1,cost 1 1,tant 0 1,cott ,cost 0 1,tant 0. 1,cott 1 0. ,cost 5 6,tant ,cott x 2 1 x. 1 x 2. Name: Class: Date:
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1 Class: Date: Practice Test (Trigonometry) 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 1, and the terminal point of t is in quadrant IV. 1 sint 0 1,cost 1 1,tant 0 0,cott1 1 0 sint 0 1,cost 1 1,tant 0 1,cott sint 0 1,cost 0 1,tant 0 1,cott Find the values of the trigonometric functions of t if tant 11 5 and cost 0. sint sint sint ,cost 5 6,tant 11 5,cost 5 6,tant 11 5,cost 5 6,tant 11 5,cott ,cott ,cott Rewrite the expression as an algebraic expression in x. tan (sin 1 x) 1 x 1 x 1 x 1 x d. x 1 4. Rewrite the expression as an algebraic expression in x. sin (cos 1 x) x 1 x 1 1 x d. 1 x e. x 1
2 5. Find the exact value of the expression. ˆ cos sin 1 1 d. 6. Simplify the expression. sin14x sin1x sinx sin6x sin7x cos6x cos7x sin1x sin6x d. sin7x sin6x e. cos7x cos6x 7. Find all solutions of the equation. cosx 0 Select the correct answer, where k is any integer: 5 k, 4 5 k 4 k, 7 4 k 4 k, 7 4 k d. 5 k, 9 5 k
3 8. Find all solutions of the equation. sinx 1 0 Select the correct answer, where k is any integer: 6 k, 11 6 k 6 k, 5 6 k 6 k, 5 6 k d. 6 k, 11 6 k 9. Find all solutions of the following equation. 4cos x 0 Select the correct answer, where k is any integer: 6 k, 5 6 k, 7 6 k, 11 6 k 6 k, 11 6 k 6 k, 5 6 k, 7 6 k, 11 6 k d. 6 k, 5 6 k 10. Find all solutions of the following equation. sec x 4 0 Select the correct answer, where k is any integer: k, k k, 5 k k, k, 4 k, 5 k d. k, k, 4 k, 5 k
4 11. Find all solutions of the following equation. 4 cos x 4 cos x + 1 = 0 Select the correct answer, where k is any integer: 4 k, 7 4 k k, 5 k 6 k, 5 6 k d. 4 k, 4 k 1. Find all solutions of the following equation. sin x = sin x + 4 Select the correct answer, where k is any integer: k 4 k, 4 k k d. k 1. Find all solutions of the following equation. ˆ cosx sinx 0 Select the correct answer, where k is any integer: k, 5 4 k k, k 5 4 k, 7 4 k d. k, k, 5 4 k, 7 4 k 4
5 14. Find all solutions of the following equation. sin x cos x sin x = 0 Select the correct answer, where k is any integer: 4 k, 4 k k, k k, 6 k d. k, k 15. Find all solutions of the following equation in the interval [0,). tan x cot x = 0 Select the correct answer, where k is any integer:,,, 4, 5 k, k d. k, k, 4 k, 5 k 16. Use an addition or subtraction formula to simplify the following equation. Then find all the solutions in the È interval 0, ˆ ÎÍ 4. cos x cos 7 x sin x sin 7 x = , 8 8 d. 16, 16 5
6 17. Use an addition or subtraction formula to simplify the following equation. Then find all the solutions in the È interval ÎÍ 0, ˆ. sin 4 x cos x cos 4 x sin x = 0 0,, 0, d. 0, 18. Plot the point that has the polar coordinates 5, ˆ 4. d. e. 6
7 19. Plot the point that has the polar coordinates, 7 6 ˆ. d. e. 7
8 0. Determine which point in the figure, P, Q, R, or S, has the given polar coordinates. 4, 5 ˆ 4 P S Q d. R 8
