTE PERID Stud Guide and Intervention (continued) Translations of Trigonometric Graphs Vertical Translations When a constant is added to a trigonometric function, the graph is shifted verticall. Vertical Shift The vertical shift of the graphs of the functions a sin 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 of a vertical shift is k. Step Determine the vertical shift, and graph the midline. Graphing Step Determine the amplitude, if it exists. Use dashed lines to indicate the maximum and Trigonometric minimum values of the function. Functions Step Determine the period of the function and graph the appropriate function. Step Determine the phase shift and translate the graph accordingl. Example State the vertical shift, equation of the midline, amplitude, and period for cos. Then graph the function. Vertical Shift: k, so the vertical shift is units down. The equation of the midline is. mplitude: a or Period: b or 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 cosine curve, adjusted to have a period of. Exercises State the vertical shift, equation of the midline, amplitude, and period for each function. Then graph the function.. cos. sin Glencoe/McGraw-Hill Glencoe lgebra

16 NME DTE PERID - Skills Practice Translations of Trigonometric Graphs State the amplitude, period, and phase shift for each function. Then graph the function.. sin ( 90). cos ( ). tan State the vertical shift, equation of the midline, amplitude, and period for each function. Then graph the function.. csc. cos. sec Lesson - State the vertical shift, amplitude, period, and phase shift of each function. Then graph the function.. cos [( )]. sin [( 90)] 9. cot Glencoe/McGraw-Hill Glencoe lgebra

17 - NME DTE PERID Practice Translations of Trigonometric Graphs State the vertical shift, amplitude, period, and phase shift for each function. Then graph the function.. tan. cos ( 0). csc ( 0) ELGY For Exercises, 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 sin t, where t is the number of ears since the stand was first cut in November, 90.. How often does the insect population reach its maximum level?. When did the population last reach its maximum?. What condition in the stand do ou think corresponds with a minimum insect population? LD PRESSURE For Exercises 9, use the following information. Jason s blood pressure is 0 over 0, meaning that the pressure oscillates between a maximum of 0 and a minimum of 0. Jason s heart rate is beats per minute. The function that represents Jason s blood pressure P can be modeled using a sine function with no phase shift.. Find the amplitude, midline, and period in seconds of the function.. Write a function that represents Jason s blood pressure P after t seconds. 9. Graph the function P Jason s lood Pressure Pressure Time t Glencoe/McGraw-Hill Glencoe lgebra

18 - NME DTE PERID Reading to Learn Mathematics Translations of Trigonometric Graphs Pre-ctivit How can translations of trigonometric graphs be used to show animal populations? Read the introduction to Lesson - at the top of page 9 in our textbook. ccording to the model given in our textbook, what would be the estimated rabbit population for Januar, 00? 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. sin ( 90) b. sin c. cos d. tan. 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. sin Lesson - b. cos ( 0) c. cos d. sec e. tan f. sin Helping You Remember. Man students have trouble remembering which of the functions sin ( ) and sin ( ) represents a shift to the left and which represents a shift to the right. Using, explain a good wa to remember which is which. Glencoe/McGraw-Hill Glencoe lgebra

19 - NME DTE PERID Enrichment Translating Graphs of Trigonometric Functions Three graphs are shown at the right: sin sin ( 0) sin Replacing with ( 0) translates the graph to the right. Replacing with translates the graph units down. = sin u 90 0 u = sin ( u 0 ) + = sin u Example Graph one ccle of cos ( 0). Step Transform the equation into the form k a cos b( h). cos ( ) Step Sketch cos. Step = cos u u Step Translate cos to obtain the desired graph. Step = cos ( u+ ) = cos ( u + ) Sketch these graphs on the same coordinate sstem.. sin ( ). sin. sin ( 90) n another piece of paper, graph one ccle of each curve.. sin ( 0). sin ( 90). cos ( 0). cos. The graphs for problems and should be the same. Use the sum formula for cosine of a sum to show that the equations are equivalent. Glencoe/McGraw-Hill Glencoe lgebra

20 - 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 expression in the equation is defined. sin Quotient Identities tan cot c os c os sin asic Trigonometric Reciprocal Identities csc sec cot sin co s ta n Identities Pthagorean Identities cos sin tan sec cot csc Example Find the value of cot if csc ; 0 0. 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. Exercises Find the value of each expression.. tan, if cot ; 0 0. csc, if cos ; cos, if sin ;0 90. sec, if sin ;0 90 Lesson -. cos, if tan ;90 0. tan, if sin ;0 90. sec, if cos ;90 0. sin, if cos ; cot, if csc ; sin, if csc 9 ; 0 0 Glencoe/McGraw-Hill 9 Glencoe lgebra

21 - NME DTE PERID Stud Guide and Intervention (continued) Trigonometric Identities Simplif Expressions The simplified form of a trigonometric expression is written as a numerical value or in terms of a single trigonometric function, if possible. n of the trigonometric identities on page 9 can be used to simplif expressions containing trigonometric functions. Example Simplif ( cos ) sec cot tan sec cos. ( cos ) sec cot tan sec cos sin c os sin co s sin c os cos co s sin sin sin Example se c cot csc sin sin sec cot csc Simplif. sin sin c os co s sin sin sin sin ( sin ) ( sin ) sin sin ( sin )( sin ) sin sin sin cos Exercises Simplif each expression. tan csc sin cot.. sec sec tan sin cos. cot tan. cot sin sec tan tan cos csc. cot sin tan csc. cot sin tan cos. tan cot sin csc cos sec. cos tan sin Glencoe/McGraw-Hill 0 Glencoe lgebra

22 - NME DTE PERID Skills Practice Trigonometric Identities Find the value of each expression.. sin, if cos and cos, if tan and 0 0. sec, if tan and cos, if tan and tan, if sin and 0 0. cos, if sec and 0 0. cos, if csc and 0 0. tan, if cos and cos, if cot and csc, if cos and cot, if csc and 0 0. tan, if sin and 0 0 Simplif each expression. Lesson -. sin sec. csc sin. cot sec. c os sec. tan cot. csc tan tan sin sin csc cot sin sin. cos. tan cos sec Glencoe/McGraw-Hill Glencoe lgebra

23 - NME DTE PERID Practice Trigonometric Identities Find the value of each expression.. sin, if cos and sec, if sin and 0 0. cot, if cos and 0 0. sin, if cot and cot, if csc and 0 0. sec, if csc and 0 0. sec, if tan and 0 0. sin, if tan and cot, if tan and cot, if cos and 0 0 Simplif each expression.. csc tan sin. tan. sin cot. cot csc. cot. csc sin cos cos cos cos. sin cos cot. 9. sec cos tan sin sin 0. ERIL PHTGRPHY The illustration shows a plane taking an aerial photograph of point. ecause the point is directl below the plane, there is no distortion in the image. For an point 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 exposure of the film reduces b (sin )(csc sin ). Express (sin )(csc sin ) in terms of cos onl.. TSUNMIS The equation a sin t represents the height of the waves passing a buo at a time t in seconds. Express a in terms of csc t. Glencoe/McGraw-Hill Glencoe lgebra

24 - 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 in our textbook. 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.) Reading the Lesson. Match each expression from the list on the left with an expression from the list on the right that is equal to it for all values for which each expression is defined. (Some of the expressions from the list on the right ma be used more than once or not at all.) a. sec tan i. sin b. cot ii. tan sin c. c os iii. d. sin cos iv. sec e. csc v. csc f. co s g. c os sin vi. cot. 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.) Lesson - a. tan, if sin and 0 0 b. sec, if tan and 90 0 Helping You Remember. 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 sine and cosine that ou learned in hapter to help ou remember the Pthagorean identit cos sin? Glencoe/McGraw-Hill Glencoe lgebra

25 - NME DTE PERID Enrichment 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 axis be toward the other focus. The polar equation of an ellipse is r ep r. Since p b and b e cos c a c, c p a a c c c c a. ecause e, a Polar xis 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 Note that e is the eccentricit of the orbit and a is the length of the semi-major axis of the ellipse. lso, a is the mean distance of the planet from the sun. The mean distance of Venus from the sun is. 0 miles and the eccentricit of its orbit is.00. Find the minimum and maximum distances of Venus from the sun. The minimum distance occurs when. r. 0 miles The maximum distance occurs when 0. r Example. 0 ( 0.00 ) 0.00 cos. 0 ( 0.00 ) 0.00 cos miles omplete each of the following.. The mean distance of Mars from the sun is. 0 miles and the eccentricit of its orbit is Find the minimum and maximum distances of Mars from the sun.. The minimum distance of Earth from the sun is 9. 0 miles and the eccentricit of its orbit is 0.0. Find the mean and maximum distances of Earth from the sun. Glencoe/McGraw-Hill Glencoe lgebra

26 - 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 expression into the form of the simpler side. Example Verif that each of the following is an identit. sin tan a. sec cos b. cos sec cot csc Transform the left side. sin cot sin os c s in sin cos sec cos cos cos sin cos cos cos cos cos cos cos cos cos Transform the left side. tan csc sin c os sin sin cos cos sec cos sec cos sec sin cos cos sec sec cos sec sec Exercises Verif that each of the following is an identit.. csc cos csc sin cot. cos cos cos sin Lesson - Glencoe/McGraw-Hill Glencoe lgebra

27 - Transform oth Sides of an Equation The following techniques can be helpful in verifing trigonometric identities. Substitute one or more basic identities to simplif an expression. Factor or multipl to simplif an expression. Multipl both numerator and denominator b the same trigonometric expression. Write each side of the identit in terms of sine and cosine onl. Then simplif each side. Example tan sin tan sec Verif that sec sin sin c co os Exercises NME DTE PERID Stud Guide and Intervention (continued) Verifing Trigonometric Identities cos sin c os s cos sin os sec tan cos c sin cos cos cos cos tan sin tan sec Verif that each of the following is an identit. sin cos sin cos tan sec. csc sec cot tan. cos cos sec tan is an identit. cos cot csc csc.. cot cot ( cos ) sin sin sec sec Glencoe/McGraw-Hill Glencoe lgebra

28 - NME DTE PERID Skills Practice Verifing Trigonometric Identities Verif that each of the following is an identit.. tan cos sin. cot tan sin. csc cos cot. cos cos. (tan )( sin ) sin cos csc. cot sec sin. tan cos. sin sin sin Lesson - Glencoe/McGraw-Hill Glencoe lgebra

29 - NME DTE PERID Practice Verifing Trigonometric Identities Verif that each of the following is an identit. sin. cos sec cos. cos sin. ( sin )( sin ) cos. tan tan sec. cos cot cot cos. (sin )(csc sec ) sec. PRJETILES 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 sin maximum height reached, and g is the acceleration due to gravit. Verif the identit gh gh sec. sin 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. Glencoe/McGraw-Hill Glencoe lgebra

30 - 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 in our textbook. For,0,or, sin sin. Does this mean that sin sin is an identit? Explain our reasoning. Reading the Lesson. Determine whether each equation is an identit or not an identit. a. sin tan cos b. sin tan sin c. c os c os s cos sin in d. cos (tan ) sin e. c os sin csc sec f. cos sin sin g. tan cos csc sin h. s ec ta n co t. Which of the following is not permitted when verifing an identit?. simplifing one side of the identit to match the other side. cross multipling if the identit is a proportion. simplifing each side of the identit separatel to get the same expression on both sides Lesson - 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? Glencoe/McGraw-Hill 9 Glencoe lgebra

31 - NME DTE PERID Enrichment 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. onsider an triangle. Let K represent the area of. Then K bc sin K b c sin Square both sides. c a b c ( cos ) b c ( cos )( cos ) b c b c a bc b c a bc b Use the law of cosines. b c a b c a a b c a b c Simplif. Let s a b c. Then s a b c a, s b a c b, s c a b c. K s(s a)(s b)(s c) Substitute. K s(s a)(s b)(s c) Heron s Formula The area of is a b c s(s a)(s b)(s c), where s. Use Heron s formula to find the area of.. a, b., c. a., b 0., c 9.. a., b 9.0, c.9. a 0., b., c 0.. a, b, c 9. a 0.0, b 0.0, c 0.0. a., b.0, c.. a.0, b 9., c.99 Glencoe/McGraw-Hill 0 Glencoe lgebra

