Download Chapter 5: Analytic Trigonometry 1. If 1 3 sin and cos 2 2 x x

Chapter 5: Analytic Trigonometry
1.
If sin x =
1
3
and cos x =
, evaluate the following function.
2
2
A)
B)
C)
D)
E)
Ans: B
Learning Objective: Evaluate trigonometric function given other trigonometric values
Section: 5.1
Page 232
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Chapter 5: Analytic Trigonometry
2.
If csc x =
4 3
and cos x < 0 , evaluate the function below.
3
A)
B)
C)
D)
E)
Ans: C
Learning Objective: Evaluate trigonometric function given other trigonometric values
Section: 5.1
3. Which of the following is equivalent to the expression below?
A)
B)
C)
D)
E)
Ans: A
Learning Objective: Simplify a trigonometric expression
Section: 5.1
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Page 233
Chapter 5: Analytic Trigonometry
4. Use fundamental identities to simplify the expression below and then determine which
of the following is not equivalent.
A)
B)
C)
D)
E)
Ans: D
Learning Objective: Simplify a trigonometric expression
Section: 5.1
5. Which of the following is equivalent to the expression below?
A)
B)
C)
D)
E)
Ans: A
Learning Objective: Simplify a trigonometric expression
Section: 5.1
Page 234
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Chapter 5: Analytic Trigonometry
6. Determine which of the following are trigonometric identities.
I. sin (θ ) + cot (θ ) cos (θ ) = csc (θ )
II. cot (θ ) − sin (θ ) cos (θ ) = 0
III. sin (θ ) + sin (θ ) cos (θ ) = csc (θ )
A) I is the only identity.
D) III is the only identity.
B) I and II are the only identities.
E) I, II, and III are identities.
C) II is the only identity.
Ans: A
Learning Objective: Identify trigonometric identities
Section: 5.1
7. Determine which of the following are trigonometric identities.
I. tan (θ ) sec (θ ) = csc (θ )
II. tan (θ ) csc (θ ) = sec (θ )
III. csc (θ ) sec (θ ) = tan (θ )
IV. tan (θ ) cos (θ ) = 1
A) II and IV are the only identities.
D) IV is the only identity.
B) II is the only identity.
E) III is the only identity.
C) II, III, and IV are the only identities.
Ans: B
Learning Objective: Identify trigonometric identities
Section: 5.1
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Page 235
Chapter 5: Analytic Trigonometry
8. Factor; then use fundamental identities to simplify the expression below and determine
which of the following is not equivalent.
A)
B)
C)
D)
E)
Ans: C
Learning Objective: Apply fundamental identities to determine non-equivalent
expression
Section: 5.1
9. Factor the expression below and use the fundamental identities to simplify.
cos 4 ( x ) − sin 4 ( x )
A)
B)
C)
D)
E)
( cos ( x ) + sin ( x ) ) ( cos ( x ) − sin ( x ) )
( cos ( x ) − sin ( x ) )
2
2
4
cos ( x ) − sin ( x )
( cos ( x ) + sin ( x ) ) ( cos ( x ) − sin ( x ) )
4 ( cos ( x ) − sin ( x ) )
Ans: D
Learning Objective: Apply fundamental identities to determine equivalent expression
Section: 5.1
Page 236
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Chapter 5: Analytic Trigonometry
10. Factor; then use fundamental identities to simplify the expression below and determine
which of the following is not equivalent.
A)
B)
C)
D)
E)
Ans: E
Learning Objective: Apply fundamental identities to determine non-equivalent
expression
Section: 5.1
11. Expand the expression below and use fundamental trigonometric identities to simplify.
A)
B)
C)
sin 2 (ω ) + cos 2 (ω )
( sin (ω ) + cos (ω ) )
2 tan (ω ) + 1
2sin (ω ) cos (ω ) + 1
2
D)
1
E)
2 cot (ω ) + 1
Ans: C
Learning Objective: Simplify a trigonometric expression
Section: 5.1
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Page 237
Chapter 5: Analytic Trigonometry
12. Multiply; then use fundamental identities to simplify the expression below and
determine which of the following is not equivalent.
A)
B)
C)
D)
E)
Ans: A
Learning Objective: Apply fundamental identities to determine non-equivalent
expression
Section: 5.1
Page 238
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Chapter 5: Analytic Trigonometry
13. Add or subtract as indicated; then use fundamental identities to simplify the expression
below and determine which of the following is not equivalent.
