Chapter 7 Resource Masters

Transcription

1 Chapter Resource Masters New York, New York Columbus, Ohio Woodland Hills, California Peoria, Illinois

2 StudentWorks TM This CD-ROM includes the entire Student Edition along with the Study Guide, Practice, and Enrichment masters. TeacherWorks TM All of the materials found in this booklet are included for viewing and printing in the Advanced Mathematical Concepts TeacherWorks CD-ROM. Copyright The McGraw-Hill Companies, Inc. All rights reserved. Printed in the United States of America. Permission is granted to reproduce the material contained herein on the condition that such material be reproduced only for classroom use; be provided to students, teachers, and families without charge; and be used solely in conjunction with Glencoe Advanced Mathematical Concepts. Any other reproduction, for use or sale, is prohibited without prior written permission of the publisher. Send all inquiries to: Glencoe/McGraw-Hill 88 Orion Place Columbus, OH 0-0 ISBN: Advanced Mathematical Concepts Chapter Resource Masters XXX

3 Vocabulary Builder vii-viii Lesson - Study Guide Practice Enrichment Lesson - Study Guide Practice Enrichment Lesson - Study Guide Practice Enrichment Lesson - Study Guide Practice Enrichment Lesson - Study Guide Practice Enrichment Lesson -6 Study Guide Practice Enrichment Contents Lesson - Study Guide Practice Enrichment Chapter Assessment Chapter Test, Form A Chapter Test, Form B Chapter Test, Form C Chapter Test, Form A Chapter Test, Form B Chapter Test, Form C Chapter Etended Response Assessment Chapter Mid-Chapter Test Chapter Quizzes A & B Chapter Quizzes C & D Chapter SAT and ACT Practice Chapter Cumulative Review SAT and ACT Practice Answer Sheet, 0 Questions A SAT and ACT Practice Answer Sheet, 0 Questions A ANSWERS A-A6 Glencoe/McGraw-Hill iii Advanced Mathematical Concepts

4 A Teacher s Guide to Using the Chapter Resource Masters The Fast File Chapter Resource system allows you to conveniently file the resources you use most often. The Chapter Resource Masters include the core materials needed for Chapter. These materials include worksheets, etensions, and assessment options. The answers for these pages appear at the back of this booklet. All of the materials found in this booklet are included for viewing and printing in the Advanced Mathematical Concepts TeacherWorks CD-ROM. Vocabulary Builder Pages vii-viii include a student study tool that presents the key vocabulary terms from the chapter. Students are to record definitions and/or eamples for each term. You may suggest that students highlight or star the terms with which they are not familiar. When to Use Give these pages to students before beginning Lesson -. Remind them to add definitions and eamples as they complete each lesson. Practice There is one master for each lesson. These problems more closely follow the structure of the Practice section of the Student Edition eercises. These eercises are of average difficulty. When to Use These provide additional practice options or may be used as homework for second day teaching of the lesson. Study Guide There is one Study Guide master for each lesson. When to 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 those students who have been absent. Enrichment There is one master for each lesson. These activities may etend the concepts in the lesson, offer a historical or multicultural look at the concepts, or widen students perspectives on the mathematics they are learning. These are not written eclusively for honors students, but are accessible for use with all levels of students. When to Use These may be used as etra credit, short-term projects, or as activities for days when class periods are shortened. Glencoe/McGraw-Hill iv Advanced Mathematical Concepts

5 Assessment Options The assessment section of the Chapter Resources Masters offers a wide range of assessment tools for intermediate and final assessment. The following lists describe each assessment master and its intended use. Intermediate Assessment A Mid-Chapter Test provides an option to assess the first half of the chapter. It is composed of free-response questions. Four free-response quizzes are included to offer assessment at appropriate intervals in the chapter. Chapter Assessments Chapter Tests Forms A, B, and C Form tests contain multiple-choice questions. Form A is intended for use with honors-level students, Form B is intended for use with averagelevel students, and Form C is intended for use with basic-level students. These tests are similar in format to offer comparable testing situations. Forms A, B, and C Form tests are composed of free-response questions. Form A is intended for use with honors-level students, Form B is intended for use with average-level students, and Form C is intended for use with basic-level students. These tests are similar in format to offer comparable testing situations. All of the above tests include a challenging Bonus question. The Etended Response Assessment includes performance assessment tasks that are suitable for all students. A scoring rubric is included for evaluation guidelines. Sample answers are provided for assessment. Continuing Assessment The SAT and ACT Practice offers continuing review of concepts in various formats, which may appear on standardized tests that they may encounter. This practice includes multiple-choice, quantitativecomparison, and grid-in questions. Bubblein and grid-in answer sections are provided on the master. The Cumulative Review provides students an opportunity to reinforce and retain skills as they proceed through their study of advanced mathematics. It can also be used as a test. The master includes free-response questions. Answers Page A is an answer sheet for the SAT and ACT Practice questions that appear in the Student Edition on page 8. Page A is an answer sheet for the SAT and ACT Practice master. These improve students familiarity with the answer formats they may encounter in test taking. The answers for the lesson-by-lesson masters are provided as reduced pages with answers appearing in red. Full-size answer keys are provided for the assessment options in this booklet. Glencoe/McGraw-Hill v Advanced Mathematical Concepts

6 Chapter Leveled Worksheets Glencoe s leveled worksheets are helpful for meeting the needs of every student in a variety of ways. These worksheets, many of which are found in the FAST FILE Chapter Resource Masters, are shown in the chart below. Study Guide masters provide worked-out eamples as well as practice problems. Each chapter s Vocabulary Builder master provides students the opportunity to write out key concepts and definitions in their own words. Practice masters provide average-level problems for students who are moving at a regular pace. Enrichment masters offer students the opportunity to etend their learning. Five Different Options to Meet the Needs of Every Student in a Variety of Ways primarily skills primarily concepts primarily applications BASIC AVERAGE ADVANCED Study Guide Vocabulary Builder Parent and Student Study Guide (online) Practice Enrichment Glencoe/McGraw-Hill vi Advanced Mathematical Concepts