9 1. Determine which point in the figure, P, Q, R, or S, has the given polar coordinates. 4, 1 ˆ 4 P Q S d. R 9
10 . Determine which point in the figure, P, Q, R, or S, has the given polar coordinates. 4, 5 ˆ 4 P R S d. Q. Find the modulus of the complex number d Find the modulus of the complex number. 1 i 5 9 d. 10
11 5. Sketch the complex number z, also sketch z, z and 1 z on the same complex plane. z 1 i d. e. 11
12 6. Sketch the complex number z and its complex conjugate z on the same complex plane. z 7 4i d. e. 1
13 7. Sketch z 1,z,z 1 z, and z 1 z on the same complex plane. z 1 i, z i d. e. 1
14 8. Write the complex number 4 4 i in polar form with argument between and. 8cos ˆ 8isin ˆ 4cos ˆ 4isin ˆ cos ˆ isin ˆ 9. Write the complex number i in polar form with argument between 0 and. 4sin 4 4icos 4 cos 4 isin 4 4 cos 4 4 isin 4 0. Write the complex number 5 i in polar form with argument between 0 and. 5cos 5isin 5cos 4 5isin 4 5cos 5isin 1. Write the complex number in polar form with argument between 0 and. cos 4 isin 4 cos isin cos isin. Write the complex number i( 8 8 i ) in polar form with argument between 0 and. 8 cos 4 8 isin 4 cos 4 isin 4 8sin 4 8icos 4 14
15 . Find the quotient z 1 z, in polar form, if: z 1 cos 5 isin 5 z cos 6 isin 6 cos isin 0 0 cos 11 isin 11 cos 0 isin 0 4. Find the product z 1 z, expressed in polar form, if: z 1 cos isin z cos 1 isin 1 cos 1 isin 1 cos 5 5 isin 1 1 cos 7 7 isin Find the quotient z 1 z, in polar form, if: z 1 7 cos75 isin75 ˆ z 7 cos60 isin60 ˆ 7 cos75 isin60 ˆ 1 cos15 isin15 ˆ cos15 isin15 ˆ 15
16 6. Write z 1 and z in polar form, and then find the product z 1 z if: ˆ z 1 i ˆ z 1 i Express your answer in polar form. cos 4 isin ˆ 4 4 cos isin ˆ cos isin ˆ 7. Write z 1 and z in polar form, and then find the quotient z 1 z ˆ z 1 5 i 5 z ( i) 5i i 5 8. Find the indicated power using DeMoivre's Theorem. ( 1 + i ) i 9. Find the indicated power using DeMoivre's Theorem. ( 1 i ) 1 64 i Find the indicated power using DeMoivre's Theorem. i i 1 1 ˆ 1 if: 16
17 41. Find the indicated power using DeMoivre's Theorem. 10 ˆ i 4 1 ˆ i 4 1 ˆ i 4 1 ˆ i 4. Find the cube roots of: 15 i 5 cos 6 isin ˆ 6, 5 cos5 6 isin 5 ˆ 6, 5 cos isin ˆ 5 cos 6 isin ˆ 6, 5 cos 5 6 isin 5 ˆ 6, 5 cos isin 5cos 6 isin ˆ 6, 5 cos7 6 isin 7 ˆ 6, 5 cos1 1 ˆ isin 6 6 ˆ Short Answer 4. Use an appropriate half-angle formula to find the exact value of the expression. sin Use an appropriate half-angle formula to find the exact value of the expression. tan Use an appropriate half-angle formula to find the exact value of the expression. sin 1 17
18 46. Simplify each expression by using a double-angle formul () () tan 1 tan = tan ˆ tan 1 tan = 47. Simplify each expression by using a half-angle formul () 1 cos0 sin ˆ () 1 cos10 = 48. Write the product as a sum. sinx sin5x 49. Write the product as a sum. 7cos6x cos7x 50. Write the sum as a product. sinx sin7x 51. Write the sum as a product. sinx sin4x 5. Find the value of the product. sin7.5 cos Simplify the expression. tanx 1 tan x 54. Find all solutions of the equation. cosx
