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Chapter 5: Analytic Trigonometry 1. If sin x = 1 3 and cos x = , evaluate the following function. 2 2 A) B) C) D) E) Ans: B Learning Objective: Evaluate trigonometric function given other trigonometric values Section: 5.1 Page 232 Copyright © Houghton Mifflin Company. All rights reserved. Chapter 5: Analytic Trigonometry 2. If csc x = 4 3 and cos x < 0 , evaluate the function below. 3 A) B) C) D) E) Ans: C Learning Objective: Evaluate trigonometric function given other trigonometric values Section: 5.1 3. Which of the following is equivalent to the expression below? A) B) C) D) E) Ans: A Learning Objective: Simplify a trigonometric expression Section: 5.1 Copyright © Houghton Mifflin Company. All rights reserved. Page 233 Chapter 5: Analytic Trigonometry 4. Use fundamental identities to simplify the expression below and then determine which of the following is not equivalent. A) B) C) D) E) Ans: D Learning Objective: Simplify a trigonometric expression Section: 5.1 5. Which of the following is equivalent to the expression below? A) B) C) D) E) Ans: A Learning Objective: Simplify a trigonometric expression Section: 5.1 Page 234 Copyright © Houghton Mifflin Company. All rights reserved. Chapter 5: Analytic Trigonometry 6. Determine which of the following are trigonometric identities. I. sin (θ ) + cot (θ ) cos (θ ) = csc (θ ) II. cot (θ ) − sin (θ ) cos (θ ) = 0 III. sin (θ ) + sin (θ ) cos (θ ) = csc (θ ) A) I is the only identity. D) III is the only identity. B) I and II are the only identities. E) I, II, and III are identities. C) II is the only identity. Ans: A Learning Objective: Identify trigonometric identities Section: 5.1 7. Determine which of the following are trigonometric identities. I. tan (θ ) sec (θ ) = csc (θ ) II. tan (θ ) csc (θ ) = sec (θ ) III. csc (θ ) sec (θ ) = tan (θ ) IV. tan (θ ) cos (θ ) = 1 A) II and IV are the only identities. D) IV is the only identity. B) II is the only identity. E) III is the only identity. C) II, III, and IV are the only identities. Ans: B Learning Objective: Identify trigonometric identities Section: 5.1 Copyright © Houghton Mifflin Company. All rights reserved. Page 235 Chapter 5: Analytic Trigonometry 8. Factor; then use fundamental identities to simplify the expression below and determine which of the following is not equivalent. A) B) C) D) E) Ans: C Learning Objective: Apply fundamental identities to determine non-equivalent expression Section: 5.1 9. Factor the expression below and use the fundamental identities to simplify. cos 4 ( x ) − sin 4 ( x ) A) B) C) D) E) ( cos ( x ) + sin ( x ) ) ( cos ( x ) − sin ( x ) ) ( cos ( x ) − sin ( x ) ) 2 2 4 cos ( x ) − sin ( x ) ( cos ( x ) + sin ( x ) ) ( cos ( x ) − sin ( x ) ) 4 ( cos ( x ) − sin ( x ) ) Ans: D Learning Objective: Apply fundamental identities to determine equivalent expression Section: 5.1 Page 236 Copyright © Houghton Mifflin Company. All rights reserved. Chapter 5: Analytic Trigonometry 10. Factor; then use fundamental identities to simplify the expression below and determine which of the following is not equivalent. A) B) C) D) E) Ans: E Learning Objective: Apply fundamental identities to determine non-equivalent expression Section: 5.1 11. Expand the expression below and use fundamental trigonometric identities to simplify. A) B) C) sin 2 (ω ) + cos 2 (ω ) ( sin (ω ) + cos (ω ) ) 2 tan (ω ) + 1 2sin (ω ) cos (ω ) + 1 2 D) 1 E) 2 cot (ω ) + 1 Ans: C Learning Objective: Simplify a trigonometric expression Section: 5.1 Copyright © Houghton Mifflin Company. All rights reserved. Page 237 Chapter 5: Analytic Trigonometry 12. Multiply; then use fundamental identities to simplify the expression below and determine which of the following is not equivalent. A) B) C) D) E) Ans: A Learning Objective: Apply fundamental identities to determine non-equivalent expression Section: 5.1 Page 238 Copyright © Houghton Mifflin Company. All rights reserved. Chapter 5: Analytic Trigonometry 13. Add or subtract as indicated; then use fundamental identities to simplify the expression below and determine which of the following is not equivalent. A) B) C) D) E) Ans: C Learning Objective: Apply fundamental identities to determine non-equivalent expression Section: 5.1 Copyright © Houghton Mifflin Company. All rights reserved. Page 239 Chapter 5: Analytic Trigonometry 14. Which of the following is equivalent to the given expression? A) B) C) D) E) Ans: B Learning Objective: Apply fundamental identities to determine equivalent expression Section: 5.1 15. sin ( y ) so that it is not in fractional form. 1 – cos ( y ) D) sin 2 – sin ( y ) tan ( y ) sin 2 + sin ( y ) tan ( y ) Rewrite the expression A) B) C) 1 – sin ( y ) tan ( y ) csc ( y ) + cot ( y ) E) 1 – cos ( y ) Ans: C Learning Objective: Rewrite fractional trigonometric expression as non-fraction Section: 5.1 Page 240 Copyright © Houghton Mifflin Company. All rights reserved. Chapter 5: Analytic Trigonometry 16. Use a graphing utility to determine which of the trigonometric functions is equal to the following expression. A) B) C) Copyright © Houghton Mifflin Company. All rights reserved. Page 241 Chapter 5: Analytic Trigonometry D) E) Ans: D Learning Objective: Identify equivalent trigonometric expressions with a graphing utility Section: 5.1 17. If x = 2 tan θ , use trigonometric substitution to write function of θ , where 0 < θ < 4 + x 2 as a trigonometric π . 2 C) 2 csc θ D) 2sec θ E) 2 tan θ A) 2sin θ B) 2 cos θ Ans: D Learning Objective: Write an algebraic expression as a trigonometric function Section: 5.1 18. Use the trigonometric substitution x = 9sec (θ ) to write the expression trigonometric function of θ , where 0 < θ < A) 9 tan (θ ) B) 81tan (θ ) π 2 C) 81sec (θ ) x 2 − 81 as a . D) 9sec (θ ) E) 9sec (θ ) − 1 Ans: A Learning Objective: Write an algebraic expression as a trigonometric function Section: 5.1 Page 242 Copyright © Houghton Mifflin Company. All rights reserved. Chapter 5: Analytic Trigonometry 19. If x = 9 tan θ , use trigonometric substitution to write function of θ , where − π 2 <θ < 81 + x 2 as a trigonometric π . 2 C) 9 tan θ D) 9sin θ E) 9 cos θ A) 9 csc θ B) 9sec θ Ans: B Learning Objective: Write an algebraic expression as a trigonometric function Section: 5.1 20. If x = 2 cot θ , use trigonometric substitution to write 4 + x 2 as a trigonometric function of θ , where 0 < θ < π . A) 2 cos θ B) 2 csc θ C) 2 cot θ D) 2sec θ E) 2sin θ Ans: B Learning Objective: Write an algebraic expression as a trigonometric function Section: 5.1 21. The rate of change of the function is given by the expression . Which of the following is its simplification? A) B) C) D) E) Ans: C Learning Objective: Simplify a trigonometric expression Section: 5.1 Copyright © Houghton Mifflin Company. All rights reserved. Page 243 Chapter 5: Analytic Trigonometry 22. Verify the identity shown below. Ans: Learning Objective: Verify a trigonometric identity Section: 5.2 23. Verify the identity shown below. Ans: Learning Objective: Verify a trigonometric identity Section: 5.2 Page 244 Copyright © Houghton Mifflin Company. All rights reserved. Chapter 5: Analytic Trigonometry 24. Verify the identity shown below. Ans: Learning Objective: Verify a trigonometric identity Section: 5.2 25. Verify the identity shown below. Ans: Learning Objective: Verify a trigonometric identity Section: 5.2 Copyright © Houghton Mifflin Company. All rights reserved. Page 245 Chapter 5: Analytic Trigonometry 26. Determine which of the following are trigonometric identities. I. II. cot ( x ) + cot ( y ) tan ( x ) + tan ( y ) + =0 tan ( x ) – tan ( y ) cot ( x ) – cot ( y ) cot ( x ) + cot ( y ) tan ( x ) + tan ( y ) + =1 tan ( x ) + tan ( y ) cot ( x ) + cot ( y ) cot ( x ) + tan ( y ) = cot ( y ) + tan ( x ) cot ( x ) tan ( y ) A) III is the only identity. D) I and II are the only identities. B) I and III are the only identities. E) I, II, and III are identities. C) II and II are the only identities. Ans: A Learning Objective: Verify a trigonometric identity Section: 5.2 III. 27. Verify the identity shown below. Ans: Learning Objective: Verify a trigonometric identity Section: 5.2 Page 246 Copyright © Houghton Mifflin Company. All rights reserved. Chapter 5: Analytic Trigonometry 28. Verify the identity shown below. Ans: Learning Objective: Verify a trigonometric identity Section: 5.2 29. Verify the identity shown below. Ans: Learning Objective: Verify a trigonometric identity Section: 5.2 Copyright © Houghton Mifflin Company. All rights reserved. Page 247 Chapter 5: Analytic Trigonometry 30. Verify the identity shown below. Ans: Learning Objective: Verify a trigonometric identity Section: 5.2 Page 248 Copyright © Houghton Mifflin Company. All rights reserved. Chapter 5: Analytic Trigonometry 31. Verify the identity shown below. Ans: Learning Objective: Verify a trigonometric identity Section: 5.2 Copyright © Houghton Mifflin Company. All rights reserved. Page 249 Chapter 5: Analytic Trigonometry 32. Verify the identity shown below. Ans: Learning Objective: Verify a trigonometric identity Section: 5.2 Page 250 Copyright © Houghton Mifflin Company. All rights reserved. Chapter 5: Analytic Trigonometry 33. Use the cofunction identities to evaluate the expression below without the aid of a calculator. sin 2 34° + sin 2 29° + sin 2 56° + sin 2 61° 1 A) 1 B) 2 C) –1 D) 0 E) 2 Ans: B Learning Objective: Evaluate trigonometric expression using cofunction identities Section: 5.2 34. Determine which of the following are trigonometric identities. 4 4 2 4 I. csc ( z ) + cot ( z ) = 1 − 2 cot ( z ) + 2 cot ( z ) 5 3 2 3 II. cot ( z ) = cot ( z ) csc ( z ) − cot ( z ) 3 2 2 4 III. cot ( z ) csc ( z ) = ( csc ( z ) − csc ( z ) ) cot ( z ) A) I, II, and III are identities. D) II and II are the only identities. B) II is the only identity. E) III is the only identity. C) I is the only identity. Ans: B Learning Objective: Verify a trigonometric identity Section: 5.2 Copyright © Houghton Mifflin Company. All rights reserved. Page 251 Chapter 5: Analytic Trigonometry 35. Which of the following is a solution to the given equation? A) B) C) D) E) Ans: D Learning Objective: Verify solution to trigonometric equation Section: 5.3 Page 252 Copyright © Houghton Mifflin Company. All rights reserved. Chapter 5: Analytic Trigonometry 36. Which of the following is a solution to the given equation? A) B) C) D) E) Ans: D Learning Objective: Verify solution to trigonometric equation Section: 5.3 Copyright © Houghton Mifflin Company. All rights reserved. Page 253 Chapter 5: Analytic Trigonometry 37. Solve the following equation. A) B) C) D) E) Ans: B Learning Objective: Solve trigonometric equation Section: 5.3 38. Solve the following equation. csc 2 ( x ) − 4 = 0 2π + π n, where n is an integer 3 3 B) π 2π + 2π n, + 2π n, where n is an integer 3 3 C) π 5π + π n, + π n, where n is an integer 6 6 D) π 2π + 2π n, + 2π n, where n is an integer 3 6 E) π 5π + 2π n, + 2π n, where n is an integer 6 6 Ans: C Learning Objective: Solve trigonometric equation Section: 5.3 A) Page 254 π + π n, Copyright © Houghton Mifflin Company. All rights reserved. Chapter 5: Analytic Trigonometry 39. Solve the following equation. A) B) C) D) E) Ans: B Learning Objective: Solve trigonometric equation Section: 5.3 Copyright © Houghton Mifflin Company. All rights reserved. Page 255 Chapter 5: Analytic Trigonometry 40. Solve the following equation. A) B) C) D) E) Ans: C Learning Objective: Solve trigonometric equation Section: 5.3 41. Find all solutions of the following equation on the interval [ 0, 2π ) . tan ( x ) + 3 = 0 D) 2π π 5π 4π , , , 3 3 3 3 E) B) π 7π , 6 6 C) 2π 5π , 3 3 Ans: C Learning Objective: Solve trigonometric equation Section: 5.3 A) Page 256 5π 11π , 6 6 π 4π , 3 3 Copyright © Houghton Mifflin Company. All rights reserved. Chapter 5: Analytic Trigonometry 42. Find all solutions of the following equation on the interval [ 0, 2π ) . csc 2 ( x ) – 2 = 0 D) 3π 4 4 E) B) π 3π 5π 7π , , , 4 4 4 4 C) π 7π , 4 4 Ans: B Learning Objective: Solve trigonometric equation Section: 5.3 A) π , 5π 7π , 4 4 3π 5π , 4 4 43. Find all solutions of the following equation in the interval [ 0, 2π ) . A) B) C) D) E) Ans: E Learning Objective: Solve trigonometric equation Section: 5.3 Copyright © Houghton Mifflin Company. All rights reserved. Page 257 Chapter 5: Analytic Trigonometry 44. Find all solutions of the following equation in the interval [ 0, 2π ) . A) B) C) D) E) Ans: A Learning Objective: Solve trigonometric equation Section: 5.3 Page 258 Copyright © Houghton Mifflin Company. All rights reserved. Chapter 5: Analytic Trigonometry 45. Find all solutions of the following equation in the interval [ 0, 2π ) . A) B) C) D) E) Ans: D Learning Objective: Solve trigonometric equation Section: 5.3 46. Approximate the solutions of the equation 2sin 2 ( x ) + 4sin ( x ) –1 = 0 by considering its graph below. Round your answer to one decimal. A) 0.2, 2.9 B) –1.0, 0.2 C) –1.0, 1.8 D) 0.2, 1.8 E) 1.8, 2.9 Ans: A Learning Objective: Approximate solutions of trigonometric equation with a graph Section: 5.3 Copyright © Houghton Mifflin Company. All rights reserved. Page 259 Chapter 5: Analytic Trigonometry 47. Approximate the solutions of the equation 2sin 2 ( x ) = 3cos ( x ) + 1 by considering its graph below. Round your answer to one decimal. A) 2.4, 3.9 B) 2.4, 3.1 C) 3.1, 3.9 D) 1.5, 5.0 E) 1.5, 2.4 Ans: D Learning Objective: Approximate solutions of trigonometric equation with a graph Section: 5.3 48. Approximate the solutions of the equation csc ( x ) + cot ( x ) = –1 by considering its graph below. Round your answer to one decimal. A) 1.9 B) 1.9, 5.9 C) 4.8 D) 5.9 E) The equation has no solution. Ans: C Learning Objective: Approximate solutions of trigonometric equation with a graph Section: 5.3 Page 260 Copyright © Houghton Mifflin Company. All rights reserved. Chapter 5: Analytic Trigonometry 49. Solve the multiple-angle equation. A) B) C) D) E) Ans: E Learning Objective: Solve multiple-angle equation Section: 5.3 Copyright © Houghton Mifflin Company. All rights reserved. Page 261 Chapter 5: Analytic Trigonometry 50. Solve the multiple-angle equation in the interval [ 0, 2π ) . A) B) C) D) E) Ans: C Learning Objective: Solve multiple-angle equation Section: 5.3 51. Solve the multi-angle equation below. sin ( 2 x ) = A) π 8 B) π 6 C) π 6 D) π 8 E) π + nπ , + nπ , π 4 π 4 + nπ , where n is an integer 3 π + 2nπ , π 4 + 2nπ , where n is an integer + 2nπ , where n is an integer + nπ , where n is an integer 6 3 Ans: E Learning Objective: Solve multiple-angle equation Section: 5.3 Page 262 + nπ , + nπ , where n is an integer π + 2nπ , 3 2 Copyright © Houghton Mifflin Company. All rights reserved. Chapter 5: Analytic Trigonometry 52. Solve the multi-angle equation below. 