7 Chapter Reading to Learn Mathematics Vocabulary Builder This is an alphabetical list of the key vocabulary terms you will learn in Chapter. As you study the chapter, complete each term s definition or description. Remember to add the page number where you found the term. Vocabulary Term countereample Found on Page Definition/Description/Eample difference identity double-angle identity half-angle identity identity normal form normal line opposite-angle identity principal value Pythagorean identity (continued on the net page) Glencoe/McGraw-Hill vii Advanced Mathematical Concepts

8 Chapter Reading to Learn Mathematics Vocabulary Builder (continued) Vocabulary Term quotient identity Found on Page Definition/Description/Eample reciprocal identity reduction identity sum identity symmetry identity trigonometric identity Glencoe/McGraw-Hill viii Advanced Mathematical Concepts

9 - Study Guide Basic Trigonometric Identities You can use the trigonometric identities to help find the values of trigonometric functions. Eample If sin,find tan. Use two identities to relate sin and tan. sin cos Pythagorean identity cos Substitute cos 6 cos 6 or for sin. Now find tan. tan sin c os Quotient identity tan tan To determine the sign of a function value, use the symmetry identities for sine and cosine. To use these identities with radian measure, replace 80 with and 60 with. Case : sin (A 60k) sin A cos (A 60k) cos A Case : sin [A 80(k )] sin A cos [A 80(k )] cos A Case : sin (60k A) sin A cos (60k A) cos A Case : sin [80(k ) A] sin A cos [80(k ) A] cos A Eample Epress tan as a trigonometric function of an angle in Quadrant I. The sum of and, which is or, is a multiple of. () Case, with A and k tan sin cos sin () cos () Quotient identity sin cos tan Symmetry identities Quotient identity Glencoe/McGraw-Hill Advanced Mathematical Concepts

10 - Practice Basic Trigonometric Identities Use the given information to determine the eact trigonometric value if If cos,find tan.. If sin,find cos.. If tan,find sin.. If tan, find cot. Epress each value as a trigonometric function of an angle in Quandrant I.. cos csc 9. sin Simplify each epression. 8. cos sin tan 9. cot A t an A 0. sin cos cos. Kite Flying Brett and Tara are flying a kite. When the string is tied to the ground, the height of the kite can be determined by the formula H L csc, where L is the length of the string and is the angle between the string and the level ground. What formula could Brett and Tara use to find the height of the kite if they know the value of sin? Glencoe/McGraw-Hill 6 Advanced Mathematical Concepts

11 - Enrichment The Physics of Soccer Recall from Lesson - that the formula for the maimum height h of a projectile is h v 0 sin g, where is the measure of the angle of elevation in degrees, v 0 is the initial velocity in feet per second, and g is the acceleration due to gravity in feet per second squared. Solve. Give answers to the nearest tenth.. A soccer player kicks a ball at an initial velocity of 60 ft/s and an angle of elevation of 0. The acceleration due to gravity is ft/s.find the maimum height reached by the ball.. With what initial velocity must you kick a ball at an angle of in order for it to reach a maimum height of 0 ft? The distance d that a projected object travels is given by the formula v 0 sin cos g d.. Find the distance traveled by the ball described in Eercise. In order to kick a ball the greatest possible distance at a given v initial velocity, a soccer player must maimize 0 sin cos d g. Since, v 0, and g are constants, this means the player must maimize sin cos. sin 0 cos 0 sin 90 cos 90 0 sin 0 cos 0 sin 80 cos sin 0 cos 0 sin 0 cos Use the patterns in the table to hypothesize a value of for which sin cos will be maimal. Use a calculator to check your hypothesis. At what angle should the player kick the ball to achieve the greatest distance? Glencoe/McGraw-Hill Advanced Mathematical Concepts

12 - Study Guide Verifying Trigonometric Identities When verifying trigonometric identities, you cannot add or subtract quantities from each side of the identity. An unverified identity is not an equation, so the properties of equality do not apply. Eample Verify that se c sin is an identity. sec Since the left side is more complicated, transform it into the epression on the right. se c sec sin (tan ) sec sin sec tan t an sin sec Simplify. sin c sin tan os sin c, sec os cos cos sin c cos os sin sin sin Multiply. The techniques that you use to verify trigonometric identities can also be used to simplify trigonometric equations. Eample Find a numerical value of one trigonometric function of if cos csc. You can simplify the trigonometric epression on the left side by writing it in terms of sine and cosine. cos csc cos sin csc sin c os Multiply. sin cot cot c os sin Therefore, if cos csc, then cot. Glencoe/McGraw-Hill 8 Advanced Mathematical Concepts

13 - Practice Verifying Trigonometric Identities Verify that each equation is an identity.. csc cot cos tan. sin y sin y sec y. sin cos ( sin cos )(sin cos ). tan cos sec sin Find a numerical value of one trigonometric function of.. sin cot 6. sin cos. cos cot 8. Physics The work done in moving an object is given by the formula W Fd cos, where d is the displacement, F is the force eerted, and is the angle between the displacement and the force. Verify that W Fd cot c is an equivalent formula. sc Glencoe/McGraw-Hill 9 Advanced Mathematical Concepts

14 - Enrichment Building from By starting with the most fundamental identity of all,, you can create new identities as comple as you would like them to be. First, think of ways to write using trigonometric identities. Some eamples are the following. cos A sec A csc A cot A cos (A 60 ) cos (60 A) Choose two such epressions and write a new identity. cos A sec A csc A cot A Now multiply the terms of the identity by the terms of another identity of your choosing, preferably one that will allow some simplification upon multiplication. cos A sec A csc A cot A sin A cos A tan A sin A sec A tan A csc A cot A Beginning with, create two trigonometric identities... Verify that each of the identities you created is an identity... Glencoe/McGraw-Hill 80 Advanced Mathematical Concepts

15 - Study Guide Sum and Difference Identities You can use the sum and difference identities and the values of the trigonometric functions of common angles to find the values of trigonometric functions of other angles. Notice how the addition and subtraction symbols are related in the sum and difference identities. Cosine function Sine function Tangent function Sum and Difference Identities cos ( ) cos cos sin sin sin ( ) sin cos cos sin tan tan tan ( ) tan tan Eample Eample Use the sum or difference identity for cosine to find the eact value of cos. 60 cos cos Symmetry identity, Case cos cos (60 ) 60 and are two common angles that differ by. cos cos 60 cos sin 60 sin Difference identity for cosine cos or 6 Find the value of sin ( y) if 0,0 y, sin, and sin y. In order to use the sum identity for sine, you need to know cos and cos y. Use a Pythagorean identity to determine the necessary values. sin cos cos sin. Pythagorean identity Since it is given that the angles are in Quadrant I, the values of sine and cosine are positive. Therefore, cos sin. cos cos y 6 or 6 9 or Now substitute these values into the sum identity for sine. sin ( y) sin cos y cos sin y or 8 Glencoe/McGraw-Hill 8 Advanced Mathematical Concepts

