Exercise Set 6.1: Sum and Difference Formulas Simplify each of the following expressions. 1. sin ( x ) 2. 3 cos x + 2 3. cos x 2 4. sin ( + x ) 5. sin 60 + sin 60 + 6. cos 60 + cos 60 + 7. cos x + cos x + 4 4 8. sin x + + sin x 6 6 9. sin 180 + sin + 180 17. Given that tan ( ) = 3 and tan ( ) = evaluate tan ( + ) . 18. Given that tan ( ) = evaluate tan ( ) . 2 , 5 4 1 and tan ( ) = , 3 2 Simplify each of the following expressions as much as possible without a calculator. ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 19. cos 55 cos 10 + sin 55 sin 10 ( ) ( ) 20. cos 75 cos 15 sin 75 sin 15 ( ( ) ) 21. sin 45 cos 15 cos 45 sin 15 22. sin 53 cos 7 + cos 53 sin 7 ( ( 10. cos 90 + + cos 90 23. cos cos sin sin 10 9 10 9 ) 24. sin cos cos sin 5 7 5 7 ) 13 25. sin 12 Answer the following. 3 11. Given that tan ( ) = 4 , evaluate tan + 4 . 12. Given that tan ( ) = 2 , evaluate tan . 4 13. Given that tan ( x ) = 2 5 , evaluate tan x 3 4 . 1 7 14. Given that tan ( x ) = , evaluate tan x + . 5 4 15. Given that tan ( x ) = 5 and tan ( y ) = 6 , evaluate tan ( x + y ) . 13 cos + cos 12 12 5 17 26. cos cos 12 12 sin 12 5 17 + sin sin 12 12 27. sin ( 2 A ) cos ( A ) cos ( 2 A ) sin ( A ) 28. cos ( 3 ) cos ( ) + sin ( 3 ) cos ( ) 29. ( ) ( ) 1 tan ( 32 ) tan ( 2 ) 30. ( ) ( ) 1 + tan ( 49 ) tan ( 4 ) tan 32 + tan 2 tan 49 tan 4 16. Given that tan ( x ) = 4 and tan ( y ) = 2 , evaluate tan ( x y ) . Math 1330, Precalculus The University of Houston Chapter 6: Trigonometric Formulas and Equations Exercise Set 6.1: Sum and Difference Formulas 38. For fractions with larger magnitude than those in Exercises 36 and 37, it can be helpful to use 5 tan tan 12 12 31. 5 1 + tan tan 12 12 larger multiples of . Rewrite each special 4 angle below so that it has a denominator of 12. 13 7 tan + tan 12 12 32. 13 7 1 tan tan 12 12 33. 34. tan ( a b ) + tan ( b ) (a) 5 4 (b) 7 4 (c) 9 4 (d) 11 4 39. Use the answers from Exercises 35 and 38 to write each fraction below as the sum of two special angles. (Hint: Each of the solutions 1 tan ( a b ) tan ( b ) contains a multiple of .) tan ( 2c + 3d ) tan ( d c ) 4 1 + tan ( 2c + 3d ) tan ( d c ) Answer the following. (a) 19 12 (b) 25 12 (c) 35 12 (d) 43 12 35. Rewrite each special angle below so that it has a denominator of 12. (a) (b) 6 2 (d) 3 (c) 4 3 (e) 4 (f) 3 5 6 36. Use the answers from Exercise 35 to write each fraction below as the sum of two special angles. (Hint: Each of the solutions contains a multiple of .) 4 (a) 5 12 (b) 7 12 (c) 13 12 (d) 11 12 37. Use the answers from Exercise 35 to write each fraction below as the difference of two special angles. (Hint: Each of the solutions contains a multiple of .) 4 5 (a) 12 (c) 7 12 Math 1330, Precalculus The University of Houston 7 (b) 12 (d) 5 12 40. Use the answers from Exercises 35 and 38 to write each fraction below as the difference of two special angles. (Hint: Each of the solutions contains a multiple of .) 