Practice Problems Test 2 Find the exact value of the expression.
2
1) sin-1 2
18) cos-1 - 6
3
Find the exact value of the expression, if possible. Do not
use a calculator.
π
19) cos-1 cos - 4
2) sin-1 (-0.5)
3) sin-1 0
4) cos-1 (1)
20) sin-1 sin 4π
7
3
5) cos-1 2
21) tan-1 tan 4π
5
3
6) cos-1 - 2
4π
22) cos-1 cos 3
7) cos-1 (-1)
π
23) cos-1 cos - 3
8) tan-1 3
6π
24) tan-1 tan 7
9) tan-1 1
10) tan-1 (-1)
25) cos (cos-1 0.6)
11) tan-1 (- 3)
26) tan (tan-1 (-9.5))
Use a sketch to find the exact value of the expression.
4
27) cos sin-1 5
Use a calculator to find the value of the expression
rounded to two decimal places.
12) tan-1 (2.7)
13) tan-1 (0.9)
14) sin-1 - 6
28) cos tan-1 5
1
6
29) cot sin-1 2
15) cos-1 3
5 26
26
3
30) sec tan-1 3
16) sin-1 (0.3)
31) tan sin-1 5
17) sin-1 3
1
2
2
41) A building 180 feet tall casts a 100 foot long
shadow. If a person looks down from the top
of the building, what is the measure of the
angle between the end of the shadow and the
vertical side of the building (to the nearest
degree)? (Assume the personʹs eyes are level
with the top of the building.)
Use a right triangle to write the expression as an algebraic
expression. Assume that x is positive and in the domain of
the given inverse trigonometric function.
x
)
32) sin(sin-1 2
x2 + 25
33) tan(sec-1 )
x
42) A radio transmission tower is 240 feet tall.
How long should a guy wire be if it is to be
attached 14 feet from the top and is to make an
angle of 23° with the ground? Give your
answer to the nearest tenth of a foot.
x2 + 25
)
34) sin(sec-1 x
Solve the right triangle shown in the figure. Round lengths
to one decimal place and express angles to the nearest
tenth of a degree.
Use the given figure to solve the problem.
43) Find the bearing from O to A.
49°
35) A = 40°, b = 51.8
57°
30°
74°
36) B = 31°, b = 55.5
37) b = 150, c = 430
38) a = 3.4 m, B = 17.9°
Solve the problem.
39) A surveyor is measuring the distance across a
small lake. He has set up his transit on one
side of the lake 130 feet from a piling that is
directly across from a pier on the other side of
the lake. From his transit, the angle between
the piling and the pier is 30°. What is the
distance between the piling and the pier to the
nearest foot?
44) Find the bearing from O to B.
44°
40) A building 190 feet tall casts a 30 foot long
shadow. If a person stands at the end of the
shadow and looks up to the top of the
building, what is the angle of the personʹs eyes
to the top of the building (to the nearest
hundredth of a degree)? (Assume the personʹs
eyes are 4 feet above ground level.)
57°
35°
75°
2
45) Find the bearing from O to C.
48°
49) A ship leaves port with a bearing of N 77° W.
After traveling 9 miles, the ship then turns 90°
and travels on a bearing of S 13° W for 24
miles. At that time, what is the bearing of the
ship from port?
Verify the identity. PRACTICE MORE PROBLEMS
FROM SECTION 5.1 IN THE BOOK!!!
(sin x + cos x)2
50)
= 1
1 + 2 sin x cos x
56°
32°
75°
Verify the identity.
51) sec 4 θ - 2 sec 2 θ tan 2 θ + tan 4 θ = 3sec4 θ-2
52) sin2 x + tan2 x + cos2 x = sec 2 θ
Verify the identity.
53) tan 2 x (1 + cos 2x) = 1 - cos 2x
46) Find the bearing from O to D.
54) sin4t = 2 sin2t cos2t
55) cos 4θ = 2 cos2 (2θ) - 1
46°
Find the exact value of the trigonometric function.
56) tan 255°
57°
40°
70°
57) sin 11π
12
Find the exact value of the expression.
tan 20° + tan 10°
58)
1 - tan 20° tan 10°
59)
Using a calculator, solve the following problems. Round
your answers to the nearest tenth.
47) A boat leaves the entrance of a harbor and
travels 93 miles on a bearing of N 43° E. How
many miles north and how many miles east
from the harbor has the boat traveled?
tan 170° - tan 50°
1 + tan 170° tan 50°
60) cos 15° cos 45° - sin 15° sin 45°
61) cos 48) A ship is 14 miles west and 42 miles south of a
harbor. What bearing should the captain set to
sail directly to harbor?
2π
π
2π
π
cos + sin sin 9
18
9
18
62) cos (155°) cos (35°) + sin (155°) sin (35°)
3
3
71) cos θ = , θ lies in quadrant IV
5
Find the exact value under the given conditions.
4
2
63) sin α = , α lies in quadrant II, and cos β = ,
5
5
β lies in quadrant I
Find sin 2
θ.
Find cos (α - β).
72) tan θ = Find the exact value under the given conditions.
4
3π
24 π
64) tan α = , π < α < ; cos β = - , < β
3
2
25 2
8
, θ lies in quadrant III
15
Find sin 2
θ.