16 - Practice Sum and Difference Identities Use sum or difference identities to find the eact value of each trigonometric function.. cos. sin (6). tan. csc 9. tan 6. sec Find each eact value if 0 and 0 y.. cos ( y) if sin and sin y 8. sin ( y) if cos 8 and cos y 9. tan ( y) if csc and cot y Verify that each equation is an identity. 0. cos (80 ) cos. sin (60 ) sin. Physics Sound waves can be modeled by equations of the form y 0 sin (t ). Determine what type of interference results when sound waves modeled by the equations y 0 sin (t 90) and y 0 sin (t 0) are combined. (Hint: Refer to the application in Lesson -.) Glencoe/McGraw-Hill 8 Advanced Mathematical Concepts

17 - Enrichment Identities for the Products of Sines and Cosines By adding the identities for the sines of the sum and difference of the measures of two angles, a new identity is obtained. sin ( ) sin cos cos sin sin ( ) sin cos cos sin (i) sin ( ) sin ( ) sin cos This new identity is useful for epressing certain products as sums. Eample Write sin cos as a sum. In the right side of identity (i) let and so that sin cos sin ( ) sin ( ). Thus, sin cos sin sin. Eample By subtracting the identities for sin ( ) and sin ( ), you obtain a similar identity for epressing a product as a difference. (ii) sin ( ) sin ( ) cos sin Verify the identity c os sin sin cos (sin sin ) sin. sin In the right sides of identities (i) and (ii) let and. Then write the following quotient. cos sin sin cos By simplifying and multiplying by the conjugate, the identity is verified. sin sin. cos sin sin cos sin ( ) sin ( ) sin ( ) sin ( ) sin sin (sin sin ) sin sin sin sin sin sin Complete.. Use the identities for cos ( ) and cos ( ) to find identities for epressing the products cos cos and sin sin as a sum or difference.. Find the value of sin 0 cos by using the identity above. Glencoe/McGraw-Hill 8 Advanced Mathematical Concepts

18 - Study Guide Double-Angle and Half-Angle Identities Eample If sin and has its terminal side in the first quadrant, find the eact value of sin. To use the double-angle identity for sin, we must first find cos. sin cos cos cos 6 cos Now find sin. sin sin cos sin Double-angle identity for sine sin, cos 8 Eample Use a half-angle identity to find the eact value of sin. sin sin 6 c o s. a Since is in Quadrant I, choose the positive sine value. Use sin cos 6 Glencoe/McGraw-Hill 8 Advanced Mathematical Concepts

19 - Practice Double-Angle and Half-Angle Identities Use a half-angle identity to find the eact value of each function.. sin 0. tan 8. cos 8 Use the given information to find sin, cos, and tan.. sin,0 90. tan, 6. sec,. sin,0 Verify that each equation is an identity. 8. sin (sin cos ) 9. cos sin sin 0. Baseball A batter hits a ball with an initial velocity v 0 of 00 feet per second at an angle to the horizontal. An outfielder catches the ball 00 feet from home plate. Find if the range of a projectile is given by the formula R v 0 sin. Glencoe/McGraw-Hill 8 Advanced Mathematical Concepts

20 - Enrichment Reading Mathematics: Using Eamples Most mathematics books, including this one, use eamples to illustrate the material of each lesson. Eamples are chosen by the authors to show how to apply the methods of the lesson and to point out places where possible errors can arise.. Eplain the purpose of Eample c in Lesson -.. Eplain the purpose of Eample in Lesson -.. Eplain the purpose of Eample in Lesson -. To make the best use of the eamples in a lesson, try following this procedure: a. When you come to an eample, stop. Think about what you have just read. If you don t understand it, reread the previous section. b. Read the eample problem. Then instead of reading the solution, try solving the problem yourself. c. After you have solved the problem or gone as far as you can go, study the solution given in the tet. Compare your method and solution with those of the authors. If necessary, find out where you went wrong. If you don t understand the solution, reread the tet or ask your teacher for help.. Eplain the advantage of working an eample yourself over simply reading the solution given in the tet. Glencoe/McGraw-Hill 86 Advanced Mathematical Concepts

21 - Study Guide Solving Trigonometric Equations When you solve trigonometric equations for principal values of, is in the interval for sin and tan. For cos, is in the interval If an equation cannot be solved easily by factoring, try writing the epressions in terms of only one trigonometric function. Eample Eample Solve tan cos cos 0 for principal values of. Epress solutions in degrees. tan cos cos 0 cos (tan ) 0 Factor. cos 0 or tan 0 Set each factor equal to tan When 90, tan is undefined, so the only principal value is. Solve tan sec tan for 0. This equation can be written in terms of tan only. tan sec tan tan (tan ) tan sec tan tan tan Simplify. tan tan 0 (tan ) 0 Factor. tan 0 Take the square root of each side. tan or When you solve for all values of, the solution should be represented as 60k or k for sin and cos and 80k or k for tan, where k is any integer. Eample Solve sin sin for all real values of. sin sin sin 0 sin sin k or k, where k is any integer The solutions are k and k. Glencoe/McGraw-Hill 8 Advanced Mathematical Concepts

22 - Practice Solving Trigonometric Equations Solve each equation for principal values of. Epress solutions in degrees.. cos cos. sin 0 Solve each equation for sec tan 0. cos cos 0 Solve each equation for 0.. sin sin 0 6. cos sin Solve each equation for all real values of.. cos cos 8. sin sin 0 9. sec 0 0. tan (tan ) 0. Aviation An airplane takes off from the ground and reaches a height of 00 feet after flying miles. Given the formula H d tan, where H is the height of the plane and d is the distance (along the ground) the plane has flown, find the angle of ascent at which the plane took off. Glencoe/McGraw-Hill 88 Advanced Mathematical Concepts