4 19 12 23 (c) 12 (a) 25 12 29 (d) 12 (b) 41. Use the answers from Exercises 35 and 38, along with their negatives to write each fraction below as the difference, x y , of two special angles, where x is negative and y is positive. (a) 7 12 (b) 13 12 (c) 29 12 (d) 41 12 42. Use the answers from Exercises 35 and 38, along with their negatives to write each fraction below as the difference, x y , of two special angles, where x is negative and y is positive. (a) 5 12 (b) 19 12 Chapter 6: Trigonometric Formulas and Equations Exercise Set 6.1: Sum and Difference Formulas 31 12 Answer the following. (c) (d) 43 12 43. Use a sum or difference formula to prove that sin ( ) = sin ( ) . 44. Use a sum or difference formula to prove that cos ( ) = cos ( ) . 45. Use a sum or difference formula to prove that tan ( ) = tan ( ) . 46. Use a sum or difference formula to prove that cos = sin ( ) . 2 Find the exact value of each of the following. ( ) ( ) 48. sin 75 ( ) ( ( ) ( ) 52. tan 165 29 62. tan 12 Answer the following. (Hint: It may help to first draw right triangles in the appropriate quadrants and label the side lengths.) 4 12 63. Suppose that sin ( ) = and sin ( ) = , 5 13 where 0 < < 2 < < . Find: (a) sin ( ) (b) cos ( + ) 64. Suppose that cos ( ) = 50. cos 255 51. tan 105 37 61. tan 12 (c) tan ( ) 47. cos 15 49. sin 195 7 60. tan 12 where 0 < < < ) 7 53. sin 12 5 54. cos 12 17 55. cos 12 11 56. sin 12 31 57. cos 12 43 58. sin 12 13 59. tan 12 Math 1330, Precalculus The University of Houston 2 8 2 and cos ( ) = , 17 3 . Find: (a) sin ( + ) (b) cos ( ) (c) tan ( + ) 65. Suppose that tan ( ) = where 5 1 and cos ( ) = , 2 2 3 < < < 2 . Find: 2 (a) sin ( + ) (b) cos ( + ) (c) tan ( ) 66. Suppose that tan ( ) = 3 and sin ( ) = where < < 1 , 4 3 and < < . Find: 2 2 (a) sin ( ) (b) cos ( ) (c) tan ( + ) Chapter 6: Trigonometric Formulas and Equations Exercise Set 6.1: Sum and Difference Formulas Evaluate the following. 1 1 67. cos tan 1 + tan 1 3 2 1 68. sin tan 1 ( 4 ) + tan 1 4 3 5 69. tan cos 1 sin 1 5 13 7 4 70. tan tan 1 + cos 1 24 5 Simplify the following. 71. cos ( A ) sin ( B ) cot ( B ) + tan ( A ) 72. tan ( A ) cos ( B ) cos ( A ) cos ( A ) cot ( A ) tan ( B ) 73. sin ( A B ) sin ( A + B ) 74. cos ( A + B ) cos ( A B ) 76. cos ( x y ) + cos ( x + y ) = 2 cos ( x ) cos ( y ) 79. 80. cos ( x y ) + cos ( x + y ) sin ( x ) sin ( y ) sin ( x y ) + sin ( x + y ) cos ( x ) cos ( y ) sin ( x y ) sin ( x + y ) cos ( x y ) cos ( x + y ) = = c b C a B 81. (a) Find sin ( A ) . (b) Find cos ( B ) . (c) Analyze the answers for (a) and (b). What do you notice? (d) What is the relationship between angles A and B? (i.e. If you knew the measure of one angle, how would you find the other?) Write answers in terms of degrees. (e) Complete the following cofunction relationships by filling in blank. Write your answers in terms of A. cos ( A ) = sin ( ________ ) 75. sin ( x y ) + sin ( x + y ) = 2sin ( x ) cos ( y ) 78. A sin ( A ) = cos ( ________ ) Prove the following. 