32 - 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 expression like sin from the known values of sine and cosine of 0 and. Sum and Difference of ngles The following identities hold true for all values of and. cos ( ) cos cos sin sin sin ( ) sin cos cos sin Example a. cos cos cos (00 ) Find the exact value of each expression. cos 00 cos sin 00 sin b. sin (0) sin (0) sin ( 0) sin cos 0 cos sin 0 Exercises Find the exact value of each expression.. sin 0. cos. cos (). cos (). sin 9. cos 0. sin (). cos 9. cos () 0. sin. cos (0). sin 9 Lesson - Glencoe/McGraw-Hill Glencoe lgebra

33 - Verif Identities You can also use the sum and difference of angles formulas to verif identities. Example NME DTE PERID Stud Guide and Intervention (continued) Sum and Difference of ngles Formulas Verif that cos sin is an identit. cos sin cos cos sin sin sin cos 0 sin () sin sin sin riginal equation Sum of ngles Formula Evaluate each expression. Simplif. Example Verif that sin cos ( ) cos is an identit. sin cos ( ) cos riginal equation sin cos cos sin cos cos sin sin cos Exercises sin 0 cos cos () sin 0 cos cos cos Sum and Difference of ngles Formulas Evaluate each expression. Simplif. Verif that each of the following is an identit.. sin (90 ) cos. cos (0 ) sin. sin cos sin. cos sin sin Glencoe/McGraw-Hill Glencoe lgebra

34 - NME DTE PERID Skills Practice Sum and Difference of ngles Formulas Find the exact value of each expression.. sin 0. cos (). sin (). cos. sin (). cos 0. cos (). sin 9. sin (9) Verif that each of the following is an identit. 0. sin (90 ) cos. sin (0 ) sin. cos (0 ) sin. cos ( 90) sin. sin cos. cos ( ) cos Lesson - Glencoe/McGraw-Hill Glencoe lgebra

35 - NME DTE PERID Practice Sum and Difference of ngles Formulas Find the exact value of each expression.. cos. cos. sin (). sin (0). sin 0. cos 0. sin. sin () 9. sin 9 Verif that each of the following is an identit. 0. cos (0 ) cos. sin (0 ) sin. sin ( ) sin ( ) sin. cos x sin x sin x. SLR ENERGY n March, the maximum amount of solar energ that falls on a square foot of ground at a certain location is given b E sin (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. ELETRIITY In Exercises and, use the following information. In a certain circuit carring alternating current, the formula i sin (0t) can be used to find the current i in amperes after t seconds.. Rewrite the formula using the sum of two angles.. Use the sum of angles formula to find the exact current at t second. Glencoe/McGraw-Hill Glencoe lgebra

36 - 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 in our textbook. onsider the functions sin x and sin x. Do the graphs of these two functions have constructive interference or destructive interference? Reading the Lesson. Match each expression from the list on the left with an expression from the list on the right that is equal to it for all values of the variables. (Some of the expressions from the list on the right ma be used more than once or not at all.) a. sin ( ) i. sin b. cos ( ) ii. sin cos cos sin c. sin (0 ) iii. cos d. sin (0 ) iv. cos cos sin sin e. cos (0 ) v. sin cos cos sin f. sin ( ) vi. cos cos sin sin g. cos (90 ) vii. sin h. cos ( ) viii. cos. Which expressions are equal to sin? (There ma be more than one correct choice.). sin cos 0 cos sin 0. sin cos 0 cos sin 0. sin 0 cos cos 0 sin D. cos 0 cos sin 0 sin 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? Lesson - Glencoe/McGraw-Hill Glencoe lgebra

37 - NME DTE PERID Enrichment Identities for the Products of Sines and osines adding the identities for the sines of the sum and difference of the measures of two angles, a new identit is obtained. (i) sin ( ) sin cos cos sin sin ( ) sin cos cos sin sin ( ) sin ( ) sin cos This new identit is useful for expressing certain products as sums. Example Write sin cos as a sum. In the identit let and so that sin cos sin ( ) sin ( ). Thus, sin cos sin sin. subtracting the identities for sin ( ) and sin ( ), a similar identit for expressing a product as a difference is obtained. (ii) sin ( ) sin ( ) cos sin Solve.. Use the identities for cos ( ) and cos ( ) to find identities for expressing the products cos cos and sin sin as a sum or difference.. Find the value of sin 0 cos without using tables.. Express cos sin as a difference. Glencoe/McGraw-Hill Glencoe lgebra

38 - 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 sin sin cos cos cos sin Formulas cos sin cos cos Lesson - Example Find the exact values of sin and cos if 9 sin and First, find the value of cos. cos sin cos sin cos 9 9 sin 0 0 cos cos 0 9 Since is in the third quadrant, cos is negative. Thus cos 0. To find sin, use the identit sin sin cos. sin sin cos The value of sin is 0. To find cos, use the identit cos sin. cos sin The value of cos is. 0 Exercises Find the exact values of sin and cos for each of the following.. sin,0 90. sin, 0 0. cos, 0 0. cos,90 0. sin, 0 0. cos,90 0 Glencoe/McGraw-Hill Glencoe lgebra

39 - NME DTE PERID Stud Guide and Intervention (continued) Double-ngle and Half-ngle Formulas Half-ngle Formulas Half-ngle The following identities hold true for all values of. Formulas sin cos cos cos Example Find the exact value of sin if sin and First find cos. cos sin cos sin cos sin cos 9 Simplif. cos Take the square root of each side. Since is in the second quadrant, cos. sin cos Half-ngle formula cos Simplif. Rationalize. Since is between 90 and 0, is between and 90. Thus sin is positive and equals. Exercises Find the exact value of sin and cos for each of the following.. cos, 0 0. cos,90 0. sin, 0 0. cos,90 0 Find the exact value of each expression b using the half-angle formulas.. cos. sin.. cos Glencoe/McGraw-Hill Glencoe lgebra

40 - NME DTE PERID Skills Practice Double-ngle and Half-ngle Formulas Find the exact values of sin, cos, sin, and cos for each of the following.. cos, sin, 0 0 Lesson -. sin 0,90 0. cos, 0 0. cos,90 0. sin,0 90 Find the exact value of each expression b using the half-angle formulas.. cos. sin 9. cos 0 0. sin. sin. cos Verif that each of the following is an identit. tan. sin tan. tan cot csc Glencoe/McGraw-Hill 9 Glencoe lgebra

41 - NME DTE PERID Practice Double-ngle and Half-ngle Formulas Find the exact values of sin, cos, sin, and cos for each of the following.. cos,0 90. sin, cos, 0 0. sin, 0 0 Find the exact value of each expression b using the half-angle formulas.. tan 0. tan. cos.. sin Verif that each of the following is an identit. 9. sin tan sin tan 0. sin cos sin cos. ERIL PHTGRPHY In aerial photograph, there is a reduction in film exposure 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 exposure for the point directl below the camera. Using the identit sin cos, verif that E 0 cos E 0 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 sin H tan. cos Glencoe/McGraw-Hill 0 Glencoe lgebra

42 - NME DTE PERID Reading to Learn Mathematics 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 9 in our textbook. Suppose that the equation for the second harmonic is sin a. Then what would be the equations for the fundamental tone (first harmonic), third harmonic, fourth harmonic, and fifth harmonic? Lesson - Reading the Lesson. Match each expression from the list on the left with all expressions from the list on the right that are equal to it for all values of. a. sin i. sin cos b. cos ii. sin c. cos iii. cos sin d. sin iv. cos v. cos. Determine whether ou would use the positive or negative square root in the half-angle identities for sin and cos in each of the following situations. (Do not actuall calculate sin and cos.) a. sin, if cos and is in Quadrant I b. cos, if cos 0.9 and is in Quadrant II c. cos, if sin 0. and is in Quadrant III d. sin, if sin 0. and is in Quadrant IV 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 using a Pthagorean identit? Glencoe/McGraw-Hill Glencoe lgebra

43 - NME DTE PERID Enrichment lternating urrent 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, its motion is along the direction of the magnetic field between the poles. Therefore, no current is generated. However, through points and D, the motion of X is perpendicular to the magnetic field. This induces maximum current in the coil. etween and, and, 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 sine curve. The actual current, i, in a household current is given b i I M sin(0t ) where I M is the maximum value of the current, t is the elapsed time in seconds, and is the angle determined b the position of the coil at time t n. X D The maximum current ma have a positive or negative value. i(amperes) t(seconds) D Example If, find a value of t for which i 0. If i 0, then I M sin (0t ) 0. i I M sin(0t ) Since I M 0, sin(0t ) 0. If ab 0 and a 0, then b 0. Let 0t s. Thus, sin s 0. s because sin 0. 0t Substitute 0t for s. 0t Substitute for. Solve for t. 0 This solution is the first positive value of t that satisfies the problem. Using the equation for the actual current in a household circuit, i I M sin(0t ), solve each problem. For each problem, find the first positive value of t.. If 0, find a value of t for. If 0, find a value of t for which which i 0. i I M.. If, find a value of t for which. If, find a value of t for which i I M. i 0. Glencoe/McGraw-Hill Glencoe lgebra

44 - NME DTE PERID 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. Example Example Find all solutions of sin 0 for the interval 0 0. sin 0 sin sin sin 0, 0, 0, 0 Solve sin cos 0 for all values of. Give our answer in both radians and degrees. sin cos 0 sin cos cos 0 cos ( sin ) 0 cos 0 or sin 0 90 k 0; k sin 0 k 0, 0 k 0; k, k Lesson - Exercises Find all solutions of each equation for the given interval.. cos cos, 0. sin cos 0, 0. cos,0 0. sin 0, 0 Solve each equation for all values of if is measured in radians.. sin 0. cos sin cos 0 Solve each equation for all values of if is measured in degrees.. cos sin. tan Glencoe/McGraw-Hill Glencoe lgebra

45 - Use Trigonometric Equations Example NME DTE PERID LIGHT Snell s law sas that sin. sin, where is the angle at which a beam of light enters water and is the angle at which the beam travels through the water. If a beam of light enters water at, at what angle does the light travel through the water? sin. sin sin. sin sin si n. sin Stud Guide and Intervention (continued) Solving Trigonometric Equations riginal equation Divide each side b.. Use a calculator. Take the arcsin of each side. The light travels through the water at an angle of approximatel 0.. Exercises. -foot pipe is propped on a -foot tall packing crate that sits on level ground. ne foot of the pipe extends above the top of the crate and the other end rests on the ground. What angle does the pipe form with the ground?. t :00 P.M. one afternoon a 0-foot statue casts a shadow that is feet long. Write an equation to find the angle of elevation of the Sun at that time. Find the angle of elevation.. conveor belt is set up to carr packages from the ground into a window feet above the ground. The angle that the conveor belt forms with the ground is. How long is the conveor belt from the ground to the window sill? SPRTS The distance a golf ball travels can be found using the formula v d 0 sin, where v 0 is the initial velocit of the ball, g is the acceleration due g to gravit (which is feet per second squared), and is the angle that the path of the ball makes with the ground.. How far will a ball travel hit 90 feet per second at an angle of?. If a ball that traveled 00 feet had an initial velocit of 0 feet per second, what angle did the path of the ball make with the ground?. Some children set up a teepee in the woods. The poles are feet long from their intersection to their bases, and the children want the distance between the poles to be feet at the base. How wide must the angle be between the poles? Glencoe/McGraw-Hill Glencoe lgebra

46 - NME DTE PERID Skills Practice Solving Trigonometric Equations Find all solutions of each equation for the given interval.. sin,0 0. cos, tan, 0 0. sin, 0. sin sin 0,. cos cos 0, 0 Solve each equation for all values of if is measured in radians.. cos cos. sin sin 0 9. sin sin cos 0 0. sin Lesson -. cos cos. tan cos Solve each equation for all values of if is measured in degrees.. sin 0. cos 0. sin 0. cos. sin. cos Solve each equation for all values of. 9. cos sin 0 0. sin sin 0. sin sin. cos sec Glencoe/McGraw-Hill Glencoe lgebra