23 - The Spectrum Enrichment In some ways, light behaves as though it were composed of waves. The wavelength of visible light ranges from about 0 cm for violet light to about 0 cm for red light. As light passes through a medium, its velocity depends upon the wavelength of the light. The greater the wavelength, the greater the velocity. Since white light, including sunlight, is composed of light of varying wavelengths, waves will pass through the medium at an infinite number of different speeds. The inde of refraction n of c the medium is defined by n v, where c is the velocity of light in a vacuum ( 0 0 cm/s), and v is the velocity of light in the medium. As you can see, the inde of refraction of a medium is not a constant. It depends on the wavelength and the velocity of light passing through it. (The inde of refraction of diamond given in the lesson is an average.). For all media, n >. Is the speed of light in a medium greater than or less than c? Eplain.. A beam of violet light travels through water at a speed of. 0 0 cm/s. Find the inde of refraction of water for violet light. The diagram shows why a prism splits white light into a spectrum. Because they travel at different velocities in the prism, waves of light of different colors are refracted different amounts.. Beams of red and violet light strike crown glass at an angle of 0. Use Snell s Law to find the difference between the angles of refraction of the two beams. violet light: n. red light: n. Glencoe/McGraw-Hill 89 Advanced Mathematical Concepts

24 -6 Normal Form of a Linear Equation Normal Form You can write the standard form of a linear equation if you are given the values of and p. Eample Study Guide The normal form of a linear equation is cos y sin p 0, where p is the length of the normal from the line to the origin and is the positive angle formed by the positive -ais and the normal. Write the standard form of the equation of a line for which the length of the normal segment to the origin is and the normal makes an angle of with the positive -ais. cos y sin p 0 Normal form cos y sin 0 and p y 0 y 0 0 Multiply each side by. The standard form of the equation is y 0 0. The standard form of a linear equation, A By C 0, can be changed to the normal form by dividing each term of the equation by A B. The sign is chosen opposite the sign of C. You can then find the length of the normal, p units, and the angle. Eample Write y 0 0 in normal form. Then find the length of the normal and the angle it makes with the positive -ais. Since C is negative, use A B to determine the normal form. A B or The normal form is y 0 0 or y 0. Therefore, cos, sin, and p. Since cos and sin are both positive, must lie in Quadrant I. tan or tan s in cos The normal segment has length units and makes an angle of with the positive -ais. Glencoe/McGraw-Hill 90 Advanced Mathematical Concepts

25 -6 Practice Normal Form of a Linear Equation Write the standard form of the equation of each line, given p, the length of the normal segment, and, the angle the normal segment makes with the positive -ais.. p, 0. p,. p, 60. p 8, 6. p, 6. p, Write each equation in normal form. Then find the length of the normal and the angle it makes with the positive -ais.. y 0 8. y 0 9. y 0 0. y. y 0. y 0 Glencoe/McGraw-Hill 9 Advanced Mathematical Concepts

26 -6 Enrichment Slopes of Perpendicular Lines The derivation of the normal form of a linear equation uses this familiar theorem, first stated in Lesson -6: Two nonvertical lines are perpendicular if and only if the slope of one is the negative reciprocal of the slope of the other. You can use trigonometric identities to prove that if lines are perpendicular, then their slopes are negative reciprocals of each other. and are perpendicular lines. and are the angles that and, respectively, make with the horizontal. Let m slope of m slope of Complete the following eercises to prove that m.. Eplain why m tan and m tan. m. According to the difference identity for the cosine function, cos ( ) cos cos sin sin. Eplain why the left side of the equation is equal to zero.. cos cos sin sin 0 sin sin cos cos sin cos cos sin Complete using the tangent function. Complete, using m and m. Glencoe/McGraw-Hill 9 Advanced Mathematical Concepts

27 - Study Guide Distance from a Point to a Line The distance from a point at (, y ) to a line with equation A By C 0 can be determined by using the formula d A By. The sign of the radical is chosen opposite A B C the sign of C. Eample Eample Find the distance between P(, ) and the line with equation y 0. First, rewrite the equation of the line in standard form. y 0 0 Then, use the formula for the distance from a point to a line. A d By C A B () () 0 d A, B, C 0,, y d 0 or Since C is negative, use A B. d. units Therefore, P is approimately. units from the line with equation y 0. Since d is positive, P is on the opposite side of the line from the origin. You can also use the formula to find the distance between two parallel lines. To do this, choose any point on one of the lines and use the formula to find the distance from that point to the other line. Find the distance between the lines with equations y and y. Since y is in slope-intercept form, you can see that it passes through the point at (0, ). Use this point to find the distance to the other line. The standard form of the other equation is y 0. A d By C A B (0) () d A, B, C, 0, y () d or Since C is negative, use A B..06 The distance between the lines is about.06 units. Glencoe/McGraw-Hill 9 Advanced Mathematical Concepts

28 - Practice Distance From a Point to a Line Find the distance between the point with the given coordinates and the line with the given equation.. (, ), y 0. (, ), y 0. (, ), y 0. (, ), 6 8y 0 Find the distance between the parallel lines with the given equations.. y 0 6. y 0 y y 8 0 Find equations of the lines that bisect the acute and obtuse angles formed by the lines with the given equations.. y 0 y y 0 0 y 6 0 Glencoe/McGraw-Hill 9 Advanced Mathematical Concepts

29 - Enrichment Deriving the Point-Line Distance Line has the equation A By C 0. Answer these questions to derive the formula given in Lesson - for the distance from P(, y ) to.. Use the equation of the line to find the coordinates of J and K, the - and y-intercepts of.. PQ is a vertical segment from P to. Find the -coordinate of Q.. Since Q is on, its coordinates must satisfy the equation of. Use your answer to Eercise to find the y-coordinate of Q.. Find PQ by finding the difference between the y-coordinates of P and Q. Write your answer as a fraction.. Triangle KJO is a right triangle. Use your answers to Eercise and the Pythagorean Theorem to find KJ. Simplify. 6. Since Q K, JKO ~ PQR. PR OJ PQ KJ Use your answers to Eercises,, and to find PR, the distance from P to. Simplify. Glencoe/McGraw-Hill 9 Advanced Mathematical Concepts

