Instructor: Koshal Dahal Test 4 date: Fri, May 1
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1 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
2 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
3 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) Use an addition or subtraction formula to find the exact value of the expression. tan( 55)
4 1. Use an addition or subtraction formula to find the exact value of the expression. sin 11 ˆ Á Use an addition or subtraction formula to find the exact value of the expression. cos ˆ Á 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)
5 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 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
6 . 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
7 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 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
9 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 = , ,
10 3. Plot the point that has the polar coordinates 5, ˆ Á. e. 10
11 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
12 37. Find the fourth term of the sequence. a n = 1 n +1 a = 5 a = 1 5 a = 5 a = 1 e. a = 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 = 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
13 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 i i 18 7 i 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
14 . 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 k 0 10 k 5 k k k 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 a n 6(n ), a a n 6 (n 1), a e. a n 6 (n 6), a
15 7. Find the common difference d of the arithmetic sequence. 5, 7, 9, 11,... n n 7 e 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 Find the fifth term of the arithmetic sequence., 10, 18, 6, e Find the fifth term of the arithmetic sequence. 5, 9, 13, 17, e
16 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 e The 0th term of an arithmetic sequence is 97, and the common difference is 5. Find a formula for the nth term (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? e Find the partial sum S n of the arithmetic sequence that satisfies the following conditions. a = 1, d =, n = e
17 57. Find the product of the numbers. 1 10,10,10,10,..., e 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 ˆ Á 3 a n ˆ Á 3 a n ˆ Á 3 a n ˆ Á 3 e. a n ˆ Á 3 n, a n 1, a n 1, a n 1, a n 1, a Determine whether the sequence 6,, 96, 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
18 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 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
19 6. Determine the nth term of the geometric sequence. 1, 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 The first term of a geometric sequence is 6, and the second term is 3. Find the fifth term
20 67. The common ratio in a geometric sequence is 3, and the fourth term is 7. Find the third term. 3 e 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 =, 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 = Find the sum S n = 1,365 S n = 10,9 S n = 1,85 S n = 5,60 e. S n = 5,61 0
21 7. Find the sum of the infinite geometric series e Find the sum of the infinite geometric series e. 5 1
22 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
23 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
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