77. Use right triangle ABC below to answer the following questions regarding cofunctions, in terms of side lengths a, b, and c. = 2 cot ( x ) cot ( y ) = 2 tan ( x ) tan ( x ) tan ( y ) tan ( x ) + tan ( y ) 1 + tan ( x ) tan ( y ) 1 tan ( x ) tan ( y ) 82. (a) Find tan ( A ) . (b) Find cot ( B ) . (c) Analyze the answers for (a) and (b). What do you notice? (d) What is the relationship between angles A and B? (i.e. If you knew the measure of one angle, how would you find the other?) Write answers in terms of degrees. (e) Complete the following cofunction relationships by filling in blank. Write your answers in terms of A. tan ( A ) = cot ( ________ ) cot ( A ) = tan ( ________ ) 83. (a) Find sec ( A ) . (b) Find csc ( B ) . (c) Analyze the answers for (a) and (b). What do you notice? Continued on the next page... Math 1330, Precalculus The University of Houston Chapter 6: Trigonometric Formulas and Equations Exercise Set 6.1: Sum and Difference Formulas (d) What is the relationship between angles A and B? (i.e. If you knew the measure of one angle, how would you find the other?) Write answers in terms of degrees. (e) Complete the following cofunction relationships by filling in the blank. Write your answers in terms of A. sec ( A ) = csc ( _______ ) csc ( A ) = sec ( _______ ) ( ) ( ) 97. sec 20 sin 70 98. cos csc 6 3 5 99. cot 2 sec2 12 12 ( ) ( ) 100. cos 2 72 + cos 2 18 84. Use a sum or difference formula to prove that ( ) sin 90 = cos ( ) . Use cofunction relationships to solve the following for acute angle x. ( ) 85. sin 75 = cos ( x ) 3 86. cos = sin ( x ) 10 2 87. sec 5 = csc ( x ) ( ) 88. csc 41 = sec ( x ) ( ) = cot ( x ) 89. tan 72 90. cot = tan ( x ) 3 Simplify the following. ( ) 91. cos 90 x csc ( x ) 92. sin x sec ( x ) 2 93. sin x tan ( x ) 2 94. csc x sin ( x ) 2 ( ) ( ) 95. sec 90 cos ( ) ( 96. tan 90 x csc 90 x Math 1330, Precalculus The University of Houston ) Chapter 6: Trigonometric Formulas and Equations Exercise Set 6.2: Double-Angle and Half-Angle Formulas Answer the following. 1. 6. formula on tan ( + ) . (a) Evaluate sin 2 . 6 (b) Evaluate 2 sin . 6 7. The sum formula for cosine yields the equation cos ( 2 ) = cos 2 ( ) sin 2 ( ) . To write cos ( 2 ) strictly in terms of the sine function, (c) Is sin 2 = 2sin ? 6 6 (a) Using the Pythagorean identity cos 2 ( ) + sin 2 ( ) = 1 , solve for cos 2 ( ) . (d) Graph f ( x ) = sin ( 2 x ) and g ( x ) = 2sin ( x ) (b) Substitute the result from part (a) into the above equation for cos ( 2 ) . on the same set of axes. (e) Is sin ( 2 x ) = 2sin ( x ) ? 8. 2. Derive the formula for tan ( 2 ) by using a sum (a) Evaluate cos 2 . 6 The sum formula for cosine yields the equation cos ( 2 ) = cos 2 ( ) sin 2 ( ) . To write cos ( 2 ) strictly in terms of the cosine function, (b) Evaluate 2 cos . 6 (a) Using the Pythagorean identity cos 2 ( ) + sin 2 ( ) = 1 , solve for sin 2 ( ) . (c) Is cos 2 = 2 cos ? 