A)
B)
C)
D)
E)
Ans: C
Learning Objective: Apply fundamental identities to determine non-equivalent
expression
Section: 5.1
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Page 239
Chapter 5: Analytic Trigonometry
14. Which of the following is equivalent to the given expression?
A)
B)
C)
D)
E)
Ans: B
Learning Objective: Apply fundamental identities to determine equivalent expression
Section: 5.1
15.
sin ( y )
so that it is not in fractional form.
1 – cos ( y )
D)
sin 2 – sin ( y ) tan ( y )
sin 2 + sin ( y ) tan ( y )
Rewrite the expression
A)
B)
C)
1 – sin ( y ) tan ( y )
csc ( y ) + cot ( y )
E)
1 – cos ( y )
Ans: C
Learning Objective: Rewrite fractional trigonometric expression as non-fraction
Section: 5.1
Page 240
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Chapter 5: Analytic Trigonometry
16. Use a graphing utility to determine which of the trigonometric functions is equal to the
following expression.
A)
B)
C)
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Page 241
Chapter 5: Analytic Trigonometry
D)
E)
Ans: D
Learning Objective: Identify equivalent trigonometric expressions with a graphing
utility
Section: 5.1
17. If x = 2 tan θ , use trigonometric substitution to write
function of θ , where 0 < θ <
4 + x 2 as a trigonometric
π
.
2
C) 2 csc θ
D) 2sec θ E) 2 tan θ
A) 2sin θ B) 2 cos θ
Ans: D
Learning Objective: Write an algebraic expression as a trigonometric function
Section: 5.1
18. Use the trigonometric substitution x = 9sec (θ ) to write the expression
trigonometric function of θ , where 0 < θ <
A) 9 tan (θ )
B) 81tan (θ )
π
2
C) 81sec (θ )
x 2 − 81 as a
.
D) 9sec (θ )
E) 9sec (θ ) − 1
Ans: A
Learning Objective: Write an algebraic expression as a trigonometric function
Section: 5.1
Page 242
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Chapter 5: Analytic Trigonometry
19. If x = 9 tan θ , use trigonometric substitution to write
function of θ , where −
π
2
<θ <
81 + x 2 as a trigonometric
π
.
2
C) 9 tan θ
D) 9sin θ E) 9 cos θ
A) 9 csc θ B) 9sec θ
Ans: B
Learning Objective: Write an algebraic expression as a trigonometric function
Section: 5.1
20. If x = 2 cot θ , use trigonometric substitution to write 4 + x 2 as a trigonometric
function of θ , where 0 < θ < π .
A) 2 cos θ B) 2 csc θ C) 2 cot θ D) 2sec θ E) 2sin θ
Ans: B
Learning Objective: Write an algebraic expression as a trigonometric function
Section: 5.1
21. The rate of change of the function
is given by the expression
.
Which of the following is its simplification?
A)
B)
C)
D)
E)
Ans: C
Learning Objective: Simplify a trigonometric expression
Section: 5.1
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Page 243
Chapter 5: Analytic Trigonometry
22. Verify the identity shown below.
Ans:
Learning Objective: Verify a trigonometric identity
Section: 5.2
23. Verify the identity shown below.
Ans:
Learning Objective: Verify a trigonometric identity
Section: 5.2
Page 244
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Chapter 5: Analytic Trigonometry
24. Verify the identity shown below.
Ans:
Learning Objective: Verify a trigonometric identity
Section: 5.2
25. Verify the identity shown below.
Ans:
Learning Objective: Verify a trigonometric identity
Section: 5.2
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Page 245
Chapter 5: Analytic Trigonometry
26. Determine which of the following are trigonometric identities.
I.
II.
cot ( x ) + cot ( y ) tan ( x ) + tan ( y )
+
=0
tan ( x ) – tan ( y ) cot ( x ) – cot ( y )
cot ( x ) + cot ( y ) tan ( x ) + tan ( y )
+
=1
tan ( x ) + tan ( y ) cot ( x ) + cot ( y )
cot ( x ) + tan ( y )
= cot ( y ) + tan ( x )
cot ( x ) tan ( y )
A) III is the only identity.
D) I and II are the only identities.
B) I and III are the only identities.
E) I, II, and III are identities.
C) II and II are the only identities.
Ans: A
Learning Objective: Verify a trigonometric identity
Section: 5.2
III.
