Download Review Test 2.tst

Trigonometry
Review Test 2
Find the exact value of the expression.
3
1) cos-1 2
8
11
Find the inverse function f-1 of the function f.
15) f(x) = 5 sin x - 7
2) tan-1 1
Use a calculator to find the value of the expression
rounded to two decimal places.
3) cos-1 (0.6)
4) sin-1 - 14) sin sin-1 16) f(x) = 7 cos x + 3
17) f(x) = 6 tan(8x)
Find the domain of the function f and of its inverse
function f-1 .
3
4
18) f(x) = 5 sin x - 7
5) tan-1 (-1.5)
19) f(x) = 3 tan x + 6
Find the exact value of the expression. Do not use a
calculator.
5π
6) sin-1 sin 7
6π
7) cos-1 cos 7
4π
8) tan-1 tan 5
4π
9) cos-1 cos 3
π
10) cos-1 cos - 4
11) cos-1 cos - 7π
6
Find the exact value, if any, of the composite function. If
there is no value, say it is ʺnot definedʺ. Do not use a
calculator.
12) tan(tan-1 (-9.6))
13) cos[cos-1 (-1.3)]
20) f(x) = 2 sin(5x)
Find the exact solution of the equation.
π
21) sin-1 x = 2
22) 3 sin-1 x = π
23) 4 cos-1 x = π
24) 3 tan-1 (2x) = π
Find the exact value of the expression.
3
25) sec sin-1 - 2
26) cos sin-1 1
2
1
27) tan cos-1 - 2
2
28) sin cos-1 - 2
29) cot-1 -1
30) csc-1 -2
46)
sin α + sin β
= sin α sin β
csc α + csc β
47)
cot x
csc x - 1
= 1 + csc x
cot x
48)
cot 2 x
1 - sin x
= csc x + 1
sin x
31) cot-1 3
Use a calculator to find the value of the expression in
radian measure rounded to two decimal places.
5
32) csc-1 2
Write the trigonometric expression as an algebraic
expression in u.
33) sin (tan-1 u)
Find the exact value of the expression.
11π
49) sin 12
34) cos (tan-1 u)
50) sin 15°
35) cos (sin-1 u)
51) sin 255°
36) cos (cot-1 u)
Simplify the expression.
cos θ
37)
+ tan θ
1 + sin θ
Establish the identity.
π
52) cos x + = -sin x
2
π
53) tan + x = -cot x
2
38) (1 + cot θ)(1 - cot θ) - csc2 θ
54) tan(θ - π) = tan θ
Establish the identity.
39) tan u(csc u - sin u) = cos u
55)
sin(α - β)
= cot β - cot α
sin α sin β
40) (sin x)(tan x cos x - cot x cos x) = 1 - 2 cos 2 x
56) cos(x - y) - cos(x + y) = 2 sin x sin y
41) (1 - cos x)(1 + cos x) = sin2 x
57)
42)
tan u - 1 1 - cot u
= tan u + 1 1 + cot u
43)
sec θ - 1
tan θ
= tan θ
sec θ + 1
44)
cot u + csc u - 1
= csc u + cot u
cot u - csc u + 1
45) csc u - sin u = cos u cot u
cos(x - y) 1 + tan x tan y
= cos(x + y) 1 - tan x tan y
Find the exact value of the expression.
3
1
58) sin cos-1 - sin-1 2
2
4
3
59) cos tan-1 - sin-1 3
5
2
1
60) sin sin-1 + cos-1 3
3
3
1
61) tan tan-1 + sin-1 4
2
Express the sum or difference as a product of sines and/or
cosines.
75) sin(10θ) + sin(4θ)
Write the trigonometric expression as an algebraic
expression containing u and v.
62) cos(sin-1 u - cos-1 v)
76) cos(8θ) - cos(2θ)
77) cos(5θ) + cos(3θ)
63) sin(tan-1 u + tan-1 v)
78) sin(8θ) - sin(4θ)
Establish the identity.
64) sec(2θ)= Establish the identity.
sin(10θ) + sin(4θ)
79)
= tan(7θ)
cos(10θ) + cos(4θ)
csc2 θ
csc2 θ - 2
u csc u + cot u
65) cot2 = 2 csc u - cot u
80)
α - β
cos α + cos β
= cot 2
sin α - sin β
66) cos(4u) = 2 cos2 (2u) - 1
Use the information given about the angle θ, 0 ≤ θ ≤ 2π, to
find the exact value of the indicated trigonometric
function.
