Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Name: ________________________ Class: ___________________ Date: __________ Practice Test3 (Trigonometry) ID: A Instructor: Koshal Dahal Multiple Choice Questions SHOW ALL WORK, EVEN FOR MULTIPLE CHOICE QUESTIONS, TO RECEIVE CREDIT. ____ ____ ____ 1. Find the values of the trigonometric functions of t if sec t 0 0 1 0 1 , cost , tan t , cot t 1 1 0 1 a. sint b. sin t 0 1 0 1 0 , cost , tan t , cot t 1 0 1 1 c. sint 0 0 1 0 0 , cost , tan t , cot t 1 1 0 1 2. Find the values of the trigonometric functions of t if tant a. sint b. sint c. sint 11 11 5 11 5 , cost , tan t , cot t 6 5 11 6 11 11 5 11 5 , cost , tan t , cot t 6 5 11 6 11 11 5 11 5 , cost , tan t , cot t 6 5 11 6 3. Rewrite the expression as an algebraic expression in x. tan (sin – 1 x) 1 a. x2 1 x b. 1 x2 ____ 1 , and the terminal point of t is in quadrant IV. 1 c. 1 x2 d. x2 1 4. Rewrite the expression as an algebraic expression in x. sin (cos – 1 x) a. b. x2 1 x–1 c. d. e. 1 x2 1–x x 1 11 and cos t 0. 5 Name: ________________________ ____ ID: A 5. Find the exact value of the expression. ÊÁ ˆ ÁÁ 1 3 ˜˜˜ Á ˜˜ cos ÁÁ sin ÁÁ 2 ˜˜˜ Ë ¯ a. b. ____ 2 1 2 c. 2 2 d. 3 2 6. Simplify the expression. sin 14x sin13x sin x sin6x a. sin7x cos 6x b. cos 7x sin 13x c. sin6x sin7x d. sin6x cos 7x e. cos 6x ____ 7. Find all solutions of the equation. 2 cos x 2 0 Select the correct answer, where k is any integer: 4 2k , 2k a. 5 5 7 b. 2k , 2k 4 4 7 c. k , k 4 4 9 d. 2k , 2k 5 5 2 Name: ________________________ ____ ID: A 8. Find all solutions of the equation. 2 sin x 1 0 Select the correct answer, where k is any integer: 11 k , k a. 6 6 5 b. k , k 6 6 5 c. 2k , 2k 6 6 11 d. 2k , 2k 6 6 ____ 9. Find all solutions of the following equation. 4 cos 2 x 3 0 Select the correct answer, where k is any integer: 5 7 11 2k , 2k , 2k , 2k a. 6 6 6 6 11 b. k , k 6 6 5 7 11 c. k , k , k , k 6 6 6 6 5 d. 2k , 2k 6 6 ____ 10. Find all solutions of the following equation. sec 2 x 4 0 Select the correct answer, where k is any integer: 2 2k , 2k a. 3 3 5 b. k , k 3 3 2 4 5 c. 2k , 2k , 2k , 2k 3 3 3 3 2 4 5 d. k , k , k , k 3 3 3 3 3 Name: ________________________ ID: A ____ 11. Find all solutions of the following equation. 