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31 Chapter Chapter Test, Form A Write the letter for the correct answer in the blank at the right of each problem.. Find an epression equivalent to sec. tan sin A. sec B. cot C. tan D. cos.. If csc and 80 0,find tan.. A. B. C. D.. Simplify tan cs. c tan A. csc B. C. tan D.. Simplify cos se cos c se. c A. tan B. cos C. cos D. cot... Find a numerical value of one trigonometric function of if. t an cot s ec co s cs. c A. csc B. sin C. csc D. sin 6. Use a sum or difference identity to find the eact value of sin. 6. A. 6 B. 6 C. 6 D. 6. Find the value of tan ( ) if cos, sin, , and A. 6 6 B. 6 6 C. D Which epression is equivalent to cos ( )? 8. A. cos B. cos C. sin D. sin 9. Which epression is not equivalent to cos? 9. A. cos sin B. cos C. sin D. sin cos 0. If cos 0.8 and 0 60,find the eact value of sin. 0. A B. 0.6 C D If csc and has its terminal side in Quadrant III, find the. eact value of tan. A. B. C. D. Glencoe/McGraw-Hill 9 Advanced Mathematical Concepts

32 Chapter Chapter Test, Form A (continued). Use a half-angle identity to find the eact value of cos 6.. A. B. C. D.. Solve sin cos 6 0 for all real values of.. A. k, k B. k, k C. k, k D. k, k. Solve cos cos 0 for principal values of.. A. 0 and 0 B. 0 C. 60 D. 60 and 00. Solve sin 0 for 0.. A. B. C. 6 D Write the equation y 0 in normal form. 6. A. y 0 B. y 0 C. y 0 D. y 0. Write the standard form of the equation of a line for which the. length of the normal is 6 and the normal makes an angle of 0 with the positive -ais. A. y 0 B. y 0 C. y 0 D. y 0 8. Find the distance between P(, ) and the line with equation 8. y. A. 9 B. 0 C D Find the distance between the lines with equations y 9 and 9. y. A. B. 0 C. 0 D Find an equation of the line that bisects the obtuse angles formed by 0. the lines with equations y and y. A. ( 0) (0 ) y 0 0 B. ( 0) (0 ) y 0 0 C. ( 0) (0 ) y 0 0 D. ( 0) (0 ) y 0 0 Bonus If and cos,find sin. Bonus: A. 8 B. 8 C. 6 D Glencoe/McGraw-Hill 98 Advanced Mathematical Concepts

33 Chapter Chapter Test, Form B Write the letter for the correct answer in the blank at the right of each problem.. Find an epression equivalent to sec sin cot csc.. A. tan B. csc C. sec D. sin. If sec and 80 0,find tan.. A. B. C. D.. Simplify ta n. tan A. csc B. C. tan D.. Simplify t an sin co. s A. tan B. cos C. cos D. sec... Find a numerical value of one trigonometric function of. if sec cot. A. csc B. sec C. sec D. csc 6. Use a sum or difference identity to find the eact value of sin A. 6 B. 6 C. 6 D. 6. Find the value of tan ( ) if cos, sin, 0 60, and A. 6 6 B. 6 6 C. 6 D Which epression is equivalent to cos ( )? 8. A. cos B. cos C. sin D. sin 9. Which epression is not equivalent to cos? 9. A. cos B. sin C. cos sin D. cos sin 0. If sin 0.6 and 90 80,find the eact value of sin. 0. A. 0.6 B C D If cos and has its terminal side in Quadrant II, find. the eact value of tan. A. B. C. D.. Use a half-angle identity to find the eact value of cos.. A. B. C. D. Glencoe/McGraw-Hill 99 Advanced Mathematical Concepts

34 Chapter Chapter Test, Form B (continued). Solve csc 0 for 0.. A. 6 and 6 B. 6 and 6 C. and D. 6 and 6. Solve cos cos 0 for principal values of.. A. 0 and 0 B. 0 C. 60 D. 0 and 60. Solve sin sin 0 for all real values of.. A. 6 k, 6 k B. 6 k, k 6 C. 6 k, k D. 6 6 k, 6 k 6. Write the equation y 0 in normal form. 6. A. y 0 B. y 0 C. y 0 D. y 0. Write the standard form of the equation of a line for which the. length of the normal is and the normal makes an angle of with the positive -ais. A. y 6 0 B. y 6 0 C. y 6 0 D. y Find the distance between P(, ) and the line with 8. equation y 0. A. 0 0 B. 0 C. 0 0 D Find the distance between the lines with equations 9. y and y. A. B. 8 C. 8 D Find an equation of the line that bisects the acute angles formed 0. by the lines with equations y 0 and y 6 0. A. ( ) ( )y 6 0 B. ( ) ( )y 6 0 C. ( ) ( )y 6 0 D. ( ) ( )y 6 0 Bonus If 90 80, epress cos in terms of tan. Bonus: A. an t B. an C. tan t D. tan Glencoe/McGraw-Hill 00 Advanced Mathematical Concepts

35 Chapter Chapter Test, Form C Write the letter for the correct answer in the blank at the right of each problem.. Find an epression equivalent to c os. sin A. tan B. cot C. sec D. csc.. If sec and 0 90,find sin.. A. B. C. D.. Simplify sec. tan A. csc B. C. tan D.. Simplify sec. csc A. tan B. cos C. D. cot... Find a numerical value of one trigonometric function of if. sin cot. A. cos B. sec C. csc D. cos 6. Use a sum or difference identity to find the eact value of sin. 6. A. 6 B. 6 C. 6 D. 6. Find the value of tan ( ) if cos, sin,0 90,. and A. 6 6 B. 6 6 C. 6 D Which epression is equivalent to cos ( )? 8. A. cos B. sin C. cos D. sin 9. Which epression is equivalent to cos for all values of? 9. A. cos sin B. cos C. sin D. sin cos 0. If cos 0.8 and 0 90,find the eact value of sin. 0. A. 9.6 B..8 C D If sin and has its terminal side in Quadrant II, find the eact. value of tan. A. B. C. D. Glencoe/McGraw-Hill 0 Advanced Mathematical Concepts