6 6 (b) Substitute the result from part (a) into the above equation for cos ( 2 ) . (d) Graph f ( x ) = cos ( 2 x ) and g ( x ) = 2cos ( x ) on the same set of axes. (e) Is cos ( x ) = 2 cos ( x ) ? Answer the following. 9. Suppose that cos ( ) = 12 3 and < < 2 . 2 13 Find: 3. (a) Evaluate tan 2 . 6 (b) Evaluate 2 tan 6 (c) Is tan 2 = 2 tan ? 6 6 (d) Graph f ( x ) = tan ( 2 x ) and g ( x ) = 2 tan ( x ) on the same set of axes. (e) Is tan ( 2 x ) = 2 tan ( x ) ? 4. Derive the formula for sin ( 2 ) by using a sum formula on sin ( + ) . (a) sin ( 2 ) (b) cos ( 2 ) (c) tan ( 2 ) 10. Suppose that tan ( ) = 3 3 and < < . 4 2 Find: (a) sin ( 2 ) (b) cos ( 2 ) (c) tan ( 2 ) 11. Suppose that sin ( ) = 2 and < < . 2 5 Find: 5. Derive the formula for cos ( 2 ) by using a sum (a) sin ( 2 ) formula on cos ( + ) . (b) cos ( 2 ) (c) tan ( 2 ) Math 1330, Precalculus The University of Houston Chapter 6: Trigonometric Formulas and Equations Exercise Set 6.2: Double-Angle and Half-Angle Formulas 12. Suppose that tan ( ) = 3 and 3 < < 2 . 2 ( ) ( 24. sin 2 67.5! cos 2 67.5! Find: ) ( ) 1 tan ( 41 ) 2 tan 41! (a) sin ( 2 ) 25. (b) cos ( 2 ) (c) tan ( 2 ) ! 2 ( 26. 1 2 cos 2 22.5! Simplify each of the following expressions as much as possible without a calculator. ( ) ( ) 13. 2 sin 15! cos 15! ) 27. 2 cos 2 1 2 28. 2 tan ( 3 ) tan 2 ( 3 ) 1 14. cos 2 sin 2 2 2 2 ( ) ! 15. 2 cos 34 1 5 16. 1 2 sin 2 12 ( 2 tan 105! 17. ( ) 1 tan 105! 2 ) 7 cos 8 ( ) ( ) 3 21. cos 8 2 2 3 sin 8 11 2 tan 12 22. 11 1 tan 2 12 7 23. 2 sin 2 12 1 Math 1330, Precalculus The University of Houston ! ( ) cos (112.5 ) 30. (a) sin 15! (b) 20. 2 sin 23! cos 23! ( ) sin ( 67.5 ) 29. (a) cos 105! (b) 18. 1 sin 2 ( x ) 7 19. 2sin 8 x x The formulas for sin and cos both contain a 2 2 sign, meaning that a choice must be made as to whether or not the sign is positive or negative. For each of the following examples, first state the quadrant in which the angle lies. Then state whether the given expression is positive or negative. (Do not evaluate the expression.) ! 15 31. (a) cos 8 13 (b) sin 12 7 32. (a) sin 8 19 (b) cos 12 Chapter 6: Trigonometric Formulas and Equations Exercise Set 6.2: Double-Angle and Half-Angle Formulas s In the text, tan is defined as: 2 sin ( s ) s tan = . 2 1 + cos ( s ) The following exercises can be used to derive this formula along with two additional formulas for s tan . 2 s 33. (a) Write the formula for sin . 2 s (b) Write the formula for cos . 2 s (c) Derive a new formula for tan ` using the 2 sin ( ) s identity tan ( ) = , where = . 2 cos ( ) Leave both the numerator and denominator in radical form. Show all work. 34. (a) This exercise will outline the derivation for: sin ( s ) s . In exercise 33, it was tan = 2 1 + cos ( s ) discovered that 1 cos ( s ) s tan = . 1 + cos ( s ) 2 Rationalize the denominator by multiplying both the numerator and denominator by 1 + cos ( s ) . Simplify the expression and s write the result for tan . 