4 cos 2 x – 4 cos x + 1 = 0 Select the correct answer, where k is any integer: 7 a. 2k , 2k 4 4 5 b. 2k , 2k 3 3 5 c. 2k , 2k 6 6 3 d. 2k , 2k 4 4 ____ 12. Find all solutions of the following equation. sin 2 x = –3 sin x + 4 Select the correct answer, where k is any integer: a. b. c. d. 2 4 k 2 2k 2k , 3 2k 4 k ____ 13. Find all solutions of the following equation. Ê cos x ÁÁÁ 2sinx Ë ˆ 2 ˜˜˜ 0 ¯ Select the correct answer, where k is any integer: 5 2k , 2k a. 2 4 3 b. 2k , 2k 2 2 5 7 c. 2k , 2k 4 4 3 5 7 d. 2k , 2k , 2k , 2k 2 2 4 4 4 Name: ________________________ ID: A ____ 14. Find all solutions of the following equation. sin x cos x – 2 sin x = 0 Select the correct answer, where k is any integer: 3 a. 2k , 2k 4 4 b. 2k , 2k c. d. k 6 3 2k , 2k 2 2 2 k , ____ 15. Find all solutions of the following equation in the interval [0,2 ). tan x – 3 cot x = 0 Select the correct answer, where k is any integer: 2 , a. 3 3 2 4 5 b. , , , 3 3 3 3 2 c. 2k , 2k 3 3 2 4 5 d. 2k , 2k , 2k , 2k 3 3 3 3 ____ 16. Use an addition or subtraction formula to simplify the following equation. Then find all the solutions in the ÍÈÍ ˆ˜ interval ÍÍÍÍ 0, ˜˜˜˜ . ÍÎ 4 ¯ cos x cos 7 x – sin x sin 7 x = 0 a. b. c. d. 16 8 , 3 8 , 3 16 8 16 5 Name: ________________________ ID: A ____ 17. Use an addition or subtraction formula to simplify the following equation. Then find all the solutions in the È interval ÍÍÍÎ 0,2 ˆ˜¯ . sin 4 x cos 3 x – cos 4 x sin 3 x = 0 a. b. 0, , 2 c. 0, 2 d. 0, ÊÁ ____ 18. Plot the point that has the polar coordinates ÁÁÁÁ 5, Ë 4 a. d. b. e. c. 6 ˆ˜ ˜˜ . ˜˜ ¯ Name: ________________________ ID: A ÁÊ 7 ____ 19. Plot the point that has the polar coordinates ÁÁÁÁ 3, 6 Ë a. d. b. e. c. 7 ˜ˆ˜ ˜˜ . ˜ ¯ Name: ________________________ ID: A ____ 20. Determine which point in the figure, P, Q, R, or S, has the given polar coordinates. ÊÁ ÁÁ 4, 25 ÁÁ 4 Ë a. b. c. d. ˆ˜ ˜˜ ˜˜ ¯ P S Q R 8 Name: ________________________ ID: A ____ 21. Determine which point in the figure, P, Q, R, or S, has the given polar coordinates. ÊÁ ÁÁ 4, 21 ÁÁ 4 Ë a. b. c. d. ˆ˜ ˜˜ ˜˜ ¯ P Q S R 9 Name: ________________________ ID: A ____ 22. Determine which point in the figure, P, Q, R, or S, has the given polar coordinates. ÊÁ ÁÁ 4, 25 ÁÁ 4 Ë a. b. c. d. ˆ˜ ˜˜ ˜˜ ¯ P R S Q ____ 23. Find the modulus of the complex number. –9 a. b. c. d. 9 0 –9 10 ____ 24. Find the modulus of the complex number. 