36 Chapter Chapter Test, Form C (continued). Use a half-angle identity to find the eact value of sin 0.. A. B. C. D.. Solve cos 0 for 0.. A. 6 and 6 B. and C. and D. 6 and 6. Solve sin sin 0 for principal values of.. A. 60 and 0 B. 0 and 0 C. 0 and 0 D. 60. Solve cos tan sin 0 for all real values of.. A. k, k B. k, k C. k, k D. k, k 6. Write the equation y 0 in normal form. 6. A. y 0 B. y 0 C. y 0 D. y 0. Write the standard form of the equation of a line for which the length. of the normal is and the normal makes an angle of with the positive -ais. A. y 8 0 B. y 8 0 C. y 8 0 D. y Find the distance between P(, ) and the line with equation 8. y 0. A. B. 0 C. D. 9. Find the distance between the lines with equations y 8 and 9. y. A. 8 B. C. D. 0. Find an equation of the line that bisects the acute angles formed by 0. the lines with equations y 0 and y 0. A. ( ) ( )y 0 B. ( ) ( )y 0 C. ( ) ( )y 0 D. ( ) ( )y 0 Bonus If 90 80, epress sin in terms of cos. Bonus: A. cos B. cos C. cos D. cos Glencoe/McGraw-Hill 0 Advanced Mathematical Concepts

37 Chapter Chapter Test, Form A. Simplify (sec tan )( sin )... If tan and 90 80,find sec... Simplify sec.. tan cot tan. Simplify sin cos tan... If tan sin,find sec sec the value of cos.. 6. Use a sum or difference identity to find the eact 6. value of sin 8.. Find the value of sin ( ) if cos, cot, 0 90, and Simplify cos If sec, find the eact value of cos If cos 0.6 and 0 60,find the eact 0. value of sin.. If sin and has its terminal side in Quadrant III,. find the eact value of tan. Glencoe/McGraw-Hill 0 Advanced Mathematical Concepts

38 Chapter Chapter Test, Form A (continued). Use a half-angle identity to find the eact value. of cos 6... Solve cos sin 0 for all real values of... Solve cos cos for principal values of. Epress. the solution(s) in degrees.. Solve sin 0 for Write the equation y 0 in normal form. 6.. Write the standard form of the equation of a line for which. the length of the normal is and the normal makes an angle of 0 with the positive -ais. 8. Find the distance between P(, ) and the line with 8. equation y Find the distance between the lines with equations 9. y and y. 0. Find an equation of the line that bisects the obtuse angles 0. formed by the lines with equations y 0 and y 6 0. Bonus If 80 0 and cos,find sin. Bonus: Glencoe/McGraw-Hill 0 Advanced Mathematical Concepts

39 Chapter Chapter Test, Form B. Simplify cos tan cos... If cot and 90 80,find sin... Simplify csc cot cos... Simplify s. in cos.. If sin sec cot, find the value of csc.. 6. Use a sum or difference identity to find the eact value 6. of cos.. Find the value of sin ( ) if tan, cot, 0 90, and Simplify sin If is an angle in the first quadrant and csc, find 9. the eact value of cos. 0. If sin 0.6 and 80 0,find the eact 0. value of sin.. If cos and has its terminal side in Quadrant IV,. find the eact value of tan. Glencoe/McGraw-Hill 0 Advanced Mathematical Concepts

40 Chapter Chapter Test, Form B (continued). Use a half-angle identity to find the eact value of. cos 0.. Solve tan 0 for 0... Solve sin 0 for principal values of. Epress. the solution(s) in degrees.. Solve cos 0 for all real values of.. 6. Write the equation y 0 in normal form. 6.. Write the standard form of the equation of a line for. which the length of the normal is and the normal makes an angle of 0 with the positive -ais. 8. Find the distance between P(, ) and the line with 8. equation y Find the distance between the lines with equations 9. y and y. 0. Find an equation of the line that bisects the acute angles 0. formed by the lines with equations y 6 0 and y 0. Bonus Epress tan in terms of sin. Bonus: sec + cot sec Glencoe/McGraw-Hill 06 Advanced Mathematical Concepts

41 Chapter. Simplify sin t. an Chapter Test, Form C.. If cos and 90 80,find cot... Simplify sec tan... Simplify sin tan cos... If tan cos,find the value of sin.. 6. Use a sum or difference identity to find the eact value 6. of cos.. Find the value of tan ( ) if cos, sin, 0 90, and Simplify sin ( ) If is an angle in the first quadrant and cos,find 9. the eact value of cos. 0. If cos 0.6 and 0 90,find the eact 0. value of sin.. If cos and has its terminal side in Quadrant II,. find the eact value of tan. Glencoe/McGraw-Hill 0 Advanced Mathematical Concepts

42 Chapter Chapter Test, Form C (continued). Use a half-angle identity to find the eact value of cos.... Solve sin 0 for 0... Solve tan 0 for principal values of. Epress. the solution(s) in degrees.. Solve s ec 0 for all real values of. csc. 6. Write the equation y 6 0 in normal form. 6.. Write the standard form of the equation of a line for. which the length of the normal is 9 and the normal makes an angle of 60 with the positive -ais. 8. Find the distance between P(, ) and the line with 8. equation y Find the distance between the lines with equations 9. y and y. 0. Find an equation of the line that bisects the acute angles 0. formed by the lines with equations y 0 and y 0. Bonus How are the lines that bisect the angles formed Bonus: by the graphs of the equations y 6 and y related to each other? Glencoe/McGraw-Hill 08 Advanced Mathematical Concepts

43 Chapter Chapter Open-Ended Assessment Instructions: Demonstrate your knowledge by giving a clear, concise solution to each problem. Be sure to include all relevant drawings and justify your answer. You may show your solution in more than one way or investigate beyond the requirements of the problem.. a. Verify that cos si n 0 is an identity. sin cos b. Why is it usually easier to transform the more complicated side of the equation into the simpler side rather than the other way around? c. Is the following method for verifying an identity correct? Why or why not? If not, write a correct verification. sec A sin A tan A sec A sin A sin A c os A cos A sec A sin A sin A cos A co sin A sin A s A sin A sin A. a. Write the equation y 6 in normal form. Then, find the length of the normal and the angle it makes with the positive -ais. Eplain how you determined the angle. b. Find the distance from a point on the line in part a to the line with equation 6 y 6 0. Tell what the sign of the distance d means. c. Will the sign of the distance from a point on the line with equation 6 y 6 0 to the line described in part a be the same as in part b? Why or why not? d. When will the sign of the distance between two parallel lines be the same regardless of which line it is measured from? Glencoe/McGraw-Hill 09 Advanced Mathematical Concepts