7π + nπ , where n is an integer 3 3 B) π 7π + 2nπ , + 2nπ , where n is an integer 3 3 C) π 7π + nπ , + nπ , where n is an integer 3 2 D) π 7π + nπ , + nπ , where n is an integer 2 2 E) π 7π + 4nπ , + 4nπ , where n is an integer 2 2 Ans: E Learning Objective: Solve multiple-angle equation Section: 5.3 A) π 2 ⎛x⎞ cos ⎜ ⎟ = ⎝2⎠ 2 + nπ , 53. Use the graph below to approximate the solutions of the equation –2 cos ( x ) – sin ( x ) = 0 on the interval [ 0, 2π ) . Round your answer to one decimal. A) –2.0, 5.9 B) 2.0, 5.9 C) 5.2, 5.9 D) 2, 5.2 E) –2, 5.2 Ans: D Learning Objective: Approximate solutions of trigonometric equation with a graph Section: 5.3 Copyright © Houghton Mifflin Company. All rights reserved. Page 263 Chapter 5: Analytic Trigonometry 54. Use a graphing utility to approximate the solutions (to three decimal places) of the given equation in the interval [ 0, 2π ) . A) B) C) D) E) Ans: E Learning Objective: Approximate solutions to trigonometric equation with graphing utility Section: 5.3 55. Use a graphing utility to approximate the solutions (to three decimal places) of the given ⎛ π π⎞ equation in the interval ⎜ − , ⎟ . ⎝ 2 2⎠ A) B) C) D) E) Ans: A Learning Objective: Approximate solutions to trigonometric equation with graphing utility Section: 5.3 Page 264 Copyright © Houghton Mifflin Company. All rights reserved. Chapter 5: Analytic Trigonometry 56. Use the graph of the function f ( x ) = – cos ( x ) + sin ( x ) to approximate the maximum points of the graph in the interval [ 0, 2π ] . Round your answer to one decimal. A) B) C) ( 2.6,1.3) , ( 6.3, –0.8) ( –0.8, 6.3) , ( 2.6,1.3) ( 2.6, –0.8 ) , ( 6.3,1.3) D) E) ( –0.8, 6.3) , (1.3, 2.6 ) (1.3, 2.6 ) , ( 6.3, –0.8) Ans: A Learning Objective: Approximate maximum or minimum of a trigonometric function Section: 5.3 57. Solve the following trigonometric equation on the interval [ 0, 2π ) . cos ( x ) + sin ( x ) = 0 5π 7π 3π 7π π 7π π 5π , , B) C) , D) , 4 4 4 4 4 4 4 4 Ans: B Learning Objective: Solve trigonometric equation Section: 5.3 A) E) 3π 5π , 4 4 58. Determine the exact value of the following expression. cos ( 240D − 0D ) 3 1 3 1 1 3 B) C) D) – E) – 2 2 2 2 2 2 Ans: D Learning Objective: Calculate exact value of expression using sum or difference formula Section: 5.4 A) – Copyright © Houghton Mifflin Company. All rights reserved. Page 265 Chapter 5: Analytic Trigonometry 59. Find the exact value of the given expression. cos ( 240° + 315° ) 1+ 3 1– 3 –1 + 3 –1 – 3 B) C) D) 2 2 2 2 2 2 2 2 Ans: D Learning Objective: Calculate exact value of expression using sum or difference formula Section: 5.4 A) 60. Find the exact value of the given expression. ⎛ 5π 7π ⎞ sin ⎜ − ⎟ 4 ⎠ ⎝ 3 – 3 +1 3 +1 – 3 –1 3 –1 B) C) D) A) 2 2 2 2 2 2 2 2 Ans: A Learning Objective: Calculate exact value of expression using sum or difference formula Section: 5.4 61. Find the exact value of the given expression using a sum or difference formula. sin 285° 3 –1 3 +1 – 3 –1 – 3 +1 B) C) D) A) 2 2 2 2 2 2 2 2 Ans: C Learning Objective: Calculate exact value of expression using sum or difference formula Section: 5.4 62. Find the exact value of the given expression using a sum or difference formula. 13π cos 12 3 +1 – 3 +1 3 –1 – 3 –1 B) C) D) A) 2 2 2 2 2 2 2 2 Ans: D Learning Objective: Calculate exact value of expression using sum or difference formula Section: 5.4 Page 266 Copyright © Houghton Mifflin Company. All rights reserved. Chapter 5: Analytic Trigonometry 63. Find the exact value of the given expression using a sum or difference formula. 7π tan 12 3 +1 3 –1 D) E) undefined A) 1 B) –1 C) 1– 3 –1 – 3 Ans: C Learning Objective: Calculate exact value of expression using sum or difference formula Section: 5.4 64. Write the given expression as the cosine of an angle. cos 30° cos 55° – sin 30° sin 55° A) cos ( 55° ) B) cos ( 85° ) C) cos ( –25° ) D) cos ( 30° ) E) cos ( –110° ) Ans: B Learning Objective: Rewrite expression using sum or difference formula Section: 5.4 65. Write the given expression as the sine of an angle. sin15° cos 55° – sin 55° cos15° A) sin ( –110° ) B) sin ( –40° ) C) sin ( 70° ) D) sin (15° ) E) sin ( 55° ) Ans: B Learning Objective: Rewrite