1 a. b. c. d. 3 i 3 5 3 2 9 2 3 3 2 3 10 Name: ________________________ ID: A ____ 25. Sketch the complex number z, also sketch 2z, –z and z 1 2i a. d. b. e. c. 11 1 z on the same complex plane. 2 Name: ________________________ ID: A ____ 26. Sketch the complex number z and its complex conjugate z on the same complex plane. z 7 4i a. d. b. e. c. 12 Name: ________________________ ID: A ____ 27. Sketch z 1 ,z 2 ,z 1 z 2 , and z 1 z 2 on the same complex plane. z 1 2 i, z 2 2 3i a. d. b. e. c. 13 Name: ________________________ ID: A ____ 28. Write the complex number 4 4 3i in polar form with argument between and . ÁÊ ˜ˆ ÁÊ ˜ˆ a. 8 cos ÁÁÁÁ ˜˜˜˜ 8i sin ÁÁÁÁ ˜˜˜˜ Ë3¯ Ë3¯ ÊÁ ˆ˜ ÊÁ ˆ˜ b. 4 cos ÁÁÁÁ ˜˜˜˜ 4i sin ÁÁÁÁ ˜˜˜˜ Ë3¯ Ë3¯ ÊÁ ˆ˜ ÊÁ ˆ˜ c. cos ÁÁÁ ˜˜˜ i sin ÁÁÁ ˜˜˜ ÁË 3 ˜¯ ÁË 3 ˜¯ ____ 29. Write the complex number 4 + 4 i in polar form with argument between 0 and 2 . 4i cos a. 4 sin b. cos c. 4 2 cos 4 i sin 4 4 4 4 4 2 i sin 4 ____ 30. Write the complex number 5 i in polar form with argument between 0 and 2 . a. 5 cos b. 5 cos c. 3 4 5 cos 2 5i sin 5i sin 5i sin 3 4 2 ____ 31. Write the complex number –2 in polar form with argument between 0 and 2 . 2i sin a. 2 cos b. 2 cos 2i sin c. 2 cos 4 2 2i sin 4 2 ____ 32. Write the complex number i( 8 – 8 i ) in polar form with argument between 0 and 2 . a. 8 2 cos b. cos c. 8 sin 4 4 4 8 2 i sin i sin 4 8i cos 4 4 14 Name: ________________________ ____ 33. Find the quotient z 1 cos z 2 cos 5 i sin ID: A z1 , in polar form, if: z2 5 i sin 6 6 11 11 cos i sin 30 30 a. b. cos c. cos 11 30 i sin i sin 11 30 ____ 34. Find the product z 1 z 2, expressed in polar form, if: z 1 cos z 2 cos 2 12 i sin i sin 2 12 i sin 12 12 5 5 cos i sin 12 12 7 7 cos i sin 12 12 cos a. b. c. ____ 35. Find the quotient z1 , in polar form, if: z2 ˆ Ê 7 ÁÁ cos 75 i sin 75 ˜˜ ¯ Ë ˆ Ê ˜ Á 3 7 Á cos 60 i sin 60 ˜ ¯ Ë 7 ÊÁ ˆ Á cos 75 i sin 60 ˜˜ 3Ë ¯ 1 ÊÁ ˆ Á cos 15 i sin 15 ˜˜ 3Ë ¯ ÊÁ ˆ 3 Á cos 15 i sin 15 ˜˜ ¯ Ë z1 z2 a. b. c. 15 Name: ________________________ ID: A ____ 36. Write z 1 and z 2 