< π Find sin (α + β).
73) sin θ = Evaluate
15
, θ lies in quadrant II
17
Find tan 2
θ.
π
65) cos ( + θ) = ?
2
Write the expression as the sine, cosine, or tangent of a
double angle. Then find the exact value of the expression.
74) 2 sin 75° cos 75°
66) sin (π - θ) = ?
75) cos2 15° - sin2 15°
Use the figure to find the exact value of the trigonometric
function.
67) Find cos 2θ.
2 tan 76)
13
5 5π
1 - tan2 12
Solve the problem.
77) If a projectile is fired at an angle θ and initial
velocity v, then the horizontal distance
traveled by the projectile is given by
1
D = v 2 sin θ cos θ. Express D as a function
16
12
68) Find sin 2θ.
5π
12
17
15 of 2θ.
Use the figure to find the exact value of the trigonometric
function.
78) Find cos 2θ.
8
69) Find tan 2θ.
5
4 13
12 3
5
Use the given information to find the exact value of the
expression.
4
70) sin θ = , θ lies in quadrant I
Find cos 2θ.
5
4
79) Find sin 2θ.
Use the information given about the angle θ, 0 ≤ θ ≤ 2π, to
find the exact value of the indicated trigonometric
function.
1
Find cos (2θ).
88) cos θ = - , csc θ < 0
7
5
4 3
80) Find tan 2θ.
17
15 5
89) csc θ = - , tan θ > 0
2
Find cos (2θ).
4 3π
90) cos θ = , < θ < 2π
5
2
Find sin (2θ).
1
91) cos θ = , csc θ > 0
4
θ
Find sin .
2
8
92) sec θ = - Use the given information to find the exact value of the
expression.
15
Find cos 2θ.
81) sin θ = , θ lies in quadrant I
17
82) cos θ = 5
, θ lies in quadrant IV
13
94) sin 75°
Find sin 2
95) sin 5
, θ lies in quadrant III
12
Find sin 2
θ.
84) sin θ = 8
, θ lies in quadrant II
17
Find tan 2
θ.
Write the expression as the sine, cosine, or tangent of a
double angle. Then find the exact value of the expression.
85) 2 sin 60° cos 60°
86) cos2 30° - sin2 30°
2 tan 87)
θ
Find sin .
2
Use the Half-angle Formulas to find the exact value of the
trigonometric function.
93) cos 22.5°
θ.
83) tan θ = 25 π
, < θ < π
24 2
5π
8
5π
1 - tan2 8
5
5π
12
Answer Key
Testname: 113REVIEWT2SP16
1)
π
4
2) - π
6
3) 0
4) 0
π
5)
6
6)
5π
6
7) π
π
8)
3
9)
π
4
10) - π
4
11) - π
3
12) 1.22
13) 0.73
14) -0.17
15) 0.84
16) 0.30
17) 0.84
18) 2.53
π
19)
4
20)
3π
7
21) - π
5
22)
2π
3
23)
π
3
24) - π
7
25) 0.6
26) -9.5
3
27)
5
28)
5 61
61
29)
1
5
6
Answer Key
Testname: 113REVIEWT2SP16
30)
2 3
3
31) 1
x 2
32)
2
33)
5
x
34)
5 x2 + 25
x2 + 25
35) B = 50°, a = 43.5, c = 67.6
36) A = 59°, a = 92.4, c = 107.8
37) A = 69.6°, B = 20.4°, a = 403
38) A = 72.1°, b = 1.1 m, c = 3.6 m
39) 75 feet
40) 80.84°
41) 29°
42) 578.4 feet
43) N 33° E
44) N 46° W
45) S 75° W
46) S 50° E
47) 68 miles north and 63.4 miles east
48) N 18.4° E
49) N 146.4° W
50) 1
51) 1
52) sec2 x
53) tan 2 x (1 + cos 2x) = 1 - cos 2x
(1 + cos 2x) = 1 - cos 2x
1 + cos 2x
54) sin 4t = sin [2(2t)] = 2 sin2t cos2t.
55) cos 4θ = cos[2(2θ)] = 2 cos2 (2θ) - 1
56)
57)
58)
3 + 2
2( 3 - 1)
4
3
3
59) - 3
1
60)
2
61) cos π
6
62) α = 155°, β = 35°
-6 + 4 21
63)
25
7
Answer Key
Testname: 113REVIEWT2SP16
64)
3
5
65) -sin θ
66) sin θ
119
67)
169
68)
240
289
69) -
120
119
70) -
7
25
71) -
24
25
72)
240
289
73)
240
161
74)
1
2
75)
3
2
3
3
76) - 77) D = 78) - 79)
1 2
v sin 2θ
32
7
25
24
25
80) - 240
161
81) - 161
289
82) - 120
169
83)
120
169
84) - 85)
86)
240
161
3
2
1
2
8
Answer Key
Testname: 113REVIEWT2SP16
87) 1
88) - 89)
47
49
17
25
90) - 24
25
91)
6
4
92)
7 2
10
93)
1
2
2 + 2
94)
1
2
2 + 3
95)
1
2
2 + 3
9