44 Chapter Chapter Mid-Chapter Test (Lessons - through -). If csc A, find the value of sin A... If tan and 90 80,find cos... Simplify csc cos cot... Simplify c sc. tan tan.. If tan csc, find the value of cos.. 6. Use a sum or difference identity to find the eact value 6. of sin 8.. Find the value of tan ( ) if csc, tan, 0 90, and If tan and 80 0,find the eact value 8. of sin. 9. If is an angle in the first quadrant and csc, find the 9. eact value of cos. 0. Use a half-angle identity to find the eact value of sin.. 0. Glencoe/McGraw-Hill 0 Advanced Mathematical Concepts

45 Chapter Chapter, Quiz A (Lessons - and -). If sec, find the value of cos... If cot and 80 0,find csc... Simplify cot sec... Simplify cos. cot.. If sec sin, find the value of cot.. Chapter Chapter, Quiz B (Lessons - and -). Use a sum or difference identity to find the eact value. of cos.. Find the value of tan ( ) if sin,cos, 0 60, and If sec and 90 80,find the eact value of sin... If cos and has its terminal side in Quadrant IV, find. the eact value of tan.. Use a half-angle identity to find the eact value of sin 6.. Glencoe/McGraw-Hill Advanced Mathematical Concepts

46 Chapter Chapter, Quiz C (Lessons - and -6). Solve sin 0 for 0... Solve cos 0 for principal values of. Epress. the solution(s) in degrees.. Solve tan 0 for all real values of... Write the equation y 8 0 in normal form... Write the standard form of the equation of a line for which. the length of the normal is and the normal makes an angle of 0 with the positive -ais. Chapter Chapter, Quiz D (Lesson -). Find the distance between P(, ) and the line with. equation y 0.. Find the distance between P(, ) and the line with. equation y 0.. Find the distance between the lines with equations. y and y.. Find the distance between the lines with equations. y 8 0 and y.. Find an equation of the line that bisects the acute angles. formed by the lines with equations y 0 and y 0. Glencoe/McGraw-Hill Advanced Mathematical Concepts

47 Chapter Chapter SAT and ACT Practice After working each problem, record the correct answer on the answer sheet provided or use your own paper. Multiple Choice. In the figure below, the measure of A is 6. If the measure of C is the measure of A, what is the measure of B? A 0 B C 6 D 6 E 68. In the figure below, three lines intersect to form a triangle. Find the sum of the measures of the marked angles. A 90 B 80 C 60 D 0 E It cannot be determined from the information given.. If y z, and y and 0, then z A B 0 C D 8 E 6. What are the roots of 69 0? A 0, 69 B 0, C 0, D 69, 69 E,. If ABC is equilateral, what is the value of ( y z) w? A 60 B 0 C 0 D 60 E It cannot be determined from the information given. 6. In the figure below, AD is parallel to BC. Find the value of. A 0 B 0 C 60 D 80 E It cannot be determined from the information given.. Sin A 0 B 6 C D E Cos 8. The lengths of the sides of a rectangle are 6 inches and 8 inches. Which of the following can be used to find, the angle that a diagonal makes with a longer side? A sin B cos C tan D tan E cos 9. Points A(, ), B(, ), and C(, ) are vertices of parallelogram ABCD. What are the coordinates of D? A (0, ) B (, ) C (, ) D (, ) E (, ) Glencoe/McGraw-Hill Advanced Mathematical Concepts

48 Chapter Chapter SAT and ACT Practice (continued) 0. The vertices of a triangle are (, ), (, 9), and (8, ). Which of the following best describes this triangle? A scalene B equilateral C right D isosceles E right, isosceles. In right ABD, CA bisects DAB. What is the value of? A 0 B 0 C 0 D 80 E None of these. In the figure below, what is the value of in terms of y? A y B y C 80 y D 80 y E 60 y. What is the greatest common factor of the terms in the epansion of (6 y 9y )(a 0ay )? A B 6y C 0a D 0ay E None of these. If y y 8 0 and 9, then y A 9 B 6 C 8 D 9 E. Which of the following could be lengths of the sides of a triangle? A, 8, B 8, 8, 6 C 8, 9, 0 D 9, 0, 00 E, 0, 6. In the rectangle ABDC below, what is the measure of ACB? A 6 B C D E It cannot be determined from the information given. 8. Quantitative Comparison A if the quantity in Column A is greater B if the quantity in Column B is greater C if the two quantities are equal D if the relationship cannot be determined from the information given Column A Column B. Side AB of triangle ABC is etended beyond B to point D. The measure of ABC 8. Angles P, Q, and R are the angles of a right triangle. 80 m P The measure of DBC Refer to the figure below. 9. Grid-In What is the value of? 0. Grid-In What is the value of y? Glencoe/McGraw-Hill Advanced Mathematical Concepts

49 Chapter Chapter Cumulative Review (Chapters -). Find the standard form of the equation of the line that. passes through (, ) and has a slope of If A and B,find AB... Given ƒ() ( ), find ƒ (). Then state whether. ƒ () is a function.. If y varies inversely as the square of and y 8 when., find y when 9.. Write a polynomial equation of least degree with roots,. i, and i. 6. Use the Remainder Theorem to find the remainder when 6. is divided by. State whether the binomial is a factor of the polynomial.. Given the triangle at the. right, find m A to the nearest tenth of a degree if b and c If a 8, b, and c, find the area of ABC to the 8. nearest tenth. 9. State the amplitude, period, and phase shift for the graph 9. of y sin( ). 0. Find the value of Cos tan. 0.. Solve sin 0 for principal values of.. Epress the solution(s) in degrees.. Find the distance between P(, ) and the line with. equation y 0. Glencoe/McGraw-Hill Advanced Mathematical Concepts

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51 Glencoe/McGraw-Hill A Advanced Mathematical Concepts SAT and ACT Practice Answer Sheet (0 Questions) / /