in polar form, and then find the product z 1 z 2 if: Ê ˆ z 1 2 ÁÁÁ 3 i ˜˜˜ Ë ¯ ÊÁ ˆ z 2 3 ÁÁ 1 i 3 ˜˜˜ Ë ¯ Express your answer in polar form. ÊÁ ˆ˜ a. 3 ÁÁÁÁ cos i sin ˜˜˜˜ 4 4¯ Ë ÊÁ ˆ˜ b. 24 ÁÁÁ cos i sin ˜˜˜ ÁË 2 2 ˜¯ ÊÁ ˆ˜ c. 2 ÁÁÁÁ cos i sin ˜˜˜˜ 2 2¯ Ë ____ 37. Write z 1 and z 2 in polar form, and then find the quotient Ê ˆ z 1 ÁÁÁ 2 5 2i 5 ˜˜˜ Ë ¯ z 2 (2 2i) a. b. c. 5i 2i 5 ____ 38. Find the indicated power using DeMoivre's Theorem. ( 1 + i ) 16 a. –256 b. 256 c. –256 i ____ 39. Find the indicated power using DeMoivre's Theorem. ( 1 – i ) 12 a. 64 i b. –64 c. 64 ____ 40. Find the indicated power using DeMoivre's Theorem. ÊÁ ÁÁ 2 2 ÁÁ ÁÁ 2 2 Á Ë a. i b. 1 c. –1 ˆ˜ 12 ˜˜ i ˜˜˜ ˜˜ ¯ 16 z1 if: z2 Name: ________________________ ID: A ____ 41. Find the indicated power using DeMoivre's Theorem. ˆ 10 ÁÊÁ 3 3 ˜˜˜˜ ÁÁ i˜ ÁÁ ÁÁ 2 2 ˜˜˜ Ë ¯ ÊÁ ˆ 3 ˜˜˜˜ ÁÁÁ 1 i˜ a. 243 ÁÁ ÁÁ 2 2 ˜˜˜ Ë ¯ ÊÁ ˆ ÁÁ 3 1 ˜˜˜ Á i ˜˜˜ b. 243 ÁÁ 2 ˜˜ ÁÁ 2 Ë ¯ ÁÊÁ 3 1 ˜ˆ˜ Á ˜ i ˜˜˜ c. 243 ÁÁÁ ÁÁ 2 2 ˜˜ Ë ¯ ____ 42. Find the cube roots of: 125 i a. b. c. ÊÁ ˆ˜ ÊÁ 3 5 5 ˆ˜˜˜ ÊÁÁÁ 3 ˆ˜˜˜ , 5 cos 5 ÁÁÁ cos i sin ˜˜˜˜ , 5 ÁÁÁÁ cos i sin i sin ˜ Á ÁË 6 6¯ Ë 6 6 ˜¯ ÁË 2 2 ˜˜¯ ÊÁ ÊÁ ÊÁ ˆ˜ 5 5 ˆ˜˜˜ ÁÁ cos 3 i sin 3 , 5 5 ÁÁÁÁ cos i sin ˜˜˜˜ , 5 ÁÁÁÁ cos i sin ˜ ÁÁ ˜ 6 6 6 6 2 2 Ë ¯ Ë ¯ Ë ÊÁ ˆ˜ ÊÁ 13 7 7 ˆ˜˜˜ ÊÁÁÁ 13 ˆ˜˜˜ , 5 cos 5 ÁÁÁÁ cos i sin ˜˜˜˜ , 5 ÁÁÁÁ cos i sin i sin ˜ Á 6 6¯ Ë 6 6 ˜¯ ÁË 6 6 ˜˜¯ Ë ˆ˜ ˜˜ ˜˜ ¯ Short Answer 43. Use an appropriate half-angle formula to find the exact value of the expression. sin15 44. Use an appropriate half-angle formula to find the exact value of the expression. tan22.5 45. Use an appropriate half-angle formula to find the exact value of the expression. sin 12 17 Name: ________________________ ID: A 46. Simplify each expression by using a double-angle formula. (a.) 2 tan2 = tan ÊÁË _ _ _ _ _ _ _ _ _ _ ˆ˜¯ 1 tan 2 2 (b.) 2tan2 = __________ 1 tan 2 2 47. Simplify each expression by using a half-angle formula. (a.) 1 cos 30 sin ÊÁË _ _ _ _ _ _ _ _ _ _ ˆ˜¯ 2 (b.) 1 cos 10 = __________ 2 48. Write the product as a sum. sin x sin5x 49. Write the product as a sum. 7cos 6x cos 7x 50. Write the sum as a product. sin2x sin7x 51. Write the sum as a product. sin3x sin4x 52. Find the value of the product. 