47 - NME DTE PERID Practice Solving Trigonometric Equations Find all solutions of each equation for the given interval.. sin cos, cos sin,0 0. cos cos, 0 0. cos cos (90 ) 0, 0. cos sin,. tan sec, Solve each equation for all values of if is measured in radians.. cos sin. cot cot 9. sin sin 0. cos sin sin. cos sin. sec Solve each equation for all values of if is measured in degrees.. sin cos cos. csc csc 0. ( cos ) cos. cos cos Solve each equation for all values of.. sin. sin 0 9. sin sin 0. cos sin 0. WVES Waves are causing a buo to float in a regular pattern in the water. The vertical position of the buo can be described b the equation h sin x. Write an expression that describes the position of the buo when its height is at its midline.. ELETRIITY The electric current in a certain circuit with an alternating current can be described b the formula i sin 0t, where i is the current in amperes and t is the time in seconds. Write an expression that describes the times at which there is no current. Glencoe/McGraw-Hill Glencoe lgebra

48 - NME 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 - at the top of page 99 in our textbook. Describe how ou could use a graphing calculator to determine the months in which the average dail high temperature is above 0F. (ssume that x represents Januar.) Specif the graphing window that ou would use. Reading the Lesson. Identif which equations have no solution. Lesson -. sin. tan sec D. csc E. cos.0 F. cot 000 G. cos H. sec. 0 I. sin Use a trigonometric identit to write the first step in the solution of each trigonometric equation. (Do not complete the solution.) a. tan cos sin,0 b. sin sin 0, 0 0 c. cos sin, 0 0 d. sin cos, 0 e. cos cos, 0 0 f. tan tan 0 Helping You Remember. good wa to remember something is to explain it to someone else. How would ou explain to a friend the difference between verifing a trigonometric identit and solving a trigonometric equation. Glencoe/McGraw-Hill Glencoe lgebra

49 - NME DTE PERID Enrichment Families of urves Use these graphs for the problems below. The Famil x n The Famil e mx. n = n =.. n = x x. Use the graph on the left to describe the relationship among the curves x, x, and x.. Graph x n for n, 0,,and 0 on the grid with x, x, and x.. Which two regions in the first quadrant contain no points of the graphs of the famil for x n?. n the right grid, graph the members of the famil e mx for which m and m.. Describe the relationship among these two curves and the -axis.. Graph e mx for m 0,,,, and. Glencoe/McGraw-Hill Glencoe lgebra

50 NME DTE PERID hapter Test, Form SRE Write the letter for the correct answer in the blank at the right of each question.. Which equation is graphed?. sin. cos. sin D. cos.. Find the amplitude of sin.... D... Find the period of cos.... D Which equation is graphed?. sin ( 0). sin ( 0). cos ( 0) D. cos ( 0).. Which equation is graphed?. cos. cos. sin D. sin ssessment. Find sin if cos and D... Find cot if tan and D... Simplif sin csc.. sin.. tan D.. 9. Simplif tan cos.. c os. cot. sin D. sec sin 9. Glencoe/McGraw-Hill 9 Glencoe lgebra

51 NME DTE PERID hapter Test, Form (continued) 0. Simplif cot sec. cos. s in. sin. csc D. sec 0.. Which expression is equivalent to sin cos? tan. cot. cos cot. cos cos D. csc.. Which expression is equivalent to csc (csc sin )?. sec. cot. tan D... Find the exact value of cos.... D... Find the exact value of sin D... Which expression is equivalent to sin (90)?. sin. sin. cos D. cos.. Find the exact value of cos if cos and D Find the exact value of sin if sin and D... Find the exact value of cos b using a half-angle formula.... D.. 9. Which is not a solution of sin? D LIGHT The length of the shadow S given b a tower that is 00 meters 00 high is S, where is the angle of inclination of the Sun. If the t an angle of inclination is, find the length of the shadow.. m. m. 00 m D. m 0. onus Verif that tan sec sin sec cos is an identit. : Glencoe/McGraw-Hill 0 Glencoe lgebra

52 NME DTE PERID hapter Test, Form SRE Write the letter for the correct answer in the blank at the right of each question.. Which equation is graphed?. sin. cos. sin D. cos.. Find the amplitude of sin.... D... Find the period of tan.... D... Which equation is graphed?. sin ( ). sin. cos D. cos.. Find the phase shift of cos. ssessment... D... Which equation is graphed?. sin. sin. cos D. cos.. Find the vertical shift of csc.... D... Find csc if cot and D.. 9. Find sin if cos and D. 9. Glencoe/McGraw-Hill Glencoe lgebra

53 NME DTE PERID hapter Test, Form (continued) 0. Simplif cos. tan. cos. sec. cos D. sin 0.. Simplif (cot csc ).... csc D. sec.. Which expression is not equivalent to?. sin cot sin sin. cos cos. sec tan D. cot sin co s.. Which expression is equivalent to tan s ec? sin. cot. cot. tan cot D. tan sec.. Find the exact value of cos.... D... Which expression is equivalent to cos?. cos. cos. sin D. sin.. Find the exact value of sin if cos and D.. 9. Find the exact value of sin if cos and D.. Find the exact value of cos 0 b using a half-angle formula..... D.. 9. Find the solutions of sin cos if , 90. 0, 0. 0, 90, 0 D. 0, 90, ILGY n insect population P in a certain area fluctuates with the t seasons. It is estimated that P, sin, where t is given in weeks. Determine the number of weeks it would take for the population to initiall reach 0,000.. weeks. 9 weeks. weeks D. weeks 0. onus Verif that cot sin cos is an identit. : csc Glencoe/McGraw-Hill Glencoe lgebra

54 NME DTE PERID hapter Test, Form SRE Write the letter for the correct answer in the blank at the right of each question.. Which equation is graphed?. sin. cos. sin D. cos.. Find the amplitude of cos.... D... Find the period of tan D... Which equation is graphed?. sin. cos. sin D. cos.. Find the phase shift of sin. ssessment... D... Which equation is graphed?. sin. sin. cos D. cos.. Find the vertical shift of sec.... D... Find sec if tan and D.. 9. Find cos if sin and D Simplif c. cot sc... tan D. sin 0. Glencoe/McGraw-Hill Glencoe lgebra

55 NME DTE PERID hapter Test, Form (continued). Simplif (sec tan ).. tan. tan. D... Which expression is equivalent to?. sin. sin sec csc. tan sec D. cot csc. sec sin sin. Which expression is equivalent to? cos cos sin. cos. sin. csc D. csc.. Find the exact value of sin ().... D... Which expression is equivalent to sin?. cos. cos. sin D. cos.. Find the exact value of cos if sin and D.. 9. Find the exact value of cos if sin and D... Find the exact value of sin 0 b using a half-angle formula.... D.. 9. Find the solutions of sin cos if , 0. 0, 0. 0, 0 D. 0, ILGY n insect population P in a certain area fluctuates with t the seasons. It is estimated that P, sin, where t is given in weeks. Determine the number of weeks it would take for the population to initiall reach,000.. weeks. weeks. 09 weeks D. weeks 0. onus Verif that csc tan tan is an identit. : Glencoe/McGraw-Hill Glencoe lgebra

56 NME DTE PERID hapter Test, Form SRE. Graph the function cos.. For Questions and, find the amplitude, if it exists, and period of each function.. sin.. tan.. State the phase shift of cos. Then graph the. function. ssessment. State the vertical shift and the equation of the midline for. cos. Then graph the function.. Find sec if sin and Find cot if csc and Simplif cos csc.. cot 9. Simplif cos. cos 9. Glencoe/McGraw-Hill Glencoe lgebra

57 NME DTE PERID hapter Test, Form (continued) 0. Verif that (cos sin ) cos sin is an identit. 0.. Verif that cot sin cos is an identit.. csc. Find the exact value of sin (9)... Find the exact value of cos... Verif that sin cos is an identit... Find the exact value of sin if cos and Find the exact value of cos if sin and Find the exact value of sin 9 b using a half-angle. formula.. Verif that sin cot csc is an identit.. 9. Solve cos cos 0 for all values of if is measured 9. in degrees. 0. USINESS The profit P for a product whose sales fluctuate 0. t with the seasons is estimated to be P sin, where t is given in weeks and P is in thousands of dollars. Determine the number of weeks it would take for the profit to initiall reach $,000. onus Find cos if sin. : Glencoe/McGraw-Hill Glencoe lgebra

58 NME DTE PERID hapter Test, Form D SRE. Graph sin.. For Questions and, find the amplitude, if it exists, and period of each function.. sin.. tan.. State the phase shift of sin. Then graph the. function. ssessment. State the vertical shift and the equation of the midline for. cos. Then graph the function.. Find csc if cos and Find tan if sec and Simplif csc tan.. sec 9. Simplif sec. sin 9. Glencoe/McGraw-Hill Glencoe lgebra

59 NME DTE PERID hapter Test, Form D (continued) 0. Verif that cos sec cos sin 0 is an identit. 0.. Verif that ta n cot sec sec sin co is an identit.. t. Find the exact value of sin... Find the exact value of cos ()... Verif that cos sin is an identit... Find the exact value of cos if cos and Find the exact value of sin if sin and Find the exact value of cos 9 b using a half-angle. formula.. Verif that cos sin ( cot csc ) is an identit.. 9. Solve sin sin 0 for all values of if is measured 9. in degrees. 0. USINESS The profit P for a product whose sales fluctuate 0. t with the seasons is estimated to be P sin, where t is given in weeks and P is in thousands of dollars. Determine the number of weeks it would take for the profit to initiall reach $0,000. onus Find cos if cos. : Glencoe/McGraw-Hill Glencoe lgebra

60 NME DTE PERID hapter Test, Form SRE. Graph csc.. Find the amplitude, if it exists, and period of each function.. cos. tan.. For Questions and, state the vertical shift, amplitude, period, and phase shift of each function. Then graph the function.. tan ( 90). ssessment. cos.. Find sec if sin and Find tan if sec and Simplif co t cos cot. cos. csc 9. Verif that c ot sin cot csc is an identit. 9. Glencoe/McGraw-Hill 9 Glencoe lgebra

61 NME DTE PERID hapter Test, Form (continued) 0. Verif that cot si n sin is an identit. 0.. Find the exact value of cos cos... Find the exact value of sin 0sin... Verif that sin cos cos. is an identit.. Verif that [sin ( )] tan cot cot tan. sin cos sin cos is an identit.. Find the exact value of sin if cos and Find the exact value of cos if sin and Find the exact value of cos b using half-angle formulas... Verif that sin sin cos is an identit.. cos 9. Solve sin cos 0 for all values of if is measured in 9. radians. 0. WVES For a short time after a wave is created b a boat, 0. its height can be modeled b h h sin t, where P h is the maximum height of the wave in feet, P is the period in seconds, and t is the propagation of the wave in seconds. If a wave has a maximum height of. feet and a period of. seconds, how long after its creation will the wave initiall reach a height of feet? Round to the nearest hundredth. sin cos onus Find the exact value of if sin and : sin 0 0. Glencoe/McGraw-Hill 90 Glencoe lgebra

62 NME DTE PERID hapter pen-ended ssessment SRE Demonstrate our knowledge b giving a clear, concise solution to each problem. e sure to include all relevant drawings and justif our answers. You ma show our solutions in more than one wa or investigate beond the requirements of the problem.. Ms. Rollins divided her students into four groups, asking each to solve the equation sin cot cos. The answers given were: Group : 0k 0, 90k 0, 0k 0 Group : 0k 0, 90k 0 Group : 90k 0 Group D: 90k 0, 0k 0 Do an of the groups have the correct solution? Explain our reasoning.. Write a trigonometric function that has no amplitude, a period of,a phase shift to the left, and a vertical shift upward. Then graph our function for 0. ssessment. Show two different methods of verifing that tan is a trigonometric identit. s in. Select a quadrant, other than Quadrant I, and values for p and q so that sin p. Use our values of p and q to find the q exact values of cos, tan, csc, sec, cot, sin, cos, sin, and cos.. Show how to find the exact value of sin 0 b each method indicated. a. using a sum of angles formula b. using a difference of angles formula c. using a double-angle formula d. using a half-angle formula Glencoe/McGraw-Hill 9 Glencoe lgebra