5
3π
θ
67) sin θ = - , < θ < 2π
Find sin .
2
2
5
68) csc θ = - 6, cos θ > 0
θ
Find cos .
2
3
69) cos θ = - , sin θ > 0
5
θ
Find cos .
2
13 π
70) sec θ = - , < θ < π
12 2
θ
Find sin .
2
Express the product as a sum containing only sines or
cosines.
71) sin(9θ) cos(5θ)
Solve the equation on the interval 0 ≤ θ < 2π.
81) 2 cos θ + 3 = 2
82) 4 sin2 θ = 1
3
θ
83) tan = 3
2
84) 2 3 sin(4θ) = 3
85) 7 csc θ - 1 = 6
Solve the equation. Give a general formula for all the
solutions.
2
86) cos θ = - 2
Use a calculator to solve the equation on the interval 0 ≤ θ
< 2π. Round the answer to two decimal places.
87) tan θ = 2.3
72) cos(3θ) cos(2θ)
88) 7 tan θ - 2 = 0
73) sin(5θ) sin(2θ)
74) cos(4θ) cos(7θ)
Solve the equation on the interval 0 ≤ θ < 2π.
89) sin2 θ + sin θ = 0
90) cos2 θ - 1 = 0
91) sin2 θ = 5(cos θ + 1)
92) cos2 θ = 3(1 - sin θ)
93) cos2 θ - sin2 θ = 1 + sin θ
94) 1 + cos θ = 2 sin2 θ
95) tan(2θ) - tan θ = 0
96) sec θ = cos θ
105) A tree casts a shadow of 26 meters when the
angle of elevation of the sun is 24°. Find the
height of the tree to the nearest meter.
Solve the triangle.
106) A = 60°, B = 100°, a = 1
107) B = 20°, C = 30°, a = 2
Two sides and an angle are given. Determine whether the
given information results in one triangle, two triangles, or
no triangle at all. Solve any triangle(s) that results.
108) a = 7, b = 9, B = 49°
97) cos(2θ) - 3 cos θ + 2 = 0
109) b = 3, c = 5, B = 70°
98) cos θ - sin θ = 0
99)
3 sin θ - cos θ = -1
Solve the right triangle using the information given.
Round answers to two decimal places, if necessary.
100) a = 8, B = 30°; Find b, c, and A.
101) a = 3, A = 20°; Find b, c, and B.
102) b = 5, c = 8; Find a, B, and A.
Solve the problem.
103) A radio transmission tower is 150 feet tall.
How long should a guy wire be if it is to be
attached 5 feet from the top and is to make an
angle of 20° with the ground? Give your
answer to the nearest tenth of a foot.
104) A straight trail with a uniform inclination of
11° leads from a lodge at an elevation of 600
feet to a mountain lake at an elevation of 9000
feet. What is the length of the trail (to the
nearest foot)?
110) a = 26, b = 20, B = 15°
Solve the problem.
111) An airplane is sighted at the same time by two
ground observers who are 3 miles apart and
both directly west of the airplane. They report
the angles of elevation as 11° and 23°. How
high is the airplane?
112) A ship sailing parallel to shore sights a
lighthouse at an angle of 10° from its direction
of travel. After traveling 5 miles farther, the
angle is 23°. At that time, how far is the ship
from the lighthouse?
113) To find the distance AB across a river, a
distance BC of 1034 m is laid off on one side of
the river. It is found that B = 104.0° and
C = 13.9°. Find AB. Round to the nearest
meter.
Solve the triangle.
114) b = 2, c = 3, A = 95°
115) a = 5, c = 10, B = 109°
116) b = 5, c = 6, A = 95°
117) a = 9, b = 9, c = 6
118) a = 8, b = 6, c = 4
Solve the problem.
119) Two points A and B are on opposite sides of a
building. A surveyor selects a third point C to
place a transit. Point C is 53 feet from point A
and 70 feet from point B. The angle ACB is 51°.
How far apart are points A and B?
120) A famous golfer tees off on a long, straight 455
yard par 4 and slices his drive 13° to the right
of the line from tee to the hole. If the drive
went 283 yards, how many yards will the
golferʹs second shot have to be to reach the
hole?