3 sin 37.5 cos 7.5 53. Simplify the expression. 2tanx 1 tan 2 x 54. Find all solutions of the equation. 2 cos x 1 0 18 Name: ________________________ ID: A 55. Find all solutions of the equation. tan 9 x 81 tan x 0 56. Find all solutions of the equation. 4 cos 2 x 4 cos x 1 0 57. Find all solutions of the equation. 4 sin x cos x 2 sin x 2 cos x 1 0 58. Consider the equation. 2 tan x 19 (a) Find all solutions of the equation. x = __________ (b) Use a calculator to solve the equation in the interval ÍÈÍ 0,2 ˆ˜ , correct to five decimal places. ÍÎ ¯ x = __________ 59. Use an addition or subtraction formula to simplify the equation. Then find all solutions in the interval ÈÍ ÍÍ 0,2 ˆ˜ . Î ¯ sin4x cos x cos 4x sin x 3 2 60. Solve the equation by first using a sum-to-product formula. sinx sin 3x 0 19 Name: ________________________ ID: A 61. If a projectile is fired with velocity v 0 at an angle , the its range, the horizontal distance it travels (in feet), is modeled by the function R ( ) v 20 sin2 32 . If v 0 2,000 ft/s, what angle (in degrees) should be chosen for the projectile to hit a target on the ground 5000 ft away? Please give the answer to five decimal places. __________ 62. A point is graphed in rectangular form. Find polar coordinates for the point, with r 0 and 0 2 . 63. A point is graphed in polar form. Find its rectangular coordinates. 20 Name: ________________________ ID: A ÁÊ 7 64. Find the rectangular coordinates for the point whose polar coordinates are ÁÁÁÁ 1, 2 Ë Ê 65. Convert the rectangular coordinates ÁÁÁ 0, Ë ˆ 7 ˜˜˜ to polar coordinates with r 0 and 0 2 . ¯ 66. Convert the equation to polar form. x2 67. Convert the polar equation to rectangular coordinates. 7 68. Convert the polar equation to rectangular coordinates. r 2 1 5sin 69. Write the complex numbers in polar form with argument between 0 and 2 . (a) 1 (b) 3 i __________ 2 2i __________ 70. Find the product z 1 z 2 and the quotient ÊÁ 2 2 z 1 7 ÁÁÁ cos i sin ÁË 3 3 z1 . Express your answer in polar form. z2 ˆ˜ Ê ˆ ˜˜ , z 2 ÁÁÁ cos i sin ˜˜˜ ˜˜ 2 ÁÁ 3 3 ˜˜¯ ¯ Ë (a) z 1 z 2 __________ (b) ˜ˆ˜ ˜˜ . ˜ ¯ z1 = __________ z2 71. Find the indicated power using DeMoivre's Theorem. ÊÁ ˆ4 ÁÁ 2 3 2i ˜˜˜ Ë ¯ 