27. Verify the identity shown below.
Ans:
Learning Objective: Verify a trigonometric identity
Section: 5.2
Page 246
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Chapter 5: Analytic Trigonometry
28. Verify the identity shown below.
Ans:
Learning Objective: Verify a trigonometric identity
Section: 5.2
29. Verify the identity shown below.
Ans:
Learning Objective: Verify a trigonometric identity
Section: 5.2
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Page 247
Chapter 5: Analytic Trigonometry
30. Verify the identity shown below.
Ans:
Learning Objective: Verify a trigonometric identity
Section: 5.2
Page 248
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Chapter 5: Analytic Trigonometry
31. Verify the identity shown below.
Ans:
Learning Objective: Verify a trigonometric identity
Section: 5.2
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Page 249
Chapter 5: Analytic Trigonometry
32. Verify the identity shown below.
Ans:
Learning Objective: Verify a trigonometric identity
Section: 5.2
Page 250
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Chapter 5: Analytic Trigonometry
33. Use the cofunction identities to evaluate the expression below without the aid of a
calculator.
sin 2 34° + sin 2 29° + sin 2 56° + sin 2 61°
1
A) 1 B) 2 C) –1 D) 0 E)
2
Ans: B
Learning Objective: Evaluate trigonometric expression using cofunction identities
Section: 5.2
34. Determine which of the following are trigonometric identities.
4
4
2
4
I. csc ( z ) + cot ( z ) = 1 − 2 cot ( z ) + 2 cot ( z )
5
3
2
3
II. cot ( z ) = cot ( z ) csc ( z ) − cot ( z )
3
2
2
4
III. cot ( z ) csc ( z ) = ( csc ( z ) − csc ( z ) ) cot ( z )
A) I, II, and III are identities.
D) II and II are the only identities.
B) II is the only identity.
E) III is the only identity.
C) I is the only identity.
Ans: B
Learning Objective: Verify a trigonometric identity
Section: 5.2
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Page 251
Chapter 5: Analytic Trigonometry
35. Which of the following is a solution to the given equation?
A)
B)
C)
D)
E)
Ans: D
Learning Objective: Verify solution to trigonometric equation
Section: 5.3
Page 252
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Chapter 5: Analytic Trigonometry
36. Which of the following is a solution to the given equation?
A)
B)
C)
D)
E)
Ans: D
Learning Objective: Verify solution to trigonometric equation
Section: 5.3
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Page 253
Chapter 5: Analytic Trigonometry
37. Solve the following equation.
A)
B)
C)
D)
E)
Ans: B
Learning Objective: Solve trigonometric equation
Section: 5.3
38. Solve the following equation.
csc 2 ( x ) − 4 = 0
2π
+ π n, where n is an integer
3
3
B)
π
2π
+ 2π n,
+ 2π n, where n is an integer
3
3
C)
π
5π
+ π n,
+ π n, where n is an integer
6
6
D)
π
2π
+ 2π n,
+ 2π n, where n is an integer
3
6
E)
π
5π
+ 2π n,
+ 2π n, where n is an integer
6
6
Ans: C
Learning Objective: Solve trigonometric equation
Section: 5.3
A)
Page 254
π
+ π n,
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Chapter 5: Analytic Trigonometry
39. Solve the following equation.
A)
B)
C)
D)
E)
Ans: B
Learning Objective: Solve trigonometric equation
Section: 5.3
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Page 255
Chapter 5: Analytic Trigonometry
40. Solve the following equation.
A)
B)
C)
D)
E)
Ans: C
Learning Objective: Solve trigonometric equation
Section: 5.3
41. Find all solutions of the following equation on the interval [ 0, 2π ) .
tan ( x ) + 3 = 0
D)
2π π 5π 4π
, ,
,
3
3 3 3
E)
B)
π 7π
,
6 6
C)
2π 5π
,
3
3
Ans: C
Learning Objective: Solve trigonometric equation
Section: 5.3
A)
Page 256
5π 11π
,
6
6
π 4π
,
3 3
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Chapter 5: Analytic Trigonometry
42. Find all solutions of the following equation on the interval [ 0, 2π ) .
csc 2 ( x ) – 2 = 0
D)
3π
4 4
E)
B)
π 3π 5π 7π
,
,
,
4
4
4 4
C)
π 7π
,
4 4
Ans: B
Learning Objective: Solve trigonometric equation
Section: 5.3
A)
