Rec/Sec: Review Test (Math1650:500) Instructor: Koshal Dahal Test date: Fri, May 1 Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Find the equivalent expression. tan x sec x cot x csc x csc x cot x csc x tan x tan x sec x e. sec x tan x. Simplify the following trigonometric expression. sec x 1 sec x 1 sin x sin x sec x 3. Simplify the following trigonometric expression. sin(z) + cos( z) + sin( z) sin z cos z sin z cos z sin z. Simplify the following trigonometric expression as much as possible. csc x sinx csc x sin x cos x sin x csc x 5. Simplify the following trigonometric expression as much as possible. sin t + cos t + tan t tan x sec x sec x tan x 1
6. Simplify the following trigonometric expression as much as possible. 1 cosx sinx sinx 1 cosx sin x cos x sin x csc x 7. Simplify the following trigonometric expression as much as possible. sec y tan y csc y csc x tan x sin x sec x 8. Find the equivalent expression. 1 tanx 1 tanx sec x csc x sinx cosx cosx sinx cosx sinx cosx sinx cosx sinx sinx cosx sinx cosx e. sinx cosx sinx cosx 9. Simplify the following trigonometric expression as much as possible. 1 csc x cotx 1 csc x cotx csc x cot x cot x csc x
10. Find the equivalent expression. 1 sinx 1 sinx 1 sinx 1 sinx tanx sec x cotx sec x cotx csc x cotx csc x e. tanx sec x 11. Make the indicated trigonometric substitution in the given algebraic expression and simplify. Assume 0 1. x, x sint 1 x sin t 1 tan t cos t 1. Use an addition or subtraction formula to find the exact value of the expression. sin( 705) 6 6 6 13. Use an addition or subtraction formula to find the exact value of the expression. tan( 55) 1 3 1 3 3 1 1 3 3
1. Use an addition or subtraction formula to find the exact value of the expression. sin 11 ˆ Á 1 6 3 6 6 3 6 15. Use an addition or subtraction formula to find the exact value of the expression. cos ˆ Á 1 6 6 6 6 16. Use an addition or subtraction formula to write the expression as a trigonometric function of one number. sin3 cos56cos3 sin56 sin( 90) cos( 180) cos( 90) sin( 90)
17. Use an addition or subtraction formula to write the expression as a trigonometric function of one number. cos 3 ˆ Á cos ˆ Á 8 sin 3 Á cos 7 ˆ Á 8 sin 7 ˆ Á 8 sin 7 ˆ Á 8 cos 7 ˆ Á 8 ˆ sin ˆ Á 8 18. Simplify the following expression as much as possible. tan 9 x ˆ Á tan(x) tan(x) cot(x) cot(x) 19. Simplify the following expression. sin u ˆ Á sin u cos u cos u sin u 0. Simplify the following expression. sin(v + x) sin(v x) cos(v)cos(x) sin(x)sin(v) cos(v)sin(x) cos(x)sin(v) 1. Simplify the following expression cos(p + z) cos(p z) cos(p)cos(z) cos(p)cos(z) sin(p)sin(z) sin(p)sin(z) 5
. Simplify the expression. tan p tan x sin(p x) cosp cosx cos(p x) cosp cosx sin(p x) cosp cosx 3. Write the following expression in terms of sine only. sin z + cos z 3 sin 6 z ˆ Á sin z ˆ Á 3 sin 6 z ˆ Á sin z ˆ Á. Write the following expression in terms of sine only. 5sinx 5 3cosx 5sin x ˆ Á 3 5sin x ˆ Á 3 10sin x ˆ Á 3 10sin x ˆ Á 3 6
5. Rewrite the expression as an algebraic expression in x. tan (sin 1 x) 1 x 1 x 1 x 1 x x 1 6. Rewrite the expression as an algebraic expression in x. sin (cos 1 x) x 1 x 1 1 x 1 x e. x 7. Find the exact value of the expression. ˆ 3 cos sin 1 Á 1 3 7
8. Simplify the expression. sin1x sin13x sinx sin6x sin7x cos6x cos7x sin13x sin6x sin7x sin6x e. cos7x cos6x 9. Find all solutions of the equation. cosx 0 Select the correct answer, where k is any integer: 5 k, 5 k k, 7 k k, 7 k 5 k, 9 5 k 30. 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 6 k, 11 6 k 8