2 (b) A detailed analysis of the signs of the trigonometric functions of s and s2 in various quadrants reveals that the symbol in part (a) is unnecessary. (This analysis is lengthy and will not be shown here.) Given this fact, rewrite the formula from part (a) without the symbol. 35. (a) In exercise 33, it was discovered that 1 cos ( s ) s tan = . 1 + cos ( s ) 2 Rationalize the numerator by multiplying both the numerator and denominator by 1 cos ( s ) . Simplify the expression and s write the new result for tan . 2 (b) A detailed analysis of the signs of the trigonometric functions of s and s2 in various quadrants reveals that the symbol in part (a) is unnecessary. (This analysis is lengthy and will not be shown here.) Given this fact, rewrite the formula from part (a) without the symbol. This gives yet another formula which can be used for s tan . 2 (c) Use the results from Exercises 33-35 to s write the three formulas for tan . Which 2 formula seems easiest to use and why? Which formula seems hardest to use and why? s 36. (a) In the text, tan is defined as: 2 sin ( s ) s . tan = 2 1 + cos ( s ) Multiply both the numerator and denominator of the right-hand side of the equation by 1 cos ( s ) . Then simplify to s obtain a formula for tan . Show all 2 work. (b) How does the result from part (a) compare to the identity obtained in part (b) of Exercise 39? (c) How does this result from part (b) compare with the formula given in the text s for tan ? 2 Math 1330, Precalculus The University of Houston Chapter 6: Trigonometric Formulas and Equations Exercise Set 6.2: Double-Angle and Half-Angle Formulas Answer the following. SHOW ALL WORK. Do not leave any radicals in the denominator, i.e. rationalize the denominator whenever appropriate. ( ) by using a sum or difference 37. (a) Find cos 75! formula. ( ) by using a half-angle (b) Find cos 75! formula. (c) Enter the results from parts (a) and (b) into a calculator and round each one to the nearest hundredth. Are they the same? ( ) ( ) 38. (a) Find sin 165! by using a sum or difference formula. (b) Find sin 165! by using a half-angle formula. (c) Enter the results from parts (a) and (b) into a calculator and round each one to the nearest hundredth. Are they the same? ( ) ( ) ( ) formula. (b) Find cos 112.5! by using a half-angle formula. (c) Find tan 112.5! by computing ( ). cos (112.5 ) Find tan (112.5 ) by using a half-angle sin 112.5! (b) ! (c) formula. ) 40. (a) Find sin 105! by using a half-angle ( (b) Find cos 105 (b) ) by using a half-angle (c) formula. ( ) ( ) (c) Find tan 105! by computing Math 1330, Precalculus The University of Houston ! ! ( ) cos ( 285 ) tan ( 285 ) ( ). cos (105 ) ! ! sin 105! (d) Find tan 105! by using a half-angle formula. ( ) cos (157.5 ) tan (157.5 ) 45. (a) sin 285! formula. ! 3 42. (a) sin 8 3 (b) cos 8 3 (c) tan 8 44. (a) sin 157.5! ! ( 13 41. (a) sin 12 13 (b) cos 12 13 (c) tan 12 11 43. (a) sin 8 11 (b) cos 8 11 (c) tan 8 39. (a) Find sin 112.5! by using a half-angle (d) Find the exact value of each of the following by using a half-angle formula. Do not leave any radicals in the denominator, i.e. rationalize the denominator whenever appropriate. ! 