72. Find the indicated power using DeMoivre's Theorem. (1 i) 8 21 ID: A Practice Test3 (Trigonometry) Answer Section Instructor: Koshal Dahal MULTIPLE CHOICE 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. ANS: ANS: ANS: ANS: ANS: ANS: ANS: ANS: ANS: ANS: ANS: ANS: ANS: ANS: ANS: ANS: ANS: ANS: ANS: ANS: ANS: ANS: ANS: ANS: ANS: ANS: ANS: ANS: ANS: ANS: ANS: ANS: ANS: ANS: ANS: ANS: ANS: ANS: ANS: ANS: A C B C B E B C A C B A D B B D D A C D D C A C B D D A C C B A C C B B C B B C PTS: PTS: PTS: PTS: PTS: PTS: PTS: PTS: PTS: PTS: PTS: PTS: PTS: PTS: PTS: PTS: PTS: PTS: PTS: PTS: PTS: PTS: PTS: PTS: PTS: PTS: PTS: PTS: PTS: PTS: PTS: PTS: PTS: PTS: PTS: PTS: PTS: PTS: PTS: PTS: 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ID: A 41. ANS: A 42. ANS: A PTS: 1 PTS: 1 SHORT ANSWER 43. ANS: 2 2 3 PTS: 1 44. ANS: 2 1 PTS: 1 45. ANS: 1 ÁÊ Á 6 4 ÁË ˆ 2 ˜˜˜ ¯ PTS: 1 46. ANS: 4; tan(4) PTS: 1 47. ANS: 15; sin(5) PTS: 1 48. ANS: 1 Ê Á cos(4x) cos(6x) ˆ˜¯ 2 Ë PTS: 1 49. ANS: 7 Ê Á cos(13x) cos(x) ˆ˜¯ 2 Ë PTS: 1 50. ANS: ÁÊ 9x ˜ˆ ÁÊ 5x ˜ˆ 2 cos ÁÁÁÁ ˜˜˜˜ sin ÁÁÁÁ ˜˜˜˜ Ë 2 ¯ Ë 2 ¯ PTS: 1 2 ID: A 51. ANS: ÊÁ 7x ˆ˜ ÊÁ x ˆ˜ 2 sin ÁÁÁÁ ˜˜˜˜ cos ÁÁÁÁ ˜˜˜˜ Ë 2 ¯ Ë2¯ PTS: 1 52. ANS: ˆ 3 ÊÁ ÁÁ 2 1 ˜˜˜ 4 Ë ¯ PTS: 1 53. ANS: sin(2x) PTS: 1 54. ANS: 2 4 x 2k , 2k 3 3 PTS: 1 55. ANS: k x 3 PTS: 1 56. ANS: x 3 2k , 5 2k 3 PTS: 1 57. ANS: 11 2 4 7 2k , 2k , 2k , 2k 6 6 3 3 PTS: 1 58. ANS: 1.46592 k ; 1.46592, 4.60751 PTS: 1 59. ANS: 2 7 8 13 14 19 4 5 26 , , , , , , , , , 15 15 15 15 15 15 15 3 3 15 PTS: 1 60. ANS: 1 x k 2 PTS: 1 3 ID: A 61. ANS: 1.14622, 88.85378 PTS: 1 62. ANS: ÊÁ ÁÁ 2 2, ÁÁ 4 Ë ˆ˜ ˜˜ ˜˜ ¯ PTS: 1 63. ANS: ÊÁ ˆ ÁÁ 5 3 5 ˜˜˜ ÁÁ ˜ ÁÁ 2 , 2 ˜˜˜ Á ˜ Ë ¯ PTS: 1 64. ANS: (0, 1) PTS: 1 65. ANS: ÊÁ ÁÁ 7 , 3 ÁÁ 2 Ë ˆ˜ ˜˜ ˜˜ ¯ PTS: 1 66. ANS: r 2sec ( ) PTS: 1 67. ANS: y0 PTS: 1 68. ANS: x 2 24y 2 20y 4 0 PTS: 1 69. ANS: ÁÊÁ ÊÁ 5 2 ÁÁÁ cos ÁÁÁÁ ÁË Ë 3 ˆ˜ Ê ˜˜ i sin ÁÁÁ 5 ˜˜ ÁÁ 3 ¯ Ë ˆ˜ ˜ˆ˜ ÁÊÁ ÊÁ 7 ˜˜ ˜˜ ; 2 ÁÁ cos ÁÁ ˜˜ ˜˜ ÁÁ ÁÁ 4 ¯¯ Ë Ë ˆ˜ Ê ˜˜ i sin ÁÁÁ 7 ˜˜ ÁÁ 4 ¯ Ë PTS: 1 70. ANS: ÁÊ ˜ˆ ˆ˜˜ 7 ÊÁÁ ÁÊ ˜ˆ 14 ÊÁË cos ( ) i sin ˆ˜¯ ; ÁÁÁ cos ÁÁÁÁ ˜˜˜˜ i sin ÁÁÁÁ ˜˜˜˜ ˜˜˜ ; 2 Á Ë3¯ Ë 3 ¯ ˜¯ Ë PTS: 1 4 ˆ˜ ˜ˆ˜ ˜˜ ˜˜ ˜˜ ˜˜ ¯¯ ID: A 71. ANS: Ê 128 ÁÁÁ 1 Ë ˆ 3 i ˜˜˜ ¯ PTS: 1 72. ANS: 1 16 PTS: 1 5