31. Find all solutions of the following equation. cos x 3 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 6 k, 5 6 k 3. Find all solutions of the following equation. cos x cos x + 1 = 0 Select the correct answer, where k is any integer: k, 7 k 3 k, 5 3 k 6 k, 5 6 k k, 3 k 33. Use an addition or subtraction formula to simplify the following equation. Then find all the solutions in the È interval 0, ˆ ÎÍ. cos x cos 7 x sin x sin 7 x = 0 16 8, 3 8 8 16, 3 16 9
3. Plot the point that has the polar coordinates 5, ˆ Á. e. 10
35. Plot the point that has the polar coordinates 3, 7 Á 6 ˆ. e. 36. Find the third term of the sequence. a n = n + 1 a 3 = 7 a 3 = 6 a 3 = 5 a 3 = 1 e. a 3 = 11
37. Find the fourth term of the sequence. a n = 1 n +1 a = 5 a = 1 5 a = 5 a = 1 e. a = 1 1 38. Find the 00th term of the sequence. a n = 10 a 00 = 1 a 00 = 10 a 00 = 00 a 00 = 10 e. a 00 = 000 39. Find the nth term of the sequence.,, 8, 16,... a n = n 1 a n = n a n = n a n = n + 1 e. a n = + n 0. Find the partial sum S 7 of the sequence. 5, 10, 15, 0,... S 7 = 10 S 7 = 50 S 7 = 80 S 7 = 11 e. S 7 = 0 1
1. Find the partial sum S 5 of the sequence. 1, 1, 1, 1,... S 5 = 0 S 5 = S 5 = S 5 = 1 e. S 5 = 1. Find the sum. 18 i 18 13 i 18 60 i 18 7 i 18 56 i 18 e. i 3. Find the sum. k 1 k k k k 99 k 1 k k 6 k 1 k k 91 k 1 k k 10 k 1 e. k k 98 k 1 13
. Write the following sum. 7 k (k 9) k 5 7 k(k 9) 5(5 9) 7(7 9) k 5 7 k(k 9) 6(6 9) 7(7 9) k 5 7 k(k 9) 6(6 9) 7(7 9) 8(8 9) k 5 7 k(k 9) 5(5 9) 6(6 9) 7(7 9) k 5 7 e. k(k 9) 5(5 9) 6(6 9) 8(8 9) k 5 5. Write the following sum using sigma notation. 5 + 10 + 15 + 0 +... + 50 50 5 k 0 10 k 5 k 0 10 5k k 1 10 5 k k 0 50 e. k k 5 6. The first term of the arithmetic sequence a is and common difference d is 6. Find the nth term and the 10th term. a n 1 6(n ), a 10 6 a n 6(n 1), a 10 58 a n 6(n ), a 10 56 a n 6 (n 1), a 10 59 e. a n 6 (n 6), a 10 55 1
7. Find the common difference d of the arithmetic sequence. 5, 7, 9, 11,... n n 7 e. 5 8. Find the first five terms and determine if the sequence is arithmeti a n 6n a 1 8, a 1, a 3 0, a 3, a 5 35 The sequence is not arithmeti a 1 8, a 1, a 3 0, a 30, a 5 8 The sequence is not arithmeti a 1 8, a 1, a 3 0, a, a 5 3 The sequence is arithmeti a 1 8, a 1, a 3 0, a 6, a 5 3 The sequence is arithmeti e. a 1 8, a 1, a 3 0, a 7, a 5 33 The sequence is arithmeti 9. If it is arithmetic, express the nth term of the sequence in the standard form a n a d(n 1) and find the common difference. a n 8n 1 a n 37(n 1), d 7 a n 36(n 1), d 6 Not an arithmetic sequence. a n 35(n 1), d 5 e. a n 38(n 1), d 8 50. Find the fifth term of the arithmetic sequence., 10, 18, 6,... 6 7 3 e. 5 51. Find the fifth term of the arithmetic sequence. 5, 9, 13, 17,... 1 17 30 5 e. 18 15
5. Find the nth term of the arithmetic sequence., + s, + s, + 3s,... s n sn s ( n 1) sn ( 1) e. n 1 sn( n 1) 53. The 1th term of an arithmetic sequence is 13 and the 5th term is 6. Find the th term. 3 1 10 e. 3 5. The 0th term of an arithmetic sequence is 97, and the common difference is 5. Find a formula for the nth term. 0 + 5(n 1) + 5(n) 5 + (n 1) + 5(n 1) e. 5 + (n + 1) 55. Which term of the arithmetic sequence 3, 8, 13,... is 73? 15 17 1 16 e. 13 56. Find the partial sum S n of the arithmetic sequence that satisfies the following conditions. a = 1, d =, n = 15 57 35 65 870 e. 61 16