63 NME DTE PERID hapter Vocabular Test/Review SRE amplitude double-angle formula half-angle formula midline phase shift trigonometric equation trigonometric identit vertical shift Tell whether each sentence is true or false. If false, replace the underlined word or words to make a true sentence.. For the graph of sin x,the vertical shift is... For the graph of cos (x ), the phase shift is... For the graph of sin x, the line is the. amplitude.. sin cos is a(n) trigonometric identit... The exact value of sin can be found b using a(n). phase shift.. cos cos sin is a(n) double-angle formula... cos cos 0 is a(n) trigonometric equation.. In our own words Define the term.. phase shift Glencoe/McGraw-Hill 9 Glencoe lgebra

64 NME DTE PERID hapter Quiz (Lessons and ) SRE For Questions and, find the amplitude, if it exists, and period of each function. Then graph the function.. cos tan.. State the phase shift of sin.. ssessment. State the vertical shift and the equation of the midline for. cos. NME DTE PERID hapter Quiz (Lessons and ) SRE For Questions and, find the value of each expression.. cos, if sin ; cot, if tan ; Simplif (tan sec )... Simplif tan. csc.. Standardized Test Practice se c t cos. cos sin. sin an. sin. D. si n Glencoe/McGraw-Hill 9 Glencoe lgebra

65 NME DTE PERID hapter Quiz (Lessons and ) SRE Find the exact value of each expression.. sin. cos (). tan 0.. Verif that each is an identit.. sin cos.. cos (0) cos. For Questions, find the exact value for each.. cos, if cos ; sin, if sin ; cos, if sin ; Find the exact value of cos b using a half-angle 9. formula. 0. Verif that cos sin tan is an identit. 0. NME DTE PERID hapter Quiz (Lesson ) SRE. Find all solutions for sin cos if Find all solutions for cos if 0... Solve cos cos for all values of if is measured in. degrees.. Solve cos sin for all values of if is measured. in radians.. LIGHT The length of the shadow s cast b a 0-foot tree. depends on the angle of inclination of the sun,. Express s as a function of. Then find the angle of inclination that produces a shadow 0 feet long. Glencoe/McGraw-Hill 9 Glencoe lgebra

66 NME DTE PERID hapter Mid-hapter Test (Lessons through ) SRE Part I For Questions, write the letter for the correct answer in the blank at the right of each question. Use the graph shown at the right.. Find the period of the function.... D... Find the amplitude of the function.... D.. For Questions and, use the graph shown at the right.. Find the phase shift of the function.... D... Find the vertical shift of the function.... D.. ssessment. Which expression is equivalent to si n sec sec cos?.. csc. sin D. cos. Part II. Graph the function cos... Find the amplitude, if it exists, and period of the function. tan.. Find sin if cos and Simplif cos sin. 9. sec 0. Simplif cot sec. 0. csc. Verif that csc cot tan is an identit.. cot Glencoe/McGraw-Hill 9 Glencoe lgebra

67 NME DTE PERID hapter umulative Review (hapters ). Solve x 0 and graph its solution set. (Lesson -). 0. Use long division to find (x x x ) (x ).. (Lesson -). Write n n in quadratic form, if possible.. Then solve. (Lesson -). Express log 0 in terms of common logarithms. Then. approximate its value to four decimal places. (Lesson 0-). Find a in a geometric series for which S n, r, and. a n. (Lesson -). From a group of students and facult members, a. committee of is selected. Find the probabilit that all are students or all are facult. (Lesson -). Six coins are tossed. Find P(at least tails). (Lesson -).. Find one angle with positive measure and one angle with. negative measure coterminal with. (Lesson -) 9. Find the exact value of sin 0. (Lesson -) P, is located on the unit circle. Find sin and 0. cos. (Lesson -). Find the amplitude, if it exists, and period of the function. cos. (Lesson -). Find tan if cos and 0 0. (Lesson -).. Find the exact value of sin if sin and (Lesson -). Solve cos sin sin for all values of if is measured. in radians. (Lesson -) Glencoe/McGraw-Hill 9 Glencoe lgebra

68 NME DTE PERID Standardized Test Practice (hapters ) For Questions and, use the bar graph that shows the height, to the nearest hundred feet, of five mountains in Vermont s Green Mountain National Forest.. What is the difference in height between the highest and lowest of the given mountains?. ft. 0 ft Part : Multiple hoice Instructions: Fill in the appropriate oval for the best answer. Romance Mtn. Mt. braham Gillespie Mtn. Robert Frost Mtn. read Loaf Mtn.. 00 ft D.,000 ft. 0 0 Height (00 feet) D. What is the mean height of the given mountains? E. 00 ft F.. ft G. 0 ft H. 0 ft. E F G H. If x 0 and z, then xz.... D. 0.. tank that holds 00 gallons of water is filled at a rate of. gallons per minute. How long, to the nearest minute, will it take the tank to fill if it alread contains gallons of water? E. min F. 9 min G. min H. min. E F G D H ssessment. In the figure, the ratio of to is :. If the area of triangle is 0 cm,then.. cm. 0 cm. 0 cm D. cm. D. Line passes through the points (, ) and (, 0). Which point does not lie on line? E. (0, ) F. (, ) G., H. (, ). E F G H. The number 0 is divisible b which of the following? I. II. III.. I onl. I and II onl. I and III onl D. I, II, and III. D. In the figure, the length of arc is. What is the length of a radius of circle? E. F. G. H.. E F G H Glencoe/McGraw-Hill 9 Glencoe lgebra 0

69 NME DTE PERID Standardized Test Practice (continued) Part : Grid In Instructions: Enter our answer b writing each digit of the answer in a column box and then shading in the appropriate oval that corresponds to that entr. 9. The probabilit of randoml selecting a white marble from a bag is. The probabilit of 0 randoml selecting a red marble is. If the bag also contains 9 blue marbles, what is the total number of marbles in the bag? 0. If the mean of x, x, x, x, x, x, and x is, what is the mode?. Find the value of n in.. the figure if m.. atherine purchased a hammer for $, a rake for $, and a shovel for $ at a local hardware store. If the state sales tax rate is %, how much change did atherine receive from the $0 she gave to the cashier? m 0 n n / /... / / / /... / / Part : Quantitative omparison Instructions: ompare the quantities in columns and. Shade in if the quantit in column is greater; if the quantit in column is greater; if the quantities are equal; or D if the relationship cannot be determined from the information given. olumn olumn. The th term of the sequence The th term of the sequence.,,,,,,,, D.. a where a. D. Regular hexagon DEF. D F x E D x Glencoe/McGraw-Hill 9 Glencoe lgebra

70 NME DTE PERID Unit Test (hapters ). Solve if 90, 0, and b 0. Round measures of sides to the nearest tenth and measures of. angles to the nearest degree.. Rewrite in radian measure... Rewrite 9 radians in degree measure... Find one angle with positive measure and one angle with. negative measure coterminal with 0. SRE. Find the exact values of the six trigonometric functions of if the terminal side of in standard position contains the point (, )... Sketch the angle with measure radians. Then label its. reference angle. For Questions 0, find the exact value of each trigonometric function.. cot. sin 0. x ssessment 9. tan () 0. sin 0cos 0. Find the area of if, b 0 feet, and c feet. Round to the nearest tenth.. In,, a, and c 0. Determine whether has no solution, one solution or two solutions. Then solve the triangle. Round to the nearest tenth For Questions and, determine whether each triangle. should be solved b beginning with the Law of Sines or Law of osines. Then solve each triangle. Round to the nearest tenth.. a, b, c 0..,, a.. P, is located on the unit circle.. Find sin and cos.. Solve x rctan ()... Verif that s ec cot csc is an identit.. sin c sc Glencoe/McGraw-Hill 99 Glencoe lgebra

71 NME DTE PERID Unit Test (continued) (hapters ). Graph the function sin.. For Questions 9 and 0, find the amplitude, if it exists, and period of each function cos 0. tan 0.. State the phase shift of cos. Then graph the. function.. State the vertical shift and the equation of the midline for cos... Find sec if sin and Simplif s in se c cos cos.. For Questions and, find the exact value of each expression... cos. sin 9.. Verif that cos sin is an identit.. For Questions and 9, use the fact that cos and 0 90 to find the exact value of each expression... sin 9. cos The profit P for a product whose sales fluctuate with the t seasons is estimated to be P sin, where t is given in weeks and P is in thousands of dollars. Determine the number of weeks it would take for the profit to initiall reach $,000. Glencoe/McGraw-Hill 900 Glencoe lgebra 0.

72 NME DTE PERID Second Semester Test (hapters ) For Questions 0, write the letter for the correct answer in the blank at the right of each question.. Find the distance between (, ) and (, ). SRE... D... Write the equation 9x in standard form.. x ( ). x ( ) 9 9. x ( ) D. x ( ) Which sstem of inequalities is graphed?. x. x x x. x D. x. x x t t. Simplif ( t ). t. ( t ) t ( t ). 9. (t ) (t ) D.. x ssessment. Determine the values of x for an holes in the graph of the rational x function f(x) x. x. x, x. x. x, x D. x.. Solve m m. 0. m 0 or m. m. m or m 0 D. m or m.. Solve log n D... Use log 0.0 and log 0. to approximate the value of log D Write an equivalent logarithmic equation for e x.. ln x. ln x. x ln D. x ln Evaluate (k ). k. 0.. D. 0. Glencoe/McGraw-Hill 90 Glencoe lgebra

73 NME DTE PERID Second Semester Test (continued) (hapters ). Find the next two terms of the geometric sequence,,,..,. 9,., 0 D.,.. Find the fifth term of the sequence in which a and a n a n n.... D. 0.. password has three letters followed b three digits. How man different passwords are possible?.,,90.,,000.,,000 D.,,000.. The odds that an event will occur are :. What is the probabilit that the event will not occur?... D... n a geometr test, of the students earned an. Find the probabilit that of randoml-selected students earned an.... D... In a surve of 0 residents, % favored the expansion of the town librar. Find the margin of sampling error.. %. %. % D. %.. In, a, b, and c 0. Find D. 9.. Find the exact value of (cos 0)(tan 0).... D.. 9. Which equation is graphed?. cos. cos. sin D. sin Find csc if cos and D. 0. Glencoe/McGraw-Hill 90 Glencoe lgebra

74 NME DTE PERID Second Semester Test (continued) (hapters ). Write an equation for the parabola with focus (, ) and. directrix.. Write an equation for a circle with center at (0, ) and. radius unit.. Find the coordinates of the vertices and foci and the. equations of the asmptotes for the hperbola with equation 9 x 9. Then graph the hperbola. x. Write the equation x x in standard. form. Then state whether the graph of the equation is a parabola, circle, ellipse, or hperbola.. Find the LM of t 0 and t 0.. ssessment. State whether the equation. r represents a direct, joint,. p or inverse variation. Then name the constant of variation. t. Solve t t.. t t For Questions 0, solve each equation... m m 9. ln (x ) log log 9 log x 0.. Express log in terms of common logarithms. Then. approximate its value to four decimal places.. The half-life of carbon- is 0 ears. scientist. unearthed a fossil whose bones contained onl % as much carbon- as the would have contained when the animal was alive. Find the constant k for carbon- for t in ears, and write the equation for modeling this exponential deca. Then determine how long ago the animal died.. Find the three arithmetic means between and 0.. Glencoe/McGraw-Hill 90 Glencoe lgebra