π
,
5π 7π
,
4
4
3π 5π
,
4
4
43. Find all solutions of the following equation in the interval [ 0, 2π ) .
A)
B)
C)
D)
E)
Ans: E
Learning Objective: Solve trigonometric equation
Section: 5.3
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Page 257
Chapter 5: Analytic Trigonometry
44. Find all solutions of the following equation in the interval [ 0, 2π ) .
A)
B)
C)
D)
E)
Ans: A
Learning Objective: Solve trigonometric equation
Section: 5.3
Page 258
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Chapter 5: Analytic Trigonometry
45. Find all solutions of the following equation in the interval [ 0, 2π ) .
A)
B)
C)
D)
E)
Ans: D
Learning Objective: Solve trigonometric equation
Section: 5.3
46. Approximate the solutions of the equation 2sin 2 ( x ) + 4sin ( x ) –1 = 0 by considering its
graph below. Round your answer to one decimal.
A) 0.2, 2.9 B) –1.0, 0.2 C) –1.0, 1.8 D) 0.2, 1.8 E) 1.8, 2.9
Ans: A
Learning Objective: Approximate solutions of trigonometric equation with a graph
Section: 5.3
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Page 259
Chapter 5: Analytic Trigonometry
47. Approximate the solutions of the equation 2sin 2 ( x ) = 3cos ( x ) + 1 by considering its
graph below. Round your answer to one decimal.
A) 2.4, 3.9 B) 2.4, 3.1 C) 3.1, 3.9 D) 1.5, 5.0 E) 1.5, 2.4
Ans: D
Learning Objective: Approximate solutions of trigonometric equation with a graph
Section: 5.3
48. Approximate the solutions of the equation csc ( x ) + cot ( x ) = –1 by considering its graph
below. Round your answer to one decimal.
A) 1.9 B) 1.9, 5.9 C) 4.8 D) 5.9 E) The equation has no solution.
Ans: C
Learning Objective: Approximate solutions of trigonometric equation with a graph
Section: 5.3
Page 260
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Chapter 5: Analytic Trigonometry
49. Solve the multiple-angle equation.
A)
B)
C)
D)
E)
Ans: E
Learning Objective: Solve multiple-angle equation
Section: 5.3
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Page 261
Chapter 5: Analytic Trigonometry
50. Solve the multiple-angle equation in the interval [ 0, 2π ) .
A)
B)
C)
D)
E)
Ans: C
Learning Objective: Solve multiple-angle equation
Section: 5.3
51. Solve the multi-angle equation below.
sin ( 2 x ) =
A)
π
8
B)
π
6
C)
π
6
D)
π
8
E)
π
+ nπ ,
+ nπ ,
π
4
π
4
+ nπ , where n is an integer
3
π
+ 2nπ ,
π
4
+ 2nπ , where n is an integer
+ 2nπ , where n is an integer
+ nπ , where n is an integer
6
3
Ans: E
Learning Objective: Solve multiple-angle equation
Section: 5.3
Page 262
+ nπ ,
+ nπ , where n is an integer
π
+ 2nπ ,
3
2
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Chapter 5: Analytic Trigonometry
52. Solve the multi-angle equation below.
7π
+ nπ , where n is an integer
3
3
B)
π
7π
+ 2nπ ,
+ 2nπ , where n is an integer
3
3
C)
π
7π
+ nπ ,
+ nπ , where n is an integer
3
2
D)
π
7π
+ nπ ,
+ nπ , where n is an integer
2
2
E)
π
7π
+ 4nπ ,
+ 4nπ , where n is an integer
2
2
Ans: E
Learning Objective: Solve multiple-angle equation
Section: 5.3
A)
π
2
⎛x⎞
cos ⎜ ⎟ =
⎝2⎠ 2
+ nπ ,
53. Use the graph below to approximate the solutions of the equation
–2 cos ( x ) – sin ( x ) = 0 on the interval [ 0, 2π ) . Round your answer to one decimal.
A) –2.0, 5.9 B) 2.0, 5.9 C) 5.2, 5.9 D) 2, 5.2 E) –2, 5.2
Ans: D
Learning Objective: Approximate solutions of trigonometric equation with a graph
Section: 5.3
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Page 263
Chapter 5: Analytic Trigonometry
54. Use a graphing utility to approximate the solutions (to three decimal places) of the given
equation in the interval [ 0, 2π ) .