17 46. (a) sin 12 17 (b) cos 12 17 (c) tan 12 Chapter 6: Trigonometric Formulas and Equations Exercise Set 6.2: Double-Angle and Half-Angle Formulas (g) Find the exact value of cos . 2 Answer the following. 47. If cos ( ) = 5 3 and < < 2 , 2 9 (h) Find the exact value of tan . 2 (a) Determine the quadrant of the terminal side of . (b) Complete the following: ___ < 2 < ___ 49. If tan ( ) = (c) Determine the quadrant of the terminal side of 2 (a) Find the exact value of sin . 2 . (b) Find the exact value of cos . 2 (d) Based on the answer to part (c), determine the sign of sin . 2 (e) Based on the answer to part (c), determine the sign of cos . 2 (f) Find the exact value of sin . 2 (c) Find the exact value of tan . 2 (a) Find the exact value of sin . 2 (b) Find the exact value of cos . 2 (h) Find the exact value of tan . 2 21 3 and < < , 2 5 (a) Determine the quadrant of the terminal side of . (c) Find the exact value of tan . 2 Prove the following. 51. (b) Complete the following: _____ < < _____ 2 (c) Determine the quadrant of the terminal side of 2 . 5 3 and < < 2 , 2 6 50. If cos ( ) = (g) Find the exact value of cos . 2 48. If sin ( ) = 7 and < < , 2 3 52. 1 cos ( 2 x ) sin ( 2 x ) cos ( 3 x ) cos ( x ) = tan ( x ) sin ( 3 x ) sin ( x ) = 2 53. 1 2 sin ( x ) 1 + 2 sin ( x ) = cos ( 2 x ) 54. cos 4 ( x ) sin 4 ( x ) = cos ( 2 x ) (d) Based on the answer to part (c), determine the sign of sin . 2 x 55. csc ( x ) cot ( x ) = tan 2 (e) Based on the answer to part (c), determine the sign of cos . 2 x 1 + tan 2 2 = sec 56. ( ) 2 x 1 tan 2 (f) Find the exact value of sin . 2 Math 1330, Precalculus The University of Houston Chapter 6: Trigonometric Formulas and Equations Math 1330 Precalculus Electronic Homework (EHW 8) Sections 6.1, 6.2, 6.3 Work the following problems and choose the correct answer. The problems that refer to the Textbook may be found at www.casa.uh.edu in the Online Textbook under the Exercises link in the appropriate section. Record your answers by logging into your CASA account, go to the EMCF link in the top menu bar, selecting the appropriate assignment in the EMCF menu and recording your choices on the EMCF form. You MUST record your answers by the deadline given. No late homework will be accepted. Your grade on this homework is out of 20 points. Problem 6.1.8 refers to problem 8 in Chapter 6, Section 1 of the online text. Section 6.1 1. Problem 6.1.8 A. 2 sin x B. 3 sin x C. 2 sin x D. sin x E. None of the above 2. Problem 6.1.48 (Find the value using sum and difference formulas.) A. B. C. D. E. 6 2 4 2 6 4 2 6 4 6 2 2 None of the above Page 1 of 9 3. Problem 6.1.56 (Hint: Express the given angle as the sum or difference of two known angles) 3 A. 5 B. 5 2 4 6 D. 4. 