75 NME DTE PERID Second Semester Test (continued) (hapters ). Find a in a geometric series for which S n, a n,. and r.. Write 0. as a fraction... Use Pascal s triangle to expand (x )... Find a counterexample to the statement. n n(n ).. How man was can ou choose three books from a locker. containing seven books? 9. Elias, lisa, and Drew each roll a die. What is the 9. probabilit that Elias rolls a, lisa rolls an even number, and Drew does not roll a or? 0. t a local gm with 00 members, 0 members take an 0. aerobics class, 00 members do weight training, and members do both weight training and take an aerobics class. What is the probabilit that a randoml-selected member takes an aerobics class or does weight training?. Determine whether the data {,,, 9,,,,,,,,,.,,, } appear to be positivel skewed, negativel skewed, or normall distributed.. n a multiple-choice quiz with eight questions, each. question has four answer choices. If Noreen randoml guesses at all eight questions, find P(more than correct).. Find the exact values of the six trigonometric functions of. if the terminal side of in standard position contains the point (, ).. Determine whether with, a 0, and b. has no solution, one solution, or two solutions. Then, if possible, solve the triangle. Round measures of sides to the nearest tenth and measures of angles to the nearest degree.. Verif that s in c csc ot tan cos is an identit... Find the exact value of cos if sin and Glencoe/McGraw-Hill 90 Glencoe lgebra

76 NME DTE PERID Final Test (hapters ) For Questions, write the letter for the correct answer in the blank at the right of each question.. The five fastest roller coasters in the world are Fujiama (Japan), Goliath (), Millennium Force (H), Steel Dragon 000 (Japan), and Superman the Escape (). The speeds, in miles per hour, of the first four coasters are,, 9, and 9, respectivel. How fast can Superman the Escape travel if the average speed of all five coasters is no more than 9 miles per hour? Source: World lmanac. no more than 00 mph. at least 9 mph SRE. at least 00 mph D. no more than 9 mph.. Write an equation of the line that passes through (9, ) and is perpendicular to the line whose equation is x.. x 9. x. x D. x.. Find x in the solution of the sstem x and x.... D. 0. ssessment. Find the coordinates of the vertices of the figure formed b x, x, and.. (0, 0), (, ), (, ). (, ), (, ), (, ). (, ), (, ), (, ) D. (, ), (, ), (, ).. Solve for.... D... The vertices of are (, ), (, ), and (, ). The triangle is rotated 90 counterclockwise. Use the rotation matrix 0 to find the 0 coordinates of.. (, ). (, ). (, ) D. (, ).. Simplif. x x. Simplif i. i 0 0. ssume that the denominator is not equal to 0... D. 0.. i. i. i D. i. Glencoe/McGraw-Hill 90 Glencoe lgebra

77 NME DTE PERID Final Test (continued) (hapters ) 9. Solve x x 0 b graphing. If exact roots cannot be found, state the consecutive integers between which the roots are located.. ; between 0 and. between 0 and ; between and., D. between 0 and ; between and Find the exact solutions to x x b using the Quadratic Formula D State the degree of 9 x x x x D. 0.. Which describes the number and tpe of roots of the equation x 0?. real root, imaginar root. real roots, imaginar roots. real roots D. real roots.. If g(x) x, find g[g( )] D... Which equation is graphed?. x x. x. x x D. x.. Write an equation for an ellipse if the endpoints of the major axis are at (, ) and (, ) and the endpoints of the minor axis are at (0, ) and (0, ). x. ( ). (x ). (x ) x D. ( ).. Find the exact solution(s) of the sstem x and x.. (, ), (, ), (, ), (, ). (, ), (, ). (, ), (, ), (, ), (, ) D. (, ), (, ). m n. Simplif n m.. m n 0n. m n m n m 0n. m n m D. n. x Glencoe/McGraw-Hill 90 Glencoe lgebra

78 NME DTE PERID Final Test (continued) (hapters ). If varies inversel as x and when x, find when x.... D.. 9. Write the equation in logarithmic form.. log. log. log D. log () Solve n 0. Round to four decimal places.. n 0.. n 0.. n. D. n Find S n for the arithmetic series in which a 9, n, and a n D. 0.. Find the sum of the infinite geometric series 9, if it exists.... D. does not exist.. Use the inomial Theorem to find the sixth term in the expansion of (m p).. m p. m p. m p D. mp. ssessment. How man four-digit numerical codes can be created if no digit ma be repeated?. 0, D bookshelf holds msteries, biographies, book of poetr, and reference books. If a book is selected at random from the shelf, find the probabilit that the book selected is a biograph or reference book.... D.. 0. Find the standard deviation of the data set to the nearest tenth. {,,,,,,,, } D.... Rewrite 00 in radian measure D Find the exact value of sin.... D.. Glencoe/McGraw-Hill 90 Glencoe lgebra

79 NME DTE PERID Final Test (continued) (hapters ) 9. Solve a and graph the solution set. 9. For Questions 0 and, use the data in the table below that shows the relationship between the distance traveled and the elapsed time for a trip. Time t (h) 0 Distance d (mi) Draw a scatter plot for the data. 0. Distance (mi) d 0 0. Use two ordered pairs to write a prediction equation. Then. use our prediction equation to predict the distance traveled in an elapsed time of hours.. lassif the sstem x 9 0 and x as consistent. and independent, consistent and dependent, or inconsistent. For Questions and, use the following information. manufacturer produces badminton and tennis rackets. The profit on each badminton racket is $0 and on each tennis racket is $. The manufacturer can make at most 00 rackets. f these, at least 00 rackets must be badminton rackets.. Let b represent the number of badminton rackets and. t represent the number of tennis rackets. Write a sstem of inequalities to represent the number of rackets that can be produced.. How man tennis rackets should the manufacturer produce. to maximize profit?. Solve the sstem of equations. x z 9. x z x z. Perform the indicated operations. If the matrix does not. exist, write impossible Evaluate 0 using expansion b minors.. Time (h) t Glencoe/McGraw-Hill 90 Glencoe lgebra

80 NME DTE PERID Final Test (continued) (hapters ) 0. Find the inverse of M, if it exists.. 9. Simplif (x 0 ) x (x ). ssume that no variable 9. equals Simplif. 0.. Write the radical t u using rational exponents... Solve x... Write a quadratic equation with and as its roots.. Write the equation in the form ax bx c 0, where a, b, and c are integers.. Write the equation x x in vertex form... Use snthetic substitution to find f() for. f(x) x x x.. List all of the possible rational zeros of. f(x) x x x. ssessment. Find the inverse of the function g(x) x... Graph x.. x 9. Write an equation for a circle if the endpoints of a diameter 9. are at (, ) and (, ). 0. Write an equation for the hperbola with vertices (0, ) and 0. (0, ) if the length of the conjugate axis is units.. Write the equation x x 9 in standard form.. Then state whether the graph of the equation is a parabola, circle, ellipse, or hperbola. Glencoe/McGraw-Hill 909 Glencoe lgebra

81 NME DTE PERID Final Test (continued) (hapters ) f g f g. Simplif... Determine the equations of an vertical asmptotes and the. x values of x for an holes in the graph of f(x) x. x. Solve m m.. m m m. Solve log n log log... In a certain lake, it is estimated that the fish population has. been doubling in size ever 0 weeks. Write an exponential growth equation of the form ae kt that models the growth of the fish population, where t is given in weeks, if the initial population was Find the eighth term of the arithmetic sequence in which. a and d.. Find the sum of the geometric series for which a 0,. a, and r. 9. Find the first three iterates x, x, x of f(x) x for an 9. initial value x How man different was can the letters of the word 0. MERI be arranged?. Three students are selected from a group of four male. students and six female students. Find the probabilit of selecting a male, a female, and another female in that order.. The heights of a group of high school students were found. to be normall distributed. The mean height was inches and the standard deviation was. inches. What percent of the students were between inches and 0 inches tall?. In,, a, and b. Determine whether. the triangle has no solution, one solution, or two solutions. Then solve the triangle. Round measure of sides to the nearest tenth and measures of angles to the nearest degree.. Find the value of cot os.. Glencoe/McGraw-Hill 90 Glencoe lgebra

82 Standardized Test Practice Student Record Sheet (Use with pages 0 of the Student Edition.) Glencoe/McGraw-Hill Glencoe lgebra NME DTE PERID nswers Select the best answer from the choices given and fill in the corresponding oval. 9 0 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 box. Then fill in the corresponding oval for that number or smbol. 9 Select the best answer from the choices given and fill in the corresponding oval. 0 D D D D D / / / / / / / / / / / / / / D D D D D D D D D D Part Short Response/Grid In Part Short Response/Grid In Part Multiple hoice Part Multiple hoice Part Quantitative omparison Part Quantitative omparison

83 nswers (Lesson -) Lesson - - 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 sine, cosine, 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 maximum and minimum values. Example Graph sin for 0 0. First make a table of values sin sin 0 sin Exercises Graph the following functions for the given domain.. cos, 0 0. tan, 0 tan cos What is the amplitude of each function?.. x x Glencoe/McGraw-Hill Glencoe lgebra - NME DTE PERID Stud Guide and Intervention (continued) Graphing Trigonometric Functions Variations of Trigonometric Functions For functions of the form a sin b and a cos b, the amplitude is a, mplitudes and the period is 0 b or b. and Periods For functions of the form a tan b, the amplitude is not defined, and the period is 0 or b b. Example a. cos Find the amplitude and period of each function. Then graph the function. First, find the amplitude. a, so the amplitude is. Next find the period Use the amplitude and period to help graph the function. cos b. tan The amplitude is not defined, and the period is Exercises Find the amplitude, if it exists, and period of each function. Then graph each function.. sin. tan amplitude: ; period or 0 no amplitude; period or Glencoe/McGraw-Hill Glencoe lgebra Glencoe/McGraw-Hill Glencoe lgebra

84 nswers (Lesson -) - NME DTE PERID Skills Practice Graphing Trigonometric Functions Glencoe/McGraw-Hill 9 Glencoe lgebra nswers Lesson - Find the amplitude, if it exists, and period of each function. Then graph each function.. cos. sin. sec ; 0 ; 0 no amplitude; tan. sin. csc no amplitude; 0 ; 0 no amplitude; tan. cos 9. sin no amplitude; 90 ; 0 ; NME DTE PERID - Practice (verage) Graphing Trigonometric Functions Find the amplitude, if it exists, and period of each function. Then graph each function.. sin. cot. cos ; 0 no amplitude; 0 ; csc. tan. sin ; 0 no amplitude; 0 no amplitude; FRE For Exercises and, use the following information. n anchoring cable exerts a force of 00 Newtons on a pole. The force has the horizontal and vertical components F x and F. ( force of one Newton (N), is the force that gives an acceleration of m/sec to a mass of kg.) 00 N F. The function F x 00 cos describes the relationship between the angle and the horizontal force. What are the amplitude and period of this function? 00; 0 F x. The function F 00 sin describes the relationship between the angle and the vertical force. What are the amplitude and period of this function? 00; 0 WETHER For Exercises 9 and 0, use the following information. The function 0 sin t, where t is in months and t 0 corresponds to pril, models the average high temperature in degrees Fahrenheit in enterville. 9. Determine the period of this function. What does this period represent? ; a calendar ear 0. What is the maximum high temperature and when does this occur? F; Jul Glencoe/McGraw-Hill 0 Glencoe lgebra Glencoe/McGraw-Hill Glencoe lgebra

85 nswers (Lesson -) Lesson - - NME 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 in our textbook. onsider the tides of the tlantic cean as a function of time. pproximatel 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 maximum and minimum points. false b. The amplitude of a function is the difference between its maximum and minimum values. false c. The amplitude of the function sin 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 g. The function sin has a period of. true h. The period of the function cot is. true i. The amplitude of the function sin is. false j. The period of the function csc is. false k. The graph of the function sin 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 0, 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 sin b and a cos b? Sample answer: The period of the functions sin 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. Glencoe/McGraw-Hill Glencoe lgebra - NME DTE PERID Enrichment lueprints 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. Example Find the unknown measures in the figure at the right. The measures x and are the legs of a right triangle. The measure of the hpotenuse is in. in. or 0 0 in. cos 0 sin 0.0 in. x 0. in. 0 " " Find the unknown measures of each of the following.. himne on roof. ir vent. Elbow joint ' ' x ' 9 ' D ' ' 0 ' 0 '.. 0 x. D. 0 0 t 9. r. Glencoe/McGraw-Hill Glencoe lgebra x r t " x Roofing Improvement 0 top view side view " metal fitting 0.09" Glencoe/McGraw-Hill Glencoe lgebra