A)
B)
C)
D)
E)
Ans: E
Learning Objective: Approximate solutions to trigonometric equation with graphing
utility
Section: 5.3
55. Use a graphing utility to approximate the solutions (to three decimal places) of the given
⎛ π π⎞
equation in the interval ⎜ − , ⎟ .
⎝ 2 2⎠
A)
B)
C)
D)
E)
Ans: A
Learning Objective: Approximate solutions to trigonometric equation with graphing
utility
Section: 5.3
Page 264
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Chapter 5: Analytic Trigonometry
56. Use the graph of the function f ( x ) = – cos ( x ) + sin ( x ) to approximate the maximum
points of the graph in the interval [ 0, 2π ] . Round your answer to one decimal.
A)
B)
C)
( 2.6,1.3) , ( 6.3, –0.8)
( –0.8, 6.3) , ( 2.6,1.3)
( 2.6, –0.8 ) , ( 6.3,1.3)
D)
E)
( –0.8, 6.3) , (1.3, 2.6 )
(1.3, 2.6 ) , ( 6.3, –0.8)
Ans: A
Learning Objective: Approximate maximum or minimum of a trigonometric function
Section: 5.3
57. Solve the following trigonometric equation on the interval [ 0, 2π ) .
cos ( x ) + sin ( x ) = 0
5π 7π
3π 7π
π 7π
π 5π
,
,
B)
C)
,
D)
,
4
4
4
4
4 4
4 4
Ans: B
Learning Objective: Solve trigonometric equation
Section: 5.3
A)
E)
3π 5π
,
4
4
58. Determine the exact value of the following expression.
cos ( 240D − 0D )
3
1
3
1
1
3
B)
C)
D) –
E) –
2
2
2
2
2 2
Ans: D
Learning Objective: Calculate exact value of expression using sum or difference
formula
Section: 5.4
A) –
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Page 265
Chapter 5: Analytic Trigonometry
59. Find the exact value of the given expression.
cos ( 240° + 315° )
1+ 3
1– 3
–1 + 3
–1 – 3
B)
C)
D)
2 2
2 2
2 2
2 2
Ans: D
Learning Objective: Calculate exact value of expression using sum or difference
formula
Section: 5.4
A)
60. Find the exact value of the given expression.
⎛ 5π 7π ⎞
sin ⎜
−
⎟
4 ⎠
⎝ 3
– 3 +1
3 +1
– 3 –1
3 –1
B)
C)
D)
A)
2 2
2 2
2 2
2 2
Ans: A
Learning Objective: Calculate exact value of expression using sum or difference
formula
Section: 5.4
61. Find the exact value of the given expression using a sum or difference formula.
sin 285°
3 –1
3 +1
– 3 –1
– 3 +1
B)
C)
D)
A)
2 2
2 2
2 2
2 2
Ans: C
Learning Objective: Calculate exact value of expression using sum or difference
formula
Section: 5.4
62. Find the exact value of the given expression using a sum or difference formula.
13π
cos
12
3 +1
– 3 +1
3 –1
– 3 –1
B)
C)
D)
A)
2 2
2 2
2 2
2 2
Ans: D
Learning Objective: Calculate exact value of expression using sum or difference
formula
Section: 5.4
Page 266
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Chapter 5: Analytic Trigonometry
63. Find the exact value of the given expression using a sum or difference formula.
7π
tan
12
3 +1
3 –1
D)
E) undefined
A) 1 B) –1 C)
1– 3
–1 – 3
Ans: C
Learning Objective: Calculate exact value of expression using sum or difference
formula
Section: 5.4
64. Write the given expression as the cosine of an angle.
cos 30° cos 55° – sin 30° sin 55°
A) cos ( 55° ) B) cos ( 85° ) C) cos ( –25° ) D) cos ( 30° )
E) cos ( –110° )
Ans: B
Learning Objective: Rewrite expression using sum or difference formula
Section: 5.4
65. Write the given expression as the sine of an angle.
sin15° cos 55° – sin 55° cos15°
A) sin ( –110° ) B) sin ( –40° ) C) sin ( 70° ) D) sin (15° )
E) sin ( 55° )
Ans: B
Learning Objective: Rewrite expression using sum or difference formula
Section: 5.4
66.
Find the exact value of sin ( u + v ) given that sin u =
8
60
and cos v = − . (Both u and v
17
61
are in Quadrant II.)