expression using sum or difference formula Section: 5.4 66. Find the exact value of sin ( u + v ) given that sin u = 8 60 and cos v = − . (Both u and v 17 61 are in Quadrant II.) D) A) 315 812 sin ( u + v ) = – sin ( u + v ) = – 1037 1037 B) E) 315 645 sin ( u + v ) = sin ( u + v ) = 1037 1037 C) 645 sin ( u + v ) = – 1037 Ans: C Learning Objective: Calculate exact value of expression using sum or difference formula Section: 5.4 Copyright © Houghton Mifflin Company. All rights reserved. Page 267 Chapter 5: Analytic Trigonometry 67. Find the exact value of tan ( u + v ) given that sin u = − 3 24 and cos v = . (Both u and v 5 25 are in Quadrant IV.) D) A) 41 89 tan ( u + v ) = tan ( u + v ) = 75 75 B) E) 38 39 tan ( u + v ) = tan ( u + v ) = – 75 25 C) 4 tan ( u + v ) = – 3 Ans: C Learning Objective: Calculate exact value of expression using sum or difference formula Section: 5.4 68. Find the exact value of cos ( u + v ) given that sin u = 7 12 and cos v = − . (Both u and v 25 13 are in Quadrant II.) A) D) 12 246 cos ( u + v ) = cos ( u + v ) = – 65 325 B) E) 36 204 cos ( u + v ) = – cos ( u + v ) = 325 325 C) 253 cos ( u + v ) = 325 Ans: C Learning Objective: Calculate exact value of expression using sum or difference formula Section: 5.4 69. Find the exact value of cos ( u − v ) given that sin u = − 8 60 and cos v = . (Both u and v 17 61 are in Quadrant IV.) D) A) 307 812 cos ( u − v ) = – cos ( u − v ) = – 1037 1037 B) E) 980 988 cos ( u − v ) = – cos ( u − v ) = 1037 1037 C) 827 cos ( u − v ) = 1037 Ans: E Learning Objective: Calculate exact value of expression using sum or difference formula Section: 5.4 Page 268 Copyright © Houghton Mifflin Company. All rights reserved. Chapter 5: Analytic Trigonometry 70. Write the given expression as an algebraic expression. A) B) C) D) E) Ans: D Learning Objective: Write trig expression as an algebraic expression Section: 5.4 71. Simplify the given expression algebraically. A) B) C) D) E) Ans: B Learning Objective: Simplify trigonometric expression using sum and difference formulas Section: 5.4 Copyright © Houghton Mifflin Company. All rights reserved. Page 269 Chapter 5: Analytic Trigonometry 72. Simplify the given expression algebraically. A) B) C) D) E) Ans: D Learning Objective: Simplify trig expression using sum and difference formulas Section: 5.4 73. Determine which of the following are trigonometric identities. I. sin ( x + y ) + sin ( x – y ) = 2sin ( x ) II. sin ( x + π ) − sin (π – x ) = 2sin ( x ) III. sin ( x + y ) + sin ( x – y ) = 2sin ( x ) sin ( y ) A) II and II are the only identities. D) I is the only identity. B) I and III are the only identities. E) III is the only identity. C) None are identities. Ans: C Learning Objective: Verify an identity using sum and difference formulas Section: 5.4 74. Verify the given identity. Ans: Learning Objective: Verify an identity using sum and difference formulas Section: 5.4 Page 270 Copyright © Houghton Mifflin Company. All rights reserved. Chapter 5: Analytic Trigonometry 75. Find all solutions of the given equation in the interval [ 0, 2π ) . A) B) C) D) E) Ans: C Learning Objective: Solve trig equation with sum/difference formulas Section: 5.4 Copyright © Houghton Mifflin Company. All rights reserved. Page 271 Chapter 5: Analytic Trigonometry 76. Use the figure below to determine the exact value of the given function. 