86 nswers (Lesson -) - NME DTE PERID Stud Guide and Intervention Translations of Trigonometric Graphs Glencoe/McGraw-Hill Glencoe lgebra nswers Lesson - 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 sin 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. Example State the amplitude, period, and phase shift for cos. Then graph the function. mplitude: a or b Period: or Phase Shift: h The phase shift is to the right since 0. Exercises State the amplitude, period, and phase shift for each function. Then graph the function.. sin ( 0). tan ; 0; 0 to the left no amplitude; ; to the right cos ( ). sin ; 0; to the right ; ; to the right NME DTE PERID Stud Guide and Intervention (continued) Translations of Trigonometric Graphs Vertical Translations When a constant is added to a trigonometric function, the graph is shifted verticall. Vertical Shift The vertical shift of the graphs of the functions a sin 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 of a vertical shift is k. Step Determine the vertical shift, and graph the midline. Graphing Step Determine the amplitude, if it exists. Use dashed lines to indicate the maximum and Trigonometric minimum values of the function. Functions Step Determine the period of the function and graph the appropriate function. Step Determine the phase shift and translate the graph accordingl. Example State the vertical shift, equation of the midline, amplitude, and period for cos. Then graph the function. Vertical Shift: k, so the vertical shift is units down. The equation of the midline is. mplitude: a or Period: or 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 cosine curve, adjusted to have a period of. Exercises State the vertical shift, equation of the midline, amplitude, and period for each function. Then graph the function.. cos. sin up; ; ; down; ; ; Glencoe/McGraw-Hill Glencoe lgebra Glencoe/McGraw-Hill Glencoe lgebra

87 nswers (Lesson -) Lesson - - NME DTE PERID Skills Practice Translations of Trigonometric Graphs State the amplitude, period, and phase shift for each function. Then graph the function.. sin ( 90). cos ( ). tan ; 0; 90 ; 0; no amplitude; ; State the vertical shift, equation of the midline, amplitude, and period for each function. Then graph the function.. csc. cos. sec ; ; ; ; ; 0 ; ; ; 0 no amplitude; State the vertical shift, amplitude, period, and phase shift of each function. Then graph the function.. cos [( )]. sin [( 90)] 9. cot ; ; 0; ; ; 0; 90 ; no amplitude; ; Glencoe/McGraw-Hill Glencoe lgebra NME DTE PERID - Practice (verage) Translations of Trigonometric Graphs 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; ELGY For Exercises, 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 sin t, where t is the number of ears since the stand was first cut in November, 90.. How often does the insect population reach its maximum level? ever 0 r. When did the population last reach its maximum? 99. 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. LD PRESSURE For Exercises 9, use the following information. Jason s blood pressure is 0 over 0, meaning that the pressure oscillates between a maximum of 0 and a minimum of 0. Jason s heart rate is beats per minute. The function that represents Jason s blood pressure P can be modeled using a sine function with no phase shift.. 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 sin 0t Graph the function P Jason s lood Pressure Time Glencoe/McGraw-Hill Glencoe lgebra Pressure t Glencoe/McGraw-Hill Glencoe lgebra

88 nswers (Lesson -) - NME DTE PERID Reading to Learn Mathematics Translations of Trigonometric Graphs Glencoe/McGraw-Hill Glencoe lgebra nswers Lesson - Pre-ctivit How can translations of trigonometric graphs be used to show animal populations? Read the introduction to Lesson - at the top of page 9 in our textbook. ccording to the model given in our textbook, what would be the estimated rabbit population for Januar, 00? 00 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. sin ( 90) to the left b. sin 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. sin amplitude change and phase shift b. cos ( 0) 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. sin amplitude change, period change, phase shift, and vertical shift Helping You Remember. Man students have trouble remembering which of the functions sin ( ) and sin ( ) represents a shift to the left and which represents a shift to the right. Using, explain a good wa to remember which is which. Sample answer: lthough sine curves are infinitel repeating periodic graphs, think of sin x starting a period or ccle at (0, 0). Then sin ( ) starts earl at (), a shift of to the left, while sin ( ) starts late at, a shift of to the right. - NME DTE PERID Enrichment Translating Graphs of Trigonometric Functions Three graphs are shown at the right: sin sin ( 0) sin Replacing with ( 0) translates the graph to the right. Replacing with translates the graph units down. = sin 90 0 = sin ( 0 ) + = sin Example Graph one ccle of cos ( 0). Step Transform the equation into the form k a cos b( h). Step = cos cos ( ) Step Sketch cos. Step Translate cos to obtain the desired graph. Step = cos ( + ) = cos ( + ) Sketch these graphs on the same coordinate sstem. See students graphs.. sin ( ). sin. sin ( 90) n another piece of paper, graph one ccle of each curve. See students graphs.. sin ( 0). sin ( 90). cos ( 0). cos. The graphs for problems and should be the same. Use the sum formula for cosine of a sum to show that the equations are equivalent. cos ( 0) (cos )(cos 0) (sin )(sin 0) (cos )() (sin )(0) cos So, cos ( 0) and cos are equivalent. Glencoe/McGraw-Hill Glencoe lgebra Glencoe/McGraw-Hill Glencoe lgebra

89 nswers (Lesson -) Lesson - - 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 expression in the equation is defined. s in Quotient Identities tan cot c o s c o s s in asic Trigonometric Reciprocal Identities csc sec cot sin co s ta n Identities Pthagorean Identities cos sin tan sec cot csc Example Find the value of cot if csc ; 0 0. cot csc Trigonometric identit cot Substitute for csc. cot cot 9 cot Square Subtract from each side.. Take the square root of each side. Since is in the third quadrant, cot is positive, Thus cot. Exercises Find the value of each expression.. tan, if cot ; 0 0. csc, if cos ;0 90. cos, if sin ;0 90. sec, if sin ;0 90. cos, if tan ;90 0. tan, if sin ; sec, if cos ;90 0. sin, if cos ; cot, if csc 9 ; sin, if csc 9 9 ; 0 0 Glencoe/McGraw-Hill 9 Glencoe lgebra - NME DTE PERID Stud Guide and Intervention (continued) Trigonometric Identities Simplif Expressions The simplified form of a trigonometric expression is written as a numerical value or in terms of a single trigonometric function, if possible. n of the trigonometric identities on page 9 can be used to simplif expressions containing trigonometric functions. Example Simplif ( cos ) sec cot tan sec cos. s c co ( cos ) sec cot tan sec cos sin c co s s sin sin sin o s in in o s cos s se Example c c ot csc si n sec cot sin Simplif. c o s co s s in si n sin csc sin sin sin sin ( sin ) sin ( sin ) ( sin )( sin ) sin cos sin sin Exercises Simplif each expression... cos tan csc sec sin cot sec tan sin cot tan cot sin cos sec tan. cos. sin tan cos sin csc cot tan cos. cot sin tan csc. csc cos. tan cot sin csc cos sec 9. tan sin cos Glencoe/McGraw-Hill 0 Glencoe lgebra Glencoe/McGraw-Hill Glencoe lgebra

90 nswers (Lesson -) - NME DTE PERID Skills Practice Trigonometric Identities Glencoe/McGraw-Hill Glencoe lgebra nswers Lesson - Find the value of each expression.. sin, if cos and cos, if tan and 0 0. sec, if tan and cos, if tan and tan, if sin and 0 0. cos, if sec and 0 0. cos, if csc and 0 0. tan, if cos and cos, if cot and csc, if cos and cot, if csc and 0 0. tan, if sin and 0 0 Simplif each expression.. sin sec tan. csc sin. cot sec csc. c o s e s cos c. tan cot cos. csc tan tan sin cos sin sin 9. sin 0. csc cot co s sin sin sin cos cos tan sec. csc. sec NME DTE PERID - Find the value of each expression.. sin, if cos Practice (verage) Trigonometric Identities and sec, if sin and 0 0. cot, if cos 0 and 0 0. sin, if cot 9 9 and cot, if csc and 0 0. sec, if csc and 0 0. sec, if tan and 0 0. sin, if tan and cot, if tan and cot, if cos and 0 0 Simplif each expression.. csc tan sec x. sin tan cos. sin cot cos. cot csc csc. cot csc. csc cos co sin cot s cos sin cos sin. sin cos cot. 9. sec cos tan csc tan sec 0. ERIL PHTGRPHY The illustration shows a plane taking an aerial photograph of point. ecause the point is directl below the plane, there is no distortion in the image. For an point 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 exposure of the film reduces b (sin )(csc sin ). Express (sin )(csc sin ) in terms of cos onl. cos. TSUNMIS The equation a sin t represents the height of the waves passing a buo at a time t in seconds. Express a in terms of csc t. a csc t Glencoe/McGraw-Hill Glencoe lgebra Glencoe/McGraw-Hill 9 Glencoe lgebra

91 nswers (Lesson -) Lesson - - 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 in our textbook. 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.) cos sin cos h 0 0 Reading the Lesson. Match each expression from the list on the left with an expression from the list on the right that is equal to it for all values for which each expression is defined. (Some of the expressions from the list on the right ma be used more than once or not at all.) a. sec tan iii i. sin b. cot v ii. tan c. c s in o s ii iii. d. sin cos iii iv. sec e. csc i v. csc f. co iv vi. cot s g. c s o in s 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 sin and 0 0 tan s c in o s ; positive b. sec, if tan and 90 0 tan sec ;negative Helping You Remember. 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 sine and cosine that ou learned in hapter to help ou remember the Pthagorean identit cos sin? Sample answer: n a unit circle, x cos and sin. The equation of the unit circle is x, so this is equivalent to the equation cos sin. Glencoe/McGraw-Hill Glencoe lgebra - NME DTE PERID Enrichment 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 axis be toward the other focus. The polar equation of an ellipse is r e p a ep.since p b cos c c c a c a and b a c, c. ecause e c, a p a a c c a a e ( e ). Polar xis 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. r a ( e e ) co s Note that e is the eccentricit of the orbit and a is the length of the semi-major axis of the ellipse. lso, a is the mean distance of the planet from the sun. Example The mean distance of Venus from the sun is. 0 miles and the eccentricit of its orbit is.00. Find the minimum and maximum distances of Venus from the sun. The minimum distance occurs when.. 0 r ( 0.00 ). 0 miles 0.00 cos The maximum distance occurs when r ( 0.00 ).0 0 miles 0.00 cos 0 omplete each of the following.. The mean distance of Mars from the sun is. 0 miles and the eccentricit of its orbit is Find the minimum and maximum distances of Mars from the sun. max. distance.9 0 mi; min. distance. 0 mi. The minimum distance of Earth from the sun is 9. 0 miles and the eccentricit of its orbit is 0.0. Find the mean and maximum distances of Earth from the sun. max. distance mi; mean distance 9. 0 mi Glencoe/McGraw-Hill Glencoe lgebra r Glencoe/McGraw-Hill 0 Glencoe lgebra

92 nswers (Lesson -) - NME DTE PERID Stud Guide and Intervention Verifing Trigonometric Identities Glencoe/McGraw-Hill Glencoe lgebra nswers Lesson - 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 expression into the form of the simpler side. Example Verif that each of the following is an identit. sin tan a. sec cos b. cos sec cot csc Transform the left side. Transform the left side. sin sec cos tan cot cos sec csc sin s in cos c o s cos s c o s in cos sec sin cos sin cos cos sin sin cos sec cos cos cos sin cos cos sec cos cos cos sec cos cos cos sec sec Exercises Verif that each of the following is an identit.. csc cos csc sin cos csc sin cos sin csc csc sin csc csc sin cos cot cos. sin ( cos ) c s sin sin cos c s sin sin cos cos cos sin sin sin cos cos (sin ) sin o in s c s o s n cos (cos ) sin cos o in s ( cos ) ( cos )( cos ) cos sin cos sin cos sin cos sin cos sin cos sin cos sin cos sin - NME DTE PERID Stud Guide and Intervention (continued) Verifing Trigonometric Identities Transform oth Sides of an Equation The following techniques can be helpful in verifing trigonometric identities. Substitute one or more basic identities to simplif an expression. Factor or multipl to simplif an expression. Multipl both numerator and denominator b the same trigonometric expression. Write each side of the identit in terms of sine and cosine onl. Then simplif each side. Example tan sin tan sec sec sin c s in o s co tan Verif that sec tan is an identit. sin tan sec sec tan s cos sin cos cos c s in o s sin cos cos sin co Exercises s c os sin cos cos cos Verif that each of the following is an identit.. csc sec cot tan cos sin sin cos sin cos cos sin sin cos sin cos sin cos sin cos tan. cos s in c o s sin cos sec cos co s cos cos cos cot. sin cos c s o in s sin cos sin csc sin sec sin sin cos cos sin csc cot. sec cot ( cos ) s sin c s o in c s o in s cos cos cos cos cos cos sin (sin ) sin sin cos cos Glencoe/McGraw-Hill Glencoe lgebra i Glencoe/McGraw-Hill Glencoe lgebra