D)
A)
315
812
sin ( u + v ) = –
sin ( u + v ) = –
1037
1037
B)
E)
315
645
sin ( u + v ) =
sin ( u + v ) =
1037
1037
C)
645
sin ( u + v ) = –
1037
Ans: C
Learning Objective: Calculate exact value of expression using sum or difference
formula
Section: 5.4
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Page 267
Chapter 5: Analytic Trigonometry
67.
Find the exact value of tan ( u + v ) given that sin u = −
3
24
and cos v =
. (Both u and v
5
25
are in Quadrant IV.)
D)
A)
41
89
tan ( u + v ) =
tan ( u + v ) =
75
75
B)
E)
38
39
tan ( u + v ) =
tan ( u + v ) = –
75
25
C)
4
tan ( u + v ) = –
3
Ans: C
Learning Objective: Calculate exact value of expression using sum or difference
formula
Section: 5.4
68.
Find the exact value of cos ( u + v ) given that sin u =
7
12
and cos v = − . (Both u and v
25
13
are in Quadrant II.)
A)
D)
12
246
cos ( u + v ) =
cos ( u + v ) = –
65
325
B)
E)
36
204
cos ( u + v ) = –
cos ( u + v ) =
325
325
C)
253
cos ( u + v ) =
325
Ans: C
Learning Objective: Calculate exact value of expression using sum or difference
formula
Section: 5.4
69.
Find the exact value of cos ( u − v ) given that sin u = −
8
60
and cos v =
. (Both u and v
17
61
are in Quadrant IV.)
D)
A)
307
812
cos ( u − v ) = –
cos ( u − v ) = –
1037
1037
B)
E)
980
988
cos ( u − v ) = –
cos ( u − v ) =
1037
1037
C)
827
cos ( u − v ) =
1037
Ans: E
Learning Objective: Calculate exact value of expression using sum or difference
formula
Section: 5.4
Page 268
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Chapter 5: Analytic Trigonometry
70. Write the given expression as an algebraic expression.
A)
B)
C)
D)
E)
Ans: D
Learning Objective: Write trig expression as an algebraic expression
Section: 5.4
71. Simplify the given expression algebraically.
A)
B)
C)
D)
E)
Ans: B
Learning Objective: Simplify trigonometric expression using sum and difference
formulas
Section: 5.4
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Page 269
Chapter 5: Analytic Trigonometry
72. Simplify the given expression algebraically.
A)
B)
C)
D)
E)
Ans: D
Learning Objective: Simplify trig expression using sum and difference formulas
Section: 5.4
73. Determine which of the following are trigonometric identities.
I. sin ( x + y ) + sin ( x – y ) = 2sin ( x )
II. sin ( x + π ) − sin (π – x ) = 2sin ( x )
III. sin ( x + y ) + sin ( x – y ) = 2sin ( x ) sin ( y )
A) II and II are the only identities.
D) I is the only identity.
B) I and III are the only identities.
E) III is the only identity.
C) None are identities.
Ans: C
Learning Objective: Verify an identity using sum and difference formulas
Section: 5.4
74. Verify the given identity.
Ans:
Learning Objective: Verify an identity using sum and difference formulas
Section: 5.4
Page 270
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Chapter 5: Analytic Trigonometry
75. Find all solutions of the given equation in the interval [ 0, 2π ) .
A)
B)
C)
D)
E)
Ans: C
Learning Objective: Solve trig equation with sum/difference formulas
Section: 5.4
Copyright © Houghton Mifflin Company. All rights reserved.
Page 271
Chapter 5: Analytic Trigonometry
76. Use the figure below to determine the exact value of the given function.
2
θ
3
A)
B)
C)
D)
E)
Ans: E
Learning Objective: Calculate exact value of a trigonometric function using a right
triangle
Section: 5.5
Page 272
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Chapter 5: Analytic Trigonometry
77. Use the graph below of the function to approximate the solutions to
2 cos ( 2 x ) − cos ( x ) = 0 in the interval [ 0, 2π ) . Round your answers to one decimal.
A) 0.6, 1.0, 1.5, 4.8
D) 0.6, 2.1, 4.2, 5.7
B) 1.0, 1.5, 4.8, 5.7
E) 0.6, 1.5, 4.2, 6.3
C) 1.0, 2.1, 4.2, 5.7
Ans: D
Learning Objective: Approximate solutions of trigonometric equation with a graph
Section: 5.5
Copyright © Houghton Mifflin Company. All rights reserved.