Answer Key
Testname: REVIEW TEST 2
1)
π
6
2)
π
4
27) - 3
2
28)
2
29)
3) 0.93
4) -0.85
5) -0.98
2π
6)
7
7)
30) - 31)
6π
7
8) - π
6
π
6
32) 0.41
u
33)
u2 + 1
π
5
9)
2π
3
34)
10)
π
4
35)
11) 3π
4
36)
5π
6
1
u2 + 1
1 - u2
u
u2 + 1
37) sec θ
12) -9.6
13) not defined
8
14)
11
15) f-1 (x) = sin-1
x + 7
5
16) f-1 (x) = cos-1
x - 3
7
38) -2 cot2 θ
39) tan u(csc u - sin u) = tan u · csc u - tan u · sin u =
1
sin u
1
sin2 u
sin u
· - · sin u = - =
cos u
cos u
cos u sin u cos u
1 - sin2 u cos2 u
= = cos u
cos u
cos u
1
x
17) f-1 (x) = tan-1
8
6
40) (sin x)(tan x cos x + cot x cos x) = sin x
sin x cos x cos 2 x
- = sin 2 x - cos 2 x
sin x
cos x
18) Domain of f: (-∞, ∞)
Domain of f-1 : [-12, -2]
= (1 - cos 2 x)- cos 2 x = 1 - 2 cos 2 x.
41) (1 - cos x)(1 + cos x) = 1 - cos2 x = sin2 x
19) Domain of f: x ≠ (2k + 1)π
; k an integer
2
Domain of f-1 : (-∞, ∞)
20) Domain of f: (-∞, ∞)
Domain of f-1 : [-2, 2]
21) x = 1
22) x = 3
2
23) x = 2
2
24) x = 3
2
25) 2
26)
3
2
1
- 1
cot u
1 - cot u
cot u
1 - cot u
1 + cot u
42)
tan u - 1
= tan u + 1
43)
sec θ - 1 sec θ - 1 sec θ + 1
sec 2 θ - 1
= · = =
tan θ
tan θ
sec θ + 1 tan θ(sec θ + 1)
1
+ 1
cot u
= 1 + cot u
cot u
tan θ
tan2 θ
= tan θ(sec θ + 1) sec θ + 1
= Answer Key
Testname: REVIEW TEST 2
44)
cot u + csc u - 1
cot u + (csc u - 1)
= =
cot u - csc u + 1
cot u - (csc u - 1)
sin ((π/2) + x)
π
=
53) tan + x = cos ((π/2) + x)
2
cot u + (csc u - 1) cot u + (csc u - 1)
· =
cot u - (csc u - 1) cot u + (csc u - 1)
1 · cos x + sin x · 0
sin (π/2) cos x + sin x cos (π/2)
= 0 · cos x - 1 · sin x
cos (π/2) cos x - sin (π/2) sin x
cot2 u + 2 cot u(csc u - 1) + (csc2 u - 2 csc u + 1)
=
cot2 u - (csc2 u - 2 csc u + 1)
= -cot x.
54) tan (θ - π) = csc2 u - 1 + 2 cot u(csc u - 1) + (csc2 u - 2 csc u + 1)
=
csc2 u - 1 - (csc2 u - 2 csc u + 1)
55)
2csc2 u - 2 csc u + 2 cot u(csc u - 1)
=
-2 + 2 csc u
2 csc u(csc u - 1) + 2 cot u(csc u - 1)
=
2 (csc u - 1)
2(csc u + cot u)(csc u - 1)
= csc u + cot u
2 (csc u - 1)
45) csc u - sin u = cos u · 46)
48)
56) cos (x - y) - cos (x + y) = cos x cos y + sin x sin y - (
cos x cos y - sin x sin y) = 2 sin x sin y.
cos (x - y) cos x cos y + sin x sin y 1/(cos x cos y)
= = 57)
cos (x + y) cos x cos y - sin x sin y 1/(cos x cos y)
· cos x cos y + sin x sin y
=
cos x cos y - sin x sin y
1 + tan x tan y
.