93 nswers (Lesson -) Lesson - - NME DTE PERID Skills Practice Verifing Trigonometric Identities Verif that each of the following is an identit.. tan cos sin tan cos sin c s in o s cos sin sin sin. cot tan cot tan c s o in s c s in o s. csc cos cot csc cos cot sin cos cot c o s cot s in cot cot sin. cos cos cos s co c o c in s s o cos s cos cos. (tan )( sin ) sin cos (tan )( sin ) sin cos tan cos sin cos c s in o s cos sin cos sin cos sin cos csc. cot sec c s s e c c sin co s cot cot c o s cot s in cot cot sin sin sin s in. tan s c c s in o tan in o tan s s tan tan tan cos. sin sin c os s in sin s s i n i sin n sin sin sin ( sin )( sin ) sin Glencoe/McGraw-Hill Glencoe lgebra NME DTE PERID - Practice (verage) Verifing Trigonometric Identities Verif that each of the following is an identit. sin cos cos. sec sin c o s c os sec cos sec sec sec cos sin. c o c c s s o o i n s s. ( sin )( sin ) cos ( sin )( sin ) cos sin cos cos cos. tan tan sec tan tan sec (tan ) sec (sec ) sec sec sec. cos cot cot cos cos cot cot cos cos cot c s cos cot o in s cos cos cot cos cot cos c o s s in cos cot cos cot cos cos sin sin (cos )( sin ) sin. (sin )(csc sec ) sec (sin )(csc sec ) sec (sin ) sin cos c s in o sec s sec tan sec sec sec. PRJETILES The square of the initial velocit of an object launched from the ground is v gh sin, where is the angle between the ground and the initial path, h is the maximum height reached, and g is the acceleration due to gravit. Verif the identit gh gh sec sin. sec g h g h gh gh s in c o s g h s ec s ec se c sec s e c. 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 Glencoe/McGraw-Hill Glencoe lgebra Glencoe/McGraw-Hill Glencoe lgebra

94 nswers (Lesson -) - NME DTE PERID Reading to Learn Mathematics Verifing Trigonometric Identities Glencoe/McGraw-Hill 9 Glencoe lgebra nswers Lesson - Pre-ctivit How can ou verif trigonometric identities? Read the introduction to Lesson - at the top of page in our textbook. For,0,or, sin sin. Does this mean that sin sin is an identit? Explain 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,sin, and sin. Reading the Lesson. Determine whether each equation is an identit or not an identit. a. sin tan identit b. sin c os not an identit ta n c. c s in o s c s o in s cos sin not an identit d. cos (tan ) identit e. c s in o s sin csc sec identit f. sin cos sin not an identit g. tan cos csc identit h. s s in e c ta n co not an identit t. Which of the following is not permitted when verifing an identit?. simplifing one side of the identit to match the other side. cross multipling if the identit is a proportion. simplifing each side of the identit separatel to get the same expression 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 sines and cosines. Then simplif each side as much as possible. - NME DTE PERID Enrichment 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. onsider an triangle. Let K represent the area of. Then K bc sin K b c s in Square both sides. c a b c ( b cos ) b c ( cos )( cos ) c b c b c a b c a b Use the law of cosines. c b b c a b c a a b c a b c Simplif. Let s a b c. Then s a b c a, s b a c b, s c a b c. K s(s a)(s b)(s c) Substitute. K s(s a)(s b)(s c) Heron s Formula The area of is a b c s(s a)(s b)(s c), where s. Use Heron s formula to find the area of.. a, b., c. a., b 0., c a., b 9.0, c.9. a 0., b., c 0..9 no such triangle. a, b, c 9. a 0.0, b 0.0, c 0.0, a., b.0, c.. a.0, b 9., c Glencoe/McGraw-Hill 0 Glencoe lgebra Glencoe/McGraw-Hill Glencoe lgebra

95 nswers (Lesson -) Lesson - - 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 expression like sin from the known values of sine and cosine of 0 and. Sum and The following identities hold true for all values of and. Difference cos ( ) cos cos sin sin of ngles sin ( ) sin cos cos sin Example a. cos Find the exact value of each expression. cos cos (00 ) cos 00 cos sin 00 sin b. sin (0) sin (0) sin ( 0) sin cos 0 cos sin 0 Exercises Find the exact value of each expression.. sin 0. cos. cos (). cos (). sin 9. cos 0. sin (). cos 9. cos () 0. sin. cos (0). sin 9 Glencoe/McGraw-Hill Glencoe lgebra - NME DTE PERID Stud Guide and Intervention (continued) Sum and Difference of ngles Formulas Verif Identities You can also use the sum and difference of angles formulas to verif identities. Example cos cos sin sin Verif that cos sin is an identit. cos sin riginal equation sin Sum of ngles Formula cos 0 sin () sin Evaluate each expression. sin sin Simplif. Example Verif that sin cos ( ) cos is an identit. sin cos ( ) cos riginal equation sin cos cos sin cos cos sin sin cos Sum and Difference of ngles Formulas sin 0 cos cos () sin 0 cos Evaluate each expression. cos cos Simplif. Exercises Verif that each of the following is an identit.. sin (90 ) cos sin 90 cos cos 90 sin cos cos 0 sin cos cos cos. cos (0 ) sin cos 0 cos sin 0 sin sin 0 cos () sin sin sin sin. sin cos sin sin cos cos sin cos cos cos sin cos. cos sin sin cos cos sin sin sin sin sin sin sin sin sin sin cos cos sin sin cos sin sin cos sin sin sin Glencoe/McGraw-Hill Glencoe lgebra Glencoe/McGraw-Hill Glencoe lgebra

96 nswers (Lesson -) - NME DTE PERID Skills Practice Sum and Difference of ngles Formulas Glencoe/McGraw-Hill Glencoe lgebra nswers Lesson - Find the exact value of each expression.. sin 0. cos (). sin (). cos. cos (). sin (). cos 0. sin 9. sin (9) Verif that each of the following is an identit. 0. sin (90 ) cos sin (90 ) cos sin 90 cos cos 90 sin cos cos 0 sin cos cos cos. sin (0 ) sin sin (0 ) sin sin 0 cos cos 0 sin sin 0 cos () sin sin sin sin. cos (0 ) sin cos (0 ) sin cos 0 cos sin 0 sin sin 0 cos () sin sin sin sin. cos ( 90) sin cos ( 90) sin cos cos 90 sin sin 90 sin (cos )(0) (sin )() sin sin sin. sin cos sin cos sin cos cos sin cos (sin )(0) (cos )() cos cos cos. cos ( ) cos cos ( ) cos cos cos sin sin cos cos 0 sin cos cos cos NME DTE PERID - Practice (verage) Sum and Difference of ngles Formulas Find the exact value of each expression.. cos. cos. sin (). sin (0). sin 0. cos 0. sin. sin () Verif that each of the following is an identit. 0. cos (0 ) cos cos (0 ) cos cos 0 cos sin 0 sin cos cos 0 sin cos cos cos. sin (0 ) sin sin (0 ) sin sin 0 cos cos 0 sin sin 0 cos sin sin sin sin 9. sin 9. sin ( ) sin ( ) sin sin ( ) sin ( ) sin cos cos sin (sin cos cos sin ) cos sin sin sin. cos x sin x sin x cos x sin x cos x cos sin x sin sin x cos cos x sin cos x sin x sin x cos x sin x. SLR ENERGY n March, the maximum amount of solar energ that falls on a square foot of ground at a certain location is given b E sin (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 ELETRIITY In Exercises and, use the following information. In a certain circuit carring alternating current, the formula i sin (0t) can be used to find the current i in amperes after t seconds. Sample answer:. Rewrite the formula using the sum of two angles. i sin (90t 0t). Use the sum of angles formula to find the exact current at t second. amperes Glencoe/McGraw-Hill Glencoe lgebra Glencoe/McGraw-Hill Glencoe lgebra

97 nswers (Lesson -) Lesson - - 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 in our textbook. onsider the functions sin x and sin x. Do the graphs of these two functions have constructive interference or destructive interference? constructive Reading the Lesson. Match each expression from the list on the left with an expression from the list on the right that is equal to it for all values of the variables. (Some of the expressions from the list on the right ma be used more than once or not at all.) a. sin ( ) v i. sin b. cos ( ) vi ii. sin cos cos sin c. sin (0 ) vii iii. cos d. sin (0 ) i iv. cos cos sin sin e. cos (0 ) iii v. sin cos cos sin f. sin ( ) ii vi. cos cos sin sin g. cos (90 ) i vii. sin h. cos ( ) iv viii. cos. Which expressions are equal to sin? (There ma be more than one correct choice.). sin cos 0 cos sin 0. sin cos 0 cos sin 0 and. sin 0 cos cos 0 sin D. cos 0 cos sin 0 sin 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 sine identities, the signs are the same on both sides. In the cosine identities, the signs are opposite on the two sides. Glencoe/McGraw-Hill Glencoe lgebra - NME DTE PERID Enrichment Identities for the Products of Sines and osines adding the identities for the sines of the sum and difference of the measures of two angles, a new identit is obtained. sin ( ) sin cos cos sin sin ( ) sin cos cos sin (i) sin ( ) sin ( ) sin cos This new identit is useful for expressing certain products as sums. Example Write sin cos as a sum. In the identit let and so that sin cos sin ( ) sin ( ). Thus, sin cos sin sin. subtracting the identities for sin ( ) and sin ( ), a similar identit for expressing a product as a difference is obtained. (ii) sin ( ) sin ( ) cos sin Solve.. Use the identities for cos ( ) and cos ( ) to find identities for expressing the products cos cos and sin sin as a sum or difference. cos cos cos ( ) cos ( ) sin sin cos ( ) cos ( ). Find the value of sin 0 cos without using tables. [sin (0 ) sin (0 )]; 0 ;. Express cos sin as a difference. cos sin sin sin cos sin sin sin Glencoe/McGraw-Hill Glencoe lgebra Glencoe/McGraw-Hill Glencoe lgebra

98 nswers (Lesson -) - NME DTE PERID Stud Guide and Intervention Double-ngle and Half-ngle Formulas Glencoe/McGraw-Hill Glencoe lgebra nswers Lesson - Double-ngle Formulas The following identities hold true for all values of. Double-ngle sin sin cos cos cos sin Formulas cos sin cos cos Example Find the exact values of sin and cos if 9 sin 0 and 0 0. First, find the value of cos. cos sin cos sin 9 9 sin cos 0 0 cos cos 0 9 Since is in the third quadrant, cos is negative. Thus cos 0. To find sin, use the identit sin sin cos. sin sin cos The value of sin is 0. To find cos, use the identit cos sin. cos sin The value of cos is 0. Exercises Find the exact values of sin and cos for each of the following.. sin,0 90,. sin, 0 0. cos, 0 0,. cos,90 0,,. sin, 0 0. cos,90 0, 9, 9 - NME DTE PERID Stud Guide and Intervention (continued) Double-ngle and Half-ngle Formulas Half-ngle Formulas Half-ngle The following identities hold true for all values of. Formulas sin cos cos cos Example Find the exact value of sin if sin and First find cos. cos sin cos sin cos sin cos Simplif. 9 cos Take the square root of each side. Since is in the second quadrant, cos. sin cos Half-ngle formula cos Simplif. Rationalize. Since is between 90 and 0, is between and 90. Thus sin is positive and equals. Exercises Find the exact value of sin and cos for each of the following.. cos, 0 0. cos,90 0, 0 0,. sin, 0 0. cos, , 0 0 0, 0 0 Find the exact value of each expression b using the half-angle formulas.. cos. sin.. cos Glencoe/McGraw-Hill Glencoe lgebra Glencoe/McGraw-Hill Glencoe lgebra