Page 273
Chapter 5: Analytic Trigonometry
78. Find the exact solutions of the given equation in the interval [ 0, 2π ) .
A)
B)
C)
D)
E)
Ans: D
Learning Objective: Solve trigonometric equation
Section: 5.5
Page 274
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Chapter 5: Analytic Trigonometry
79. Find the exact solutions of the given equation in the interval [ 0, 2π ) .
A)
B)
C)
D)
x=0
E)
Ans: B
Learning Objective: Solve trigonometric equation
Section: 5.5
80. Use a double-angle formula to find the exact value of cos 2u when
7
π
<u <π .
sin u = , where
25
2
D)
A)
478
527
cos 2u = –
cos 2u =
625
625
E)
B)
168
1152
cos 2u =
cos 2u = –
625
625
C)
336
cos 2u =
625
Ans: D
Learning Objective: Calculate exact value of a trigonometric function using a doubleangle formula
Section: 5.5
Copyright © Houghton Mifflin Company. All rights reserved.
Page 275
Chapter 5: Analytic Trigonometry
81. Use a double-angle formula to find the exact value of tan 2u when
12
3π
.
cos u = − , where π < u <
13
2
D)
A)
5
130
tan 2u = –
tan 2u =
6
119
E)
B)
312
5
tan 2u = –
tan 2u =
25
12
C)
120
tan 2u =
119
Ans: C
Learning Objective: Calculate exact value of a trigonometric function using a doubleangle formula
Section: 5.5
82. Use a double angle formula to rewrite the following expression.
A)
B)
C)
sin ( –14x )
2sin ( –7x )
–7 cos ( 2x )
–14sin ( x ) cos ( x )
D)
E)
–7 sin ( 2x )
cos ( –14x )
Ans: D
Learning Objective: Rewrite expression as a double angle
Section: 5.5
83. Use a double angle formula to rewrite the given expression.
8cos 2 x − 4
A) 4 cos 2x B) 8cos 2x C) 2 cos 4x D) 4 cos 4x E) 2 cos8x
Ans: A
Learning Objective: Rewrite expression as a double angle
Section: 5.5
Page 276
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Chapter 5: Analytic Trigonometry
84. Determine which of the following are trigonometric identities.
1
(1+ 4 cos ( 2 x ) + cos ( 4 x ) )
4
1
II. cos 4 ( x ) = ( 3 + 4sin ( 2 x ) – sin ( 4 x ) )
8
1
III. cos 4 ( x ) = ( 3 + 4 cos ( 2 x ) + cos ( 4 x ) )
8
A) None are identities.
D) I and III are the only identities.
B) III is the only identity.
E) I, II, and III are identities.
C) I and II are the only identities.
Ans: B
Learning Objective: Verify a trigonometric identity
Section: 5.5
I. cos 4 ( x ) =
85. Use the power-reducing formulas to rewrite the given expression in terms of the first
power of the cosine.
A)
B)
C)
D)
E)
Ans: E
Learning Objective: Rewrite expression using power-reducing formulas
Section: 5.5
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Page 277
Chapter 5: Analytic Trigonometry
86. Use the figure below to find the exact value of the given trigonometric expression.
sin
θ
2
θ
6
8
(figure not necessarily to scale)
3
3
3 10
10
1
B)
C)
D)
E)
A)
10
2
10
10
10 10
Ans: D
Learning Objective: Calculate exact value of a trigonometric function using a right
triangle
Section: 5.5
Page 278
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Chapter 5: Analytic Trigonometry
87. Use the figure below to find the exact value of the given trigonometric expression.
7
θ
24
A)
B)
C)
D)
E)
Ans: C
Learning Objective: Calculate exact value of a trigonometric function using a right
triangle
Section: 5.5
88. Use the half-angle formulas to determine the exact value of the following.
cos ( 22.5D )
A) –
2+ 3
2
B)
2– 2
2
C) –
2– 2
2
D)
3– 3
2
E)
2+ 2
2
Ans: E
Learning Objective: Calculate exact value of a trigonometric function using a halfangle formula
Section: 5.5
Copyright © Houghton Mifflin Company. All rights reserved.
Page 279
Chapter 5: Analytic Trigonometry
89. Use the half-angle formulas to determine the exact value of the given trigonometric
expression.