2 θ 3 A) B) C) D) E) Ans: E Learning Objective: Calculate exact value of a trigonometric function using a right triangle Section: 5.5 Page 272 Copyright © Houghton Mifflin Company. All rights reserved. Chapter 5: Analytic Trigonometry 77. Use the graph below of the function to approximate the solutions to 2 cos ( 2 x ) − cos ( x ) = 0 in the interval [ 0, 2π ) . Round your answers to one decimal. A) 0.6, 1.0, 1.5, 4.8 D) 0.6, 2.1, 4.2, 5.7 B) 1.0, 1.5, 4.8, 5.7 E) 0.6, 1.5, 4.2, 6.3 C) 1.0, 2.1, 4.2, 5.7 Ans: D Learning Objective: Approximate solutions of trigonometric equation with a graph Section: 5.5 Copyright © Houghton Mifflin Company. All rights reserved. Page 273 Chapter 5: Analytic Trigonometry 78. Find the exact solutions of the given equation in the interval [ 0, 2π ) . A) B) C) D) E) Ans: D Learning Objective: Solve trigonometric equation Section: 5.5 Page 274 Copyright © Houghton Mifflin Company. All rights reserved. Chapter 5: Analytic Trigonometry 79. Find the exact solutions of the given equation in the interval [ 0, 2π ) . A) B) C) D) x=0 E) Ans: B Learning Objective: Solve trigonometric equation Section: 5.5 80. Use a double-angle formula to find the exact value of cos 2u when 7 π <u <π . sin u = , where 25 2 D) A) 478 527 cos 2u = – cos 2u = 625 625 E) B) 168 1152 cos 2u = cos 2u = – 625 625 C) 336 cos 2u = 625 Ans: D Learning Objective: Calculate exact value of a trigonometric function using a doubleangle formula Section: 5.5 Copyright © Houghton Mifflin Company. All rights reserved. Page 275 Chapter 5: Analytic Trigonometry 81. Use a double-angle formula to find the exact value of tan 2u when 12 3π . cos u = − , where π < u < 13 2 D) A) 5 130 tan 2u = – tan 2u = 6 119 E) B) 312 5 tan 2u = – tan 2u = 25 12 C) 120 tan 2u = 119 Ans: C Learning Objective: Calculate exact value of a trigonometric function using a doubleangle formula Section: 5.5 82. Use a double angle formula to rewrite the following expression. A) B) C) sin ( –14x ) 2sin ( –7x ) –7 cos ( 2x ) –14sin ( x ) cos ( x ) D) E) –7 sin ( 2x ) cos ( –14x ) Ans: D Learning Objective: Rewrite expression as a double angle Section: 5.5 83. Use a double angle formula to rewrite the given expression. 8cos 2 x − 4 A) 4 cos 2x B) 8cos 2x C) 2 cos 4x D) 4 cos 4x E) 2 cos8x Ans: A Learning Objective: Rewrite expression as a double angle Section: 5.5 Page 276 Copyright © Houghton Mifflin Company. All rights reserved. Chapter 5: Analytic Trigonometry 84. Determine which of the following are trigonometric identities. 1 (1+ 4 cos ( 2 x ) + cos ( 4 x ) ) 4 1 II. cos 4 ( x ) = ( 3 + 4sin ( 2 x ) – sin ( 4 x ) ) 8 1 III. cos 4 ( x ) = ( 3 + 4 cos ( 2 x ) + cos ( 4 x ) ) 8 A) None are identities. D) I and III are the only identities. B) III is the only identity. E) I, II, and III are identities. C) I and II are the only identities. Ans: B Learning Objective: Verify a trigonometric identity Section: 5.5 I. cos 4 ( x ) = 85. Use the power-reducing formulas to rewrite the given expression in terms of the first power of the cosine. A) B) C) D) E) Ans: E Learning Objective: Rewrite expression using power-reducing formulas Section: 5.5 Copyright © Houghton Mifflin Company. All rights reserved. Page 277 Chapter 5: Analytic Trigonometry 86. Use the figure below to find the exact value of the given trigonometric expression. sin θ 2 θ 6 8 (figure not necessarily to scale) 3 3 3 10 10 1 B) C) D) E) A) 10 2 10 10 10 10 Ans: D Learning Objective: Calculate exact value of a trigonometric function using a right triangle Section: 5.5 Page 278 Copyright © Houghton Mifflin Company. All rights reserved. Chapter 5: Analytic Trigonometry 87. Use the figure below to find the exact value of the given trigonometric expression. 7 θ 24 A) B) C) D) E) Ans: C Learning Objective: Calculate exact value of a trigonometric function using a right triangle Section: 5.5 88. Use the half-angle formulas to determine the exact value of the following. cos ( 22.5D ) A) – 2+ 3 2 B) 2– 2 2 C) – 2– 2 2 D) 3– 3 2 E) 2+ 2 2 Ans: E Learning Objective: Calculate exact value of a trigonometric function using a halfangle formula Section: 5.5 Copyright © Houghton Mifflin Company. All rights reserved. Page 279 Chapter 5: Analytic Trigonometry 89. Use the half-angle formulas to determine the exact value of the given trigonometric expression. A) B) C) D) E) Ans: A Learning Objective: Calculate exact value of a trigonometric function using a halfangle formula Section: 5.5 90. Use the half-angle formula to simplify the given expression. 