52 Glencoe/McGraw-Hill A Advanced Mathematical Concepts SAT and ACT Practice Answer Sheet (0 Questions) / / / /

53 Answers (Lesson -) - Practice Basic Trigonometric Identities Use the given information to determine the eact trigonometric value if If cos, find tan.. If sin, find cos.. If tan, find sin.. If tan, find cot. Epress each value as a trigonometric function of an angle in Quandrant I.. cos csc 9. sin cos 8 csc sin Simplify each epression. 8. cos sin tan 9. c t o a t A 0. sin n A cos cos sec cot A cos. Kite Flying Brett and Tara are flying a kite. When the string is tied to the ground, the height of the kite can be determined by the formula L H csc, where L is the length of the string and is the angle between the string and the level ground. What formula could Brett and Tara use to find the height of the kite if they know the value of sin? H L sin Glencoe/McGraw-Hill 6 Advanced Mathematical Concepts - Enrichment The Physics of Soccer Recall from Lesson - that the formula for the maimum height h of a projectile is, where is the measure of the angle of h v 0 sin g elevation in degrees, v 0 is the initial velocity in feet per second, and g is the acceleration due to gravity in feet per second squared. Solve. Give answers to the nearest tenth.. A soccer player kicks a ball at an initial velocity of 60 ft/s and an angle of elevation of 0. The acceleration due to gravity is ft/s.find the maimum height reached by the ball.. ft. With what initial velocity must you kick a ball at an angle of in order for it to reach a maimum height of 0 ft? 6. ft/s The distance d that a projected object travels is given by the v formula 0 sin cos d g.. Find the distance traveled by the ball described in Eercise. 0.8 ft In order to kick a ball the greatest possible distance at a given initial velocity, a soccer player must maimize d v 0 sin cos g. Since, v 0, and g are constants, this means the player must maimize sin cos. sin 0 cos 0 sin 90 cos 90 0 sin 0 cos 0 sin 80 cos sin 0 cos 0 sin 0 cos Use the patterns in the table to hypothesize a value of for which sin cos will be maimal. Use a calculator to check your hypothesis. At what angle should the player kick the ball to achieve the greatest distance? Glencoe/McGraw-Hill Advanced Mathematical Concepts Glencoe/McGraw-Hill A Advanced Mathematical Concepts

54 Answers (Lesson -) - Practice Verifying Trigonometric Identities Verify that each equation is an identity.. cs c cot t an cos c s c cot t an sin s n c o s c o s s n c o s s s in in c o s cos c o s co s cos sin. sin y sin y sin y sin y sec y sin y sin y sin y co sec y s y. sin cos ( sin cos )(sin cos ) sin cos (sin cos )(sin sin cos cos ) (sin cos )( sin cos ) ( sin cos )(sin cos ). tan cos si tan c sec n o s s in s in c o sin (cos )( sin ) s c sec o s s in sin sin cos (cos )( sin ) Find a numerical value of one trigonometric function of.. sin cot 6. sin cos. cos cot cos tan csc or sin 8. Physics The work done in moving an object is given by the formula W Fd cos, where d is the displacement, F is the force eerted, and is the angle between the displacement and the force. Verify that W Fd c c W Fd c c o s t c is an equivalent formula. s o t c Fd Fd c s c s o in s sin o in s sin Fd cos Glencoe/McGraw-Hill 9 Advanced Mathematical Concepts - Enrichment Building from By starting with the most fundamental identity of all,, you can create new identities as comple as you would like them to be. First, think of ways to write using trigonometric identities. Some eamples are the following. cos A sec A csc A cot A cos (A 60 ) cos (60 A) Choose two such epressions and write a new identity. cos A sec A csc A cot A Now multiply the terms of the identity by the terms of another identity of your choosing, preferably one that will allow some simplification upon multiplication. cos A sec A csc A cot A sin A cos A tan A sin A sec A tan A csc A cot A Beginning with, create two trigonometric identities. Answers will vary... Verify that each of the identities you created is an identity... Glencoe/McGraw-Hill 80 Advanced Mathematical Concepts i i Glencoe/McGraw-Hill A Advanced Mathematical Concepts

55 Answers (Lesson -) - Practice Sum and Difference Identities Use sum or difference identities to find the eact value of each trigonometric function.. cos. sin (6). tan 6 6. csc 9. tan 6. sec 6 6 Find each eact value if 0 and 0 y.. cos ( y) if sin 6 6 and sin y 8. sin ( y) if cos 8 8 and cos y 9. tan ( y) if csc and cot y 6 6 Verify that each equation is an identity. 0. cos (80 ) cos. sin (60 ) sin cos (80 ) sin (60 ) cos 80 cos sin 80 sin sin 60 cos cos 60 sin () cos 0 sin 0 cos sin cos sin. Physics Sound waves can be modeled by equations of the form y 0 sin (t ). Determine what type of interference results when sound waves modeled by the equations y 0 sin (t 90) and y 0 sin (t 0) are combined. (Hint: Refer to the application in Lesson -.) The interference is destructive. The waves cancel each other completely. Glencoe/McGraw-Hill 8 Advanced Mathematical Concepts - Enrichment Identities for the Products of Sines and Cosines By adding the identities for the sines of the sum and difference of the measures of two angles, a new identity is obtained. sin ( ) sin cos cos sin sin ( ) sin cos cos sin (i) sin ( ) sin ( ) sin cos This new identity is useful for epressing certain products as sums. Eample Write sin cos as a sum. In the right side of identity (i) let and so that sin cos sin ( ) sin ( ). Thus, sin cos sin sin. By subtracting the identities for sin ( ) and sin ( ), you obtain a similar identity for epressing a product as a difference. (ii) sin ( ) sin ( ) cos sin Eample Verify the identity c si o n s s c in o s (sin sin ) sin sin. In the right sides of identities (i) and (ii) let and. Then write the following quotient. cos sin sin cos By simplifying and multiplying by the conjugate, the identity is verified. cos sin sin cos sin ( ) sin ( ) sin ( ) sin ( ) sin sin sin sin. sin sin sin sin (sin sin ) sin sin Complete.. Use the identities for cos ( ) and cos ( ) to find identities for epressing the products cos cos and sin sin as a sum or difference. cos ( ) cos cos sin sin cos ( ) cos cos sin sin cos cos cos ( ) cos ( ) sin sin cos ( ) cos ( ). Find the value of sin 0 cos by using the identity above. sin 0 cos (sin 80 sin 0 ) (0 ) or 0. Glencoe/McGraw-Hill 8 Advanced Mathematical Concepts Glencoe/McGraw-Hill A Advanced Mathematical Concepts