A)
B)
C)
D)
E)
Ans: A
Learning Objective: Calculate exact value of a trigonometric function using a halfangle formula
Section: 5.5
90. Use the half-angle formula to simplify the given expression.
1 + cos 20 x
2
A) cos 40x B) cos10x C) cos 20x D) cos80x E) cos 5x
Ans: B
Learning Objective: Rewrite expression using half-angle formulas
Section: 5.5
Page 280
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Chapter 5: Analytic Trigonometry
91. Find all solutions of the given equation in the interval [ 0, 2π ) .
A)
B)
C)
D)
E)
Ans: C
Learning Objective: Solve trigonometric equation using half-angle formulas
Section: 5.5
92. Use the product-to-sum formula to write the given product as a sum or difference.
8sin
A)
B)
C)
π
8
sin
4sin
π
π
8
16
4 − 4 cos
4 + 4 cos
D)
π
4
π
E)
–4sin
4sin
π
16
π
8
+ 4 cos
π
8
16
Ans: B
Learning Objective: Rewrite expression with a product-to-sum formula
Section: 5.5
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Page 281
Chapter 5: Analytic Trigonometry
93. Use the product-to-sum formula to write the given product as a sum or difference.
4sin
A)
B)
C)
π
12
2sin
2sin
cos
π
12
π
12
+ 2 cos
π
6
2 + 2 cos
π
D)
12
E)
2 − 2 cos
–2sin
π
π
π
24
24
24
Ans: B
Learning Objective: Rewrite expression with a product-to-sum formula
Section: 5.5
94. Use the product-to-sum formulas to write the expression below as a sum or difference.
sin ( 6θ ) cos ( 4θ )
D)
1
1
cos ( 2θ ) + cos (10θ ) )
(
( cos ( 2θ ) − cos (10θ ) )
2
2
B)
E)
1
1
sin ( 2θ ) + cos (10θ ) )
(
( sin (10θ ) − sin ( 2θ ) )
2
2
C)
1
( sin (10θ ) + sin ( 2θ ) )
2
Ans: C
Learning Objective: Rewrite expression with a product-to-sum formula
Section: 5.5
A)
95. Use the sum-to-product formulas to write the given expression as a product.
sin 9θ − sin 7θ
2sin 8θ cos θ
−2 cos8θ cos θ
A)
D)
2 cos8θ cos θ
2 cos8θ sin θ
B)
E)
−2sin 8θ sin θ
C)
Ans: E
Learning Objective: Rewrite expression with a product-to-sum formula
Section: 5.5
96. Use the sum-to-product formulas to write the given expression as a product.
cos 6θ − cos 4θ
−2sin 5θ sin θ
2sin 5θ cos θ
A)
D)
2 cos 5θ cos θ
−2 cos 5θ cos θ
B)
E)
2 cos 5θ sin θ
C)
Ans: A
Learning Objective: Rewrite expression with a sum-to-product formula
Section: 5.5
Page 282
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Chapter 5: Analytic Trigonometry
97. Use the sum-to-product formulas to find the exact value of the given expression.
A)
B)
C)
D)
E)
Ans: C
Learning Objective: Evaluate expression using sum-to-product formulas
Section: 5.5
98. Find all solutions of the given equation in the interval [ 0, 2π ) .
A)
B)
C)
D)
E)
Ans: D
Learning Objective: Solve trigonometric equations using sum-to-product formulas
Section: 5.5
Copyright © Houghton Mifflin Company. All rights reserved.
Page 283
Chapter 5: Analytic Trigonometry
99. Verify the given identity.
Ans:
Learning Objective: Verify an identity using sum-to-product formulas
Section: 5.5
100. Determine which of the following are trigonometric identities.
I.
II.
cos ( 4 x ) – cos ( 2 x )
= – sin ( x )
2sin ( 3 x )
cos ( 4 x ) – cos ( x )
= – sin ( 2 x )
sin ( 3x ) − sin ( x )
cos ( 6 x ) – cos ( 2 x )
= – sin ( 3x )
sin ( 4 x ) + sin ( 2 x )
A) I is the only identity.
D) I and II are the only identities.
B) II and II are the only identities.
E) III is the only identity.
C) None are identities.
Ans: A
Learning Objective: Verify an identity using sum-to-product formulas
Section: 5.5
III.
Page 284
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Chapter 5: Analytic Trigonometry
101. Verify the given identity.
Ans:
Learning Objective: Verify an identity using sum-to-product formulas
Section: 5.5
Copyright © Houghton Mifflin Company. All rights reserved.
Page 285