57. Find the product of the numbers. 1 10,10,10,10,..., 10 380 10 380 19 10 10 190 e. 10 19 3 19 10 58. Find the nth term of the geometric sequence with given first term a and common ratio r. What is the fifth term? a 7 3, r 1 3 a n 7 3 1 ˆ Á 3 a n 7 3 1 ˆ Á 3 a n 1 3 1 ˆ Á 3 a n 7 3 1 ˆ Á 3 e. a n 7 3 1 ˆ Á 3 n, a 5 1 3 n 1, a 5 7 81 n 1, a 5 7 81 n 1, a 5 7 81 n 1, a 5 7 3 59. Determine whether the sequence 6,, 96, 38... is geometri If it is geometric, find the common ratio. Geometric sequence, r = 6 Geometric sequence, r 1 Not a geometric sequence. Geometric sequence, r = e. Geometric sequence, r 1 5 17
60. Determine whether the sequence is geometri 8,,, 1,... If it is geometric, find the common ratio. Geometric, 1 Not geometri Geometric, 1 Geometric, e. Geometric, 61. Determine whether the sequence is geometri If it is geometric, find the common ratio. e, e 7, e 10, e 13,... Geometric, r = e 3 Not geometri Geometric, r = 3 Geometric, r = 1 e 3 e. Geometric, r = e 6. Find the first five terms of the sequence and determine if it is geometri If it is geometric express the nth term of the sequence in the standard form a n ar n 1. a n ( 1) 3 n 3, 9, 7, 81, 3; ; it is not geometri 3, 9, 7, 81, 3; a n 3() n 1 3, 9, 7, 8, 3; a n 3() n 1 3, 9, 7, 81, 3; a n 3(3) n 1 e. 3, 9, 7, 8, 3; a n 3(3) n 1 63. Determine the common ratio, the 6th term, and the nth term of the geometric sequence. 5, 0, 80, 30,... Common ratio 5, the 6th term 5,10, and the nth term n 1 Common ratio, the 6th term 5,10, and the nth term 5 n 1 Common ratio, the 6th term 1,500, and the nth term 5 n 1 Common ratio 5, the 6th term 1,500, and the nth term 5 n 1 e. Common ratio, the 6th term 5,10, and the nth term n 1 18
6. Determine the nth term of the geometric sequence. 1, 11,11, 11 11,... 11 n 1 n 1 1 ˆ Á 11 n 1 1 ˆ Á 11 Not a geometric series. e. n 1 ˆ 11 Á 65. Determine the nth term of the geometric sequence. x, x 5, x 3 5, x 15,... x n 5 x n 5 n x n 1 e. x n 1 5 n 1 x n 5 n 1 66. The first term of a geometric sequence is 6, and the second term is 3. Find the fifth term. 3 3 8 6 8 19
67. The common ratio in a geometric sequence is 3, and the fourth term is 7. Find the third term. 3 e. 3 7 6 3 1 7 68. Which term of the geometric sequence 5, 0, 80,... is 080? 7th 13th 6th 8th e. 9th 69. Find the partial sum S n of the geometric sequence that satisfies the given conditions. a = 3, r =, n = 6 S n =,09 S n = 16,383 S n = 8,190 S n =,09 e. S n =,095 70. Find the partial sum S n of the geometric sequence that satisfies the given conditions. a 16, a 6 6, n S n = 60 S n = 6 S n = 8 S n = 9 e. S n = 30 71. Find the sum. 1 + + 16 +... + 096 S n = 1,365 S n = 10,9 S n = 1,85 S n = 5,60 e. S n = 5,61 0
7. Find the sum of the infinite geometric series. 1 1 3 1 9 1 7... 5 1 3 e. 3 73. Find the sum of the infinite geometric series. 1 1 5 1 5 1 15... 5 6 11 1 6 5 1 6 e. 5 1
Review Test Answer Section MULTIPLE CHOICE 1. ANS: D. ANS: C 3. ANS: B. ANS: B 5. ANS: B 6. ANS: D 7. ANS: C 8. ANS: B 9. ANS: A 10. ANS: E 11. ANS: C 1. ANS: C 13. ANS: C 1. ANS: A 15. ANS: B 16. ANS: A 17. ANS: A 18. ANS: C 19. ANS: B 0. ANS: C 1. ANS: D. ANS: C 3. ANS: B. ANS: D 5. ANS: B 6. ANS: C 7. ANS: B 8. ANS: E 9. ANS: B 30. ANS: C 31. ANS: A 3. ANS: B 33. ANS: D 3. ANS: A 35. ANS: C 36. ANS: A 37. ANS: B 38. ANS: B 39. ANS: C 0. ANS: A 1
1. ANS: E. ANS: B 3. ANS: E. ANS: D 5. ANS: C 6. ANS: B 7. ANS: A 8. ANS: D 9. ANS: E 50. ANS: C 51. ANS: A 5. ANS: D 53. ANS: E 5. ANS: D 55. ANS: A 56. ANS: B 57. ANS: E 58. ANS: E 59. ANS: D 60. ANS: C 61. ANS: A 6. ANS: D 63. ANS: B 6. ANS: E 65. ANS: E 66. ANS: B 67. ANS: E 68. ANS: A 69. ANS: E 70. ANS: E 71. ANS: E 7. ANS: E 73. ANS: A