19 55. Find all solutions of the equation. tan 9 x 81tanx Find all solutions of the equation. 4cos x 4cosx Find all solutions of the equation. 4sinx cosx sinx cosx Consider the equation. tanx 19 (a) Find all solutions of the equation. x = È (b) Use a calculator to solve the equation in the interval ÎÍ 0, ˆ, correct to five decimal places. x = 59. Use an addition or subtraction formula to simplify the equation. Then find all solutions in the interval È ÎÍ 0, ˆ. sin4x cosx cos4x sinx 60. Solve the equation by first using a sum-to-product formul sinx sinx 0 19
20 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 sin 0. If v 0,000 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. 6. A point is graphed in rectangular form. Find polar coordinates for the point, with r 0 and A point is graphed in polar form. Find its rectangular coordinates. 0
21 64. Find the rectangular coordinates for the point whose polar coordinates are 1, 7 ˆ. ˆ 65. Convert the rectangular coordinates 0, 7 to polar coordinates with r 0 and Convert the equation to polar form. x 67. Convert the polar equation to rectangular coordinates Convert the polar equation to rectangular coordinates. r 1 5sin 69. Write the complex numbers in polar form with argument between 0 and. (a) 1 i (b) i 70. Find the product z 1 z and the quotient z 1 z. Express your answer in polar form. z 1 7 cos isin ˆ, z cos isin ˆ (a) z 1 z (b) z 1 z = 71. Find the indicated power using DeMoivre's Theorem. ˆ i 4 7. Find the indicated power using DeMoivre's Theorem. (1 i) 8 1
22 Practice Test (Trigonometry) Answer Section Instructor: Koshal Dahal MULTIPLE CHOICE 1. ANS: A. ANS: C. ANS: B 4. ANS: C 5. ANS: B 6. ANS: E 7. ANS: B 8. ANS: C 9. ANS: A 10. ANS: C 11. ANS: B 1. ANS: A 1. ANS: D 14. ANS: B 15. ANS: B 16. ANS: D 17. ANS: D 18. ANS: A 19. ANS: C 0. ANS: D 1. ANS: D. ANS: C. ANS: A 4. ANS: C 5. ANS: B 6. ANS: D 7. ANS: D 8. ANS: A 9. ANS: C 0. ANS: C 1. ANS: B. ANS: A. ANS: C 4. ANS: C 5. ANS: B 6. ANS: B 7. ANS: C 8. ANS: B 9. ANS: B 40. ANS: C 1
23 41. ANS: A 4. ANS: A SHORT ANSWER 4. ANS: 44. ANS: ANS: ˆ 46. ANS: 4; tan(4) 47. ANS: 15; sin(5) 48. ANS: 1 cos(4x) cos(6x) ˆ 49. ANS: 7 cos(1x) cos(x) ˆ 50. ANS: cos 9x ˆ 5x ˆ sin
24 51. ANS: sin 7x ˆ cos x ˆ 5. ANS: 4 1 ˆ 5. ANS: sin(x) 54. ANS: x k, 4 k 55. ANS: x k 56. ANS: x k, 5 k 57. ANS: 7 6 k, 11 6 k, k, 4 k 58. ANS: k ; , ANS: 15, 15, 7 15, 8 15, 1 15, 14 15, 19 15, 4, 5, ANS: x 1 k
25 61. ANS: 1.146, ANS:, ˆ 4 6. ANS: 5, 5 ˆ 64. ANS: (0, 1) 65. ANS: 7, ˆ 66. ANS: r sec( ) 67. ANS: y ANS: x 4y 0y ANS: cos 5 ˆ i sin 5 ˆ ˆ ; cos 7 4 ˆ i sin 7 4 ˆ ˆ 70. ANS: 14 cos( ) i sin ˆ; 7 cos ˆ i sin ˆ ˆ ; 4
26 71. ANS: ˆ 18 1 i 7. ANS:
Ê 7, 45 Ê 7 Ë 7 Ë. Time: 100 minutes. Name: Class: Date:
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7, 48 7, 6 7, 45. Name: Class: Date: Multiple Choice Questions
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