99 nswers (Lesson -) Lesson - - NME DTE PERID Skills Practice Double-ngle and Half-ngle Formulas Find the exact values of sin, cos, sin, and cos for each of the following.. cos,, sin,,,, 0 0,,. sin 0, 0,90 0. cos 9,, 9, 0 0 0, 9,,. cos,90 0. sin,, 0,, 9, 9 9,0 90, Find the exact value of each expression b using the half-angle formulas.. cos. sin 9. cos 0 0. sin. sin. cos Verif that each of the following is an identit.. sin t n n a t a sin sin cos ta se c ta ta n n n s in sin cos cos c o s sin cos sin cos Glencoe/McGraw-Hill 9 Glencoe lgebra NME DTE PERID - Practice (verage) Double-ngle and Half-ngle Formulas Find the exact values of sin, cos, sin, and cos for each of the following.. cos,0 90. sin 0 9, 9 9,, 0 9,,90 0 9,,. cos, 0 0. sin, 0 0,,, 0 9, 9,, Find the exact value of each expression b using the half-angle formulas.. tan 0. tan. cos.. sin Verif that each of the following is an identit. 9. sin tan s in ta n cos tan s in ; ta n t a n s in cos t a n t a n ; cos cos t a n t a n 0. sin cos sin cos sin cos sin cos sin () cos sin cos sin cos cos sin cos ( sin cos )(cos ) cos sin cos cos sin cos cos sin cos. ERIL PHTGRPHY In aerial photograph, there is a reduction in film exposure 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 exposure for the point directl below the camera. Using the identit sin cos, verif that E 0 cos E 0 cos E 0 cos E 0 (cos ) E 0 ( sin ) E 0 si n E 0 c os E 0 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 s c in o s Hsin cos H s in c os c os H ( cos ) co H s c in o s s H tan. s in Htan. Glencoe/McGraw-Hill 0 Glencoe lgebra Glencoe/McGraw-Hill Glencoe lgebra

100 nswers (Lesson -) - NME DTE PERID Reading to Learn Mathematics Double-ngle and Half-ngle Formulas Glencoe/McGraw-Hill Glencoe lgebra nswers Lesson - Pre-ctivit How can trigonometric functions be used to describe music? Read the introduction to Lesson - at the top of page 9 in our textbook. Suppose that the equation for the second harmonic is sin a. Then what would be the equations for the fundamental tone (first harmonic), third harmonic, fourth harmonic, and fifth harmonic? sin 0.a; sin.a; sin a; sin.a Reading the Lesson. Match each expression from the list on the left with all expressions from the list on the right that are equal to it for all values of. a. sin v i. sin cos b. cos ii and iii ii. sin c. cos iv iii. cos sin d. sin i iv. cos v. cos. Determine whether ou would use the positive or negative square root in the half-angle identities for sin and cos in each of the following situations. (Do not actuall calculate sin and cos.) a. sin, if cos and is in Quadrant I positive b. cos, if cos 0.9 and is in Quadrant II positive c. cos, if sin 0. and is in Quadrant III negative d. sin, if sin 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 using a Pthagorean identit? Sample answer: Just remember the identit cos cos sin. Using the Pthagorean identit cos sin, ou can substitute either sin for cos or cos for sin to get the other two identities for cos. - NME DTE PERID Enrichment lternating urrent 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, its motion is along the direction of the magnetic field between the poles. Therefore, no current is generated. However, through points and D, the motion of X is perpendicular to the magnetic field. The maximum current ma have a positive This induces maximum current in the coil. etween or negative value. and, and, 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 sine curve. The actual current, i, in a household current is given b i I M sin(0t ) where I M is the maximum value of the current, t is the elapsed time in seconds, and is the angle determined b the position of the coil at time t n. X i(amperes) D t(seconds) D Example If,find a value of t for which i 0. If i 0, then I M sin (0t ) 0. i I M sin(0t ) Since I M 0, sin(0t ) 0. If ab 0 and a 0, then b 0. Let 0t s. Thus, sin s 0. s because sin 0. 0t Substitute 0t for s. 0t Substitute for. Solve for t. 0 This solution is the first positive value of t that satisfies the problem. Using the equation for the actual current in a household circuit, i I M sin(0t ), solve each problem. For each problem, find the first positive value of t.. If 0, find a value of t for. If 0, find a value of t for which which i 0. t 0 i I M. t 0. If, find a value of t for which. If, find a value of t for which i I M. t 0 i 0. t 0 Glencoe/McGraw-Hill Glencoe lgebra Glencoe/McGraw-Hill 9 Glencoe lgebra

101 nswers (Lesson -) Lesson - - NME DTE PERID 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. Example Example Find all solutions of sin 0 for the interval 0 0. sin 0 sin sin sin 0, 0, 0, 0 Solve sin cos 0 for all values of. Give our answer in both radians and degrees. sin cos 0 sin cos cos 0 cos ( sin ) 0 cos 0 or sin 0 sin 90 k 0; 0 k 0, k 0 k 0; k, k Exercises Find all solutions of each equation for the given interval.. cos cos, 0. sin cos 0, 0,, 0,,,. cos,0 0. sin 0, 0,, 9,, Solve each equation for all values of if is measured in radians.. sin 0. cos sin cos 0 k, k, k k, k, k Solve each equation for all values of if is measured in degrees.. cos sin. tan k 90. k 0,. k 0 Glencoe/McGraw-Hill Glencoe lgebra - NME DTE PERID Stud Guide and Intervention (continued) Solving Trigonometric Equations Use Trigonometric Equations Example LIGHT Snell s law sas that sin. sin, where is the angle at which a beam of light enters water and is the angle at which the beam travels through the water. If a beam of light enters water at, at what angle does the light travel through the water? sin. sin riginal equation sin. sin sin si n. Divide each side b.. sin 0.0 Use a calculator. 0. Take the arcsin of each side. The light travels through the water at an angle of approximatel 0.. Exercises. -foot pipe is propped on a -foot tall packing crate that sits on level ground. ne foot of the pipe extends above the top of the crate and the other end rests on the ground. What angle does the pipe form with the ground?.9. t :00 P.M. one afternoon a 0-foot statue casts a shadow that is feet long. Write an equation to find the angle of elevation of the Sun at that time. Find the angle of elevation. tan 0 ;.. conveor belt is set up to carr packages from the ground into a window feet above the ground. The angle that the conveor belt forms with the ground is. How long is the conveor belt from the ground to the window sill?. ft SPRTS The distance a golf ball travels can be found using the formula v d 0 sin, where v 0 is the initial velocit of the ball, g is the acceleration due g to gravit (which is feet per second squared), and is the angle that the path of the ball makes with the ground.. How far will a ball travel hit 90 feet per second at an angle of?.9 ft. If a ball that traveled 00 feet had an initial velocit of 0 feet per second, what angle did the path of the ball make with the ground?. or.. Some children set up a teepee in the woods. The poles are feet long from their intersection to their bases, and the children want the distance between the poles to be feet at the base. How wide must the angle be between the poles?. Glencoe/McGraw-Hill Glencoe lgebra Glencoe/McGraw-Hill 0 Glencoe lgebra

102 nswers (Lesson -) - NME DTE PERID Skills Practice Solving Trigonometric Equations Glencoe/McGraw-Hill Glencoe lgebra nswers Lesson - Find all solutions of each equation for the given interval.. sin,0 0,. cos, tan, 0 0,. sin, 0,. sin sin 0,,. cos cos 0, 0, Solve each equation for all values of if is measured in radians.. cos cos. sin sin 0 0 k, k, and k k 9. sin sin cos 0 0. sin k k. cos cos. tan cos k k, k k, Solve each equation for all values of if is measured in degrees.. sin 0. cos 0 0 k 0 and 0 k 0 0 k 0 and 0 k 0. sin 0. cos k 0 and k 0 k 90. sin. cos 0 k 0 and 0 k 0 90 k 0 Solve each equation for all values of. 9. cos sin 0 0. sin sin 0 k and k, or k and k, or 0 k 0 and 0 k 0 k 0 and 0 k 0. sin sin. cos sec k,or 90 k 0 k, or k 0 NME DTE PERID - Practice (verage) Solving Trigonometric Equations Find all solutions of each equation for the given interval.. sin cos, cos sin,0 0 90, 0, 90,, 0. cos cos, 0 0. cos cos (90 ) 0, 0 0, 0, 00,. cos sin,. tan sec,, Solve each equation for all values of if is measured in radians.. cos sin. cot cot k k and k 9. sin sin 0. cos sin sin k, k, and k k. cos sin. sec k k Solve each equation for all values of if is measured in degrees.. sin cos cos. csc csc 0 90 k 0 0 k 0, 90k 0, and 0 k 0. ( cos ). cos cos cos 0 k 0 and 0 k 0 90 k 0 and 0 k 0 Solve each equation for all values of.. sin k and k,. sin 0 k and k, or 0 k 0 and 0 k 0 or 0 k 0 and 0 k 0 9. sin sin k, 0. cos sin 0 k and or 0 k 0 or k 0 and 0 k 0 k,. WVES Waves are causing a buo to float in a regular pattern in the water. The vertical position of the buo can be described b the equation h sin x. Write an expression that describes the position of the buo when its height is at its midline. k or k 0. ELETRIITY The electric current in a certain circuit with an alternating current can be described b the formula i sin 0t, where i is the current in amperes and t is the time in seconds. Write an expression that describes the times at which there is no current. 0.kt Glencoe/McGraw-Hill Glencoe lgebra Glencoe/McGraw-Hill Glencoe lgebra

103 nswers (Lesson -) Lesson - - NME 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 - at the top of page 99 in our textbook. Describe how ou could use a graphing calculator to determine the months in which the average dail high temperature is above 0F. (ssume that x represents Januar.) Specif the graphing window that ou would use. Sample answer: Graph the functions. sin (0.x.) 0.9 (using radian mode) and 0 on the same screen. Use the window [, ] b [0, 00] with Xscl and Yscl. Note the x values for which the curve is above the horizontal line. Reading the Lesson. Identif which equations have no solution., E, and G. sin. tan sec D. csc E. cos.0 F. cot 000 G. cos H. sec. 0 I. sin Use a trigonometric identit to write the first step in the solution of each trigonometric equation. (Do not complete the solution.) a. tan cos sin,0 tan b. sin sin 0, 0 0 (sin ) 0 c. cos sin, 0 0 sin sin d. sin cos, 0 sin 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 explain it to someone else. How would ou explain 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 expression 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 expressions, but it also requires finding the angles for which a trigonometric function has a particular value. Glencoe/McGraw-Hill Glencoe lgebra - NME DTE PERID Enrichment Families of urves Use these graphs for the problems below. The Famil x n The Famil e mx n = 0 n = n = n = n = n = n = 0 m = m = m = m = m = m = m = m = m = m = m = x x. Use the graph on the left to describe the relationship among the curves x, x, and x. For n and n, the graphs are reflections of one another in the line with equation x.. Graph x n for n, 0,,and 0 on the grid with x, x, and x. See students graphs.. Which two regions in the first quadrant contain no points of the graphs of the famil for x n? {(x, )x > and 0 < < } and {(x, )0 < x < and > }. n the right grid, graph the members of the famil e mx for which m and m. See students graphs.. Describe the relationship among these two curves and the -axis. the graphs for m and m are reflections in the -axis.. Graph e mx for m 0,,,, and. See students graphs. Glencoe/McGraw-Hill Glencoe lgebra Glencoe/McGraw-Hill Glencoe lgebra