1 - tan x tan y
cos u
= cos u cot u
sin u
58) 0
24
59)
25
60)
2 + 2 10
9
cot x
cot x cot x
csc 2 x - 1
= = =
1 + csc x (1+ csc x) cot x (1+ csc x) cot x
61)
9 + 4 3
12 - 3 3
(csc x + 1)(csc x - 1) csc x - 1
= .
cot x
(1+ csc x) cot x
62) v 1 - u2 + u 1 - v2
u + v
63)
2
u + 1 · v2 + 1
(sin α + sin β) · 47)
sin α cos β - cos α sin β sin α cos β
sin(α - β)
= = sin α sin β
sin α sin β
sin α sin β
cos α
cos α sin β cos β
= - = cot β - cot α
sin β
sin α
sin α sin β
1
1 - sin2 u cos2 u
- sin u = = =
sin u
sin u
sin u
sin α + sin β
sin α + sin β
sin α + sin β
= = =
1
sin β + sin α
1
csc α + csc β
+ sin α sin β
sin α sin β
tan θ - tan π
tan θ - 0
= = tan θ
1 + tan θ tan π 1 + tan θ · 0
sin α sin β
= sin α sin β
sin β + sin α
cot 2 x
csc2 x - 1 (csc x + 1)(csc x - 1)
= = = csc x csc x + 1
csc x + 1
csc x + 1
sin x 1 - sin x
1
- = .
1 = sin x
sin x sin x
49)
2( 3 - 1)
4
50)
2( 3 - 1)
4
51) -
2( 3 - 1)
4
52) cos x + π
π
π
= cos x cos - sin x sin = (cos x)(0) 2
2
2
(sin x)(1) = - sin x.
64) sec(2θ) = 1
1
= = cos(2θ) 1 - 2 sin2 θ
1
sin2 θ
1
- 2
sin2 θ
=
csc2 θ
csc2 θ - 2
u
65) cot2 = 2
1
u
tan2 2
= 1 + cos u
csc u + cot u
= 1 - cos u
csc u - cot u
66) cos(4u) = cos[2(2u)] = 2 cos2 (2u) - 1
67) 5 - 2 5
10
68) - 6 + 30
12
Answer Key
Testname: REVIEW TEST 2
69)
5
5
93) 0, π, 7π 11π
, 6
6
70)
5 26
26
π
5π
94) , π, 3
3
71)
1
[sin(14θ) + sin(4θ)]
2
72)
1
[ cos θ + cos(5θ)]
2
95) 0, π
96) 0, π
π 5π
97) 0, , 3 3
73)
1
[ cos(3θ) - cos(7θ)]
2
98)
74)
1
[cos(3θ) + cos(11θ)]
2
99) 0, 75) 2 sin(7θ) cos(3θ)
76) -2 sin(5θ) sin(3θ)
77) 2 cos(4θ) cos θ
78) 2 sin(2θ) cos(6θ)
2 sin (7θ) cos (3θ)
sin (7θ)
sin (10θ) + sin (4θ)
= = 79)
cos (10θ) + cos (4θ) 2 cos (7θ) cos (3θ) cos (7θ)
= tan (7θ)
80)
cos α + cos β
= sin α - sin β
cot 2 cos α + β
α - β
cos 2
2
2 sin α - β
α + β
cos 2
2
α - β
2
81)
2π 4π
, 3
3
82)
π 5π 7π 11π
, , , 6 6
6
6
83)
π
3
84)
π π 2π 7π 7π 13π 5π 19π
, , , , , , , 12 6 3 12 6
12
3
12
85)
π
2
86) θ θ = 3π
5π
+ 2kπ, θ = + 2kπ
4
4
87) 1.16, 4.30
88) 0.28, 3.42
3π
89) 0, π, 2
90) 0, π
91) π
π
92)
2
cos α - β
2
sin α - β
2
= =
π 5π
, 4 4
4π
3
100) b = 4.62
c = 9.24
A = 60°
101) b = 8.24
c = 8.77
B = 70°
102) a = 6.24
B = 38.68°
A = 51.32°
103) 424.0 ft
104) 44,023 ft
105) 12 m
106) C = 20°, b = 1.14, c = 0.39
107) A = 130°, b = 0.89, c = 1.31
108) one triangle
A = 35.94°, C = 95.06°, c = 11.88
109) no triangle
110) two triangles
A1 = 19.66°, C1 = 145.34°, c1 = 43.95 or
111)
112)
113)
114)
115)
116)
117)
118)
119)
120)
A2 = 160.34°, C2 = 4.66°, c2 = 6.28
1.08 mi
3.86 mi
281 m
a = 3.75, B = 32.1°, C = 52.9°
b = 12.6, A = 22°, C = 49°
a = 8.14, B = 37.7°, C = 47.3°
A = 70.5°, B = 70.5°, C = 39°
A = 104.5°, B = 46.6°, C = 28.9°
55.1 ft
190.2 yd