3 4 C. E. 2 4 2 4 None of the above Problem 6.1.64 a and b A. a: 30 8 5 51 b: 16 15 5 51 B. a: 30 8 5 51 b: 16 15 5 51 C. a: 30 8 5 51 b: 16 15 5 51 D. a: 16 15 5 51 b: 30 8 5 51 E. a: 16 15 5 51 b: 30 8 5 51 F. None of the above 5. Problem 6.1.66 a A. B. 1 3 15 4 10 1 3 15 4 10 Page 2 of 9 1 3 15 4 10 C. 1 3 15 4 10 D. E. None of the above Section 6.2 6. Problem 6.2.10 a and b A. sin 2 6 5 cos 2 8 5 B. sin 2 24 25 cos 2 7 25 C. sin 2 24 25 cos 2 7 25 D. sin 2 24 25 cos 2 7 25 E. sin 2 24 25 cos 2 1 7 cos 2 4 5 F. None of the above 7. Problem 6.2.12 a and b 3 5 A. sin 2 B. sin 2 3 5 cos 2 4 5 C. sin 2 3 10 5 cos 2 2 10 5 D, sin 2 cos 2 4 5 E. sin 2 3 5 3 10 5 cos 2 10 5 F. None of the above Page 3 of 9 8. 9. Problem 6.2.42 (a) and (b) 2 2 A. a: B. a: C. a: D. a: E. a: F. None of the above 2 2 2 b: 2 2 2 2 2 2 2 2 b: b: 2 2 2 2 2 2 b: 2 2 2 b: 2 2 2 2 2 2 2 2 Problem 6.2.50 a 3 A. 12 B. C. 3 6 33 6 33 6 D. E. 3 6 F. None of the above 10. Given A. is an acute angle with cos 1 , find the value of 4 tan 2 tan 2 . 5 15 14 Page 4 of 9 B. 2 15 35 C. 12 15 35 D. 2 15 35 E. None of the above Note: For questions 1 - 4, let \"k\" be an integer. 11. Problem 6.3.6 b and c A. b: B. b: C. b: D. b: E. b: 6 3 3 3 , 5 6 c: , 2 3 c: , 2 3 c: , 2 3 c: c: 3 6 3 3 3 3 2k , 5 6 2k k , 2 3 k 2 , 2 3 2 2k , 2 3 2k 2k F. None of these 12. Problem 6.3.10 b and c A. b: 5 11 , 6 6 c: 5 6 k B. b: 2 5 , 3 3 c: 2 3 k C. b: 4 3 3 c: , 3 k Page 5 of 9 4 3 D. b: E. None of the above 13. 3 , c: 3 2k , 4 3 c: 2k , 5 3 2k , 2k , 5 3 2k 2k , 5 3 2k , 2k Problem 6.3.16 b and c 5 3 , 3 2 A. b: B. b: 5 3 3 C. b: , D. b: E. b: 3 , , c: 5 , 3 c: 2 4 , ,0 3 3 c: 5 , 3 c: 3 3 , 3 3 3 2 3 3 4 3 2k , 2 , 5 3 3 2 2k 2k 2k , 0 2k 2 , 2k F. None of the above 14. Problem 6.3.30 Use radians instead of degrees. 0 x 2 A. B. C. D. E. 0, 2 2 , 3 , 2 , 3 2 No solution. F. None of the above Page 6 of 9 15. Problem 6.3.38 0 x 2 A. 7 11 , 6 6 B. 7 11 19 23 31 35 , , , , , 18 18 18 18 18 18 C. 7 11 19 23 , , , 18 18 18 18 D. E. 18 5 13 17 25 29 , , , , 18 18 18 18 18 , No solution. F. None of these Problem 6.3.40 16. A. 3 , 2 2 B. 5 7 , 4 4 0 x 2 3 2 C. , D. 0, E. No solution. 2 F. None of these Page 7 of 9 17. Problem 6.3.46 A. 6 5 7 11 , , 6 6 6 , B. 11 13 23 , , 12 12 12 12 C. 5 3 3 D. 5 6 6 E. 2 3 , , , F. None of these 18. Problem 6.3.48 Use radians instead of degrees. 0 A. B. C. D. E. x 2 7 13 19 25 31 , , , , 18 18 18 18 18 18 , 9 9 , 4 7 10 13 16 , , , , 9 9 9 9 9 , 4 10 13 , , 9 9 9 3 9 F. None of these 19. Solve over the interval 0, A. x 1 2 arcsin ; x 4 3 B. x arcsin 2 ; 3 x 2 : 3sin(4 x) 1 2 1 2 arcsin 4 3 arcsin 2 3 Page 8 of 9 1 ; 12 C. x arcsin D. x 1 1 arcsin ; 4 3 1 1 arcsin ; 4 3 F. None of these E. x 20. x x arcsin 4 x 1 12 1 1 arcsin 4 3 1 1 arcsin 4 3 Find all solutions in the interval 0, 2 : cos 2 (2 x) sin 2 (2 x) 1 2 (Hint: Use an identity to simplify the left side.) A. 5 6 6 B. 2 4 , 3 3 C. 5 7 11 , , 12 12 12 12 D. 5 7 11 , , 6 6 6 6 E. 5 12 12 , , , , F. None of these Page 9 of 9