1 + cos 20 x 2 A) cos 40x B) cos10x C) cos 20x D) cos80x E) cos 5x Ans: B Learning Objective: Rewrite expression using half-angle formulas Section: 5.5 Page 280 Copyright © Houghton Mifflin Company. All rights reserved. Chapter 5: Analytic Trigonometry 91. Find all solutions of the given equation in the interval [ 0, 2π ) . A) B) C) D) E) Ans: C Learning Objective: Solve trigonometric equation using half-angle formulas Section: 5.5 92. Use the product-to-sum formula to write the given product as a sum or difference. 8sin A) B) C) π 8 sin 4sin π π 8 16 4 − 4 cos 4 + 4 cos D) π 4 π E) –4sin 4sin π 16 π 8 + 4 cos π 8 16 Ans: B Learning Objective: Rewrite expression with a product-to-sum formula Section: 5.5 Copyright © Houghton Mifflin Company. All rights reserved. Page 281 Chapter 5: Analytic Trigonometry 93. Use the product-to-sum formula to write the given product as a sum or difference. 4sin A) B) C) π 12 2sin 2sin cos π 12 π 12 + 2 cos π 6 2 + 2 cos π D) 12 E) 2 − 2 cos –2sin π π π 24 24 24 Ans: B Learning Objective: Rewrite expression with a product-to-sum formula Section: 5.5 94. Use the product-to-sum formulas to write the expression below as a sum or difference. sin ( 6θ ) cos ( 4θ ) D) 1 1 cos ( 2θ ) + cos (10θ ) ) ( ( cos ( 2θ ) − cos (10θ ) ) 2 2 B) E) 1 1 sin ( 2θ ) + cos (10θ ) ) ( ( sin (10θ ) − sin ( 2θ ) ) 2 2 C) 1 ( sin (10θ ) + sin ( 2θ ) ) 2 Ans: C Learning Objective: Rewrite expression with a product-to-sum formula Section: 5.5 A) 95. Use the sum-to-product formulas to write the given expression as a product. sin 9θ − sin 7θ 2sin 8θ cos θ −2 cos8θ cos θ A) D) 2 cos8θ cos θ 2 cos8θ sin θ B) E) −2sin 8θ sin θ C) Ans: E Learning Objective: Rewrite expression with a product-to-sum formula Section: 5.5 96. Use the sum-to-product formulas to write the given expression as a product. cos 6θ − cos 4θ −2sin 5θ sin θ 2sin 5θ cos θ A) D) 2 cos 5θ cos θ −2 cos 5θ cos θ B) E) 2 cos 5θ sin θ C) Ans: A Learning Objective: Rewrite expression with a sum-to-product formula Section: 5.5 Page 282 Copyright © Houghton Mifflin Company. All rights reserved. Chapter 5: Analytic Trigonometry 97. Use the sum-to-product formulas to find the exact value of the given expression. A) B) C) D) E) Ans: C Learning Objective: Evaluate expression using sum-to-product formulas Section: 5.5 98. Find all solutions of the given equation in the interval [ 0, 2π ) . A) B) C) D) E) Ans: D Learning Objective: Solve trigonometric equations using sum-to-product formulas Section: 5.5 Copyright © Houghton Mifflin Company. All rights reserved. Page 283 Chapter 5: Analytic Trigonometry 99. Verify the given identity. Ans: Learning Objective: Verify an identity using sum-to-product formulas Section: 5.5 100. Determine which of the following are trigonometric identities. I. II. cos ( 4 x ) – cos ( 2 x ) = – sin ( x ) 2sin ( 3 x ) cos ( 4 x ) – cos ( x ) = – sin ( 2 x ) sin ( 3x ) − sin ( x ) cos ( 6 x ) – cos ( 2 x ) = – sin ( 3x ) sin ( 4 x ) + sin ( 2 x ) A) I is the only identity. D) I and II are the only identities. B) II and II are the only identities. E) III is the only identity. C) None are identities. Ans: A Learning Objective: Verify an identity using sum-to-product formulas Section: 5.5 III. Page 284 Copyright © Houghton Mifflin Company. All rights reserved. Chapter 5: Analytic Trigonometry 101. Verify the given identity. Ans: Learning Objective: Verify an identity using sum-to-product formulas Section: 5.5 Copyright © Houghton Mifflin Company. All rights reserved. Page 285