Download Math 113 Test II Practice Problems spring 2011.tst

Math 113 Test II Practice Problems - Sections 4.7, 4.8, 5.1, 5.2, 5.3, 5.5
Spring 2011
Find the exact value of the expression.
3
1) sin-1 2
Solve the right triangle shown in the figure. Round
lengths to one decimal place and express angles to the
nearest tenth of a degree.
2
2) cos-1 - 2
3
3) tan-1 3
12) A = 40°, b = 46.6
Use a calculator to find the value of the expression
rounded to two decimal places.
4) tan-1 (-1.8)
13) b = 110, c = 410
14) A = 51.9°, c = 51.2
2
5) sin-1 5
Solve the problem.
15) From a boat on the lake, the angle of elevation
to the top of a cliff is 24°22ʹ. If the base of the
cliff is 747 feet from the boat, how high is the
cliff (to the nearest foot)?
2
6) cos-1 - 3
Find the exact value of the expression, if possible. Do not
use a calculator.
6π
7) sin-1 sin 7
16) A building 290 feet tall casts a 100 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 5 feet above ground level.)
π
8) cos-1 cos - 3
Using a calculator, solve the following problems. Round
your answers to the nearest tenth.
17) A boat leaves the entrance of a harbor and
travels 87 miles on a bearing of N 26° E. How
many miles north and how many miles east
from the harbor has the boat traveled?
10π
9) tan-1 tan 11
Use a sketch to find the exact value of the expression.
3
10) cos sin-1 5
Verify the identity.
18) sec 4 x + sec2 x tan2 x - 2 tan 4 x = 3sec2 x - 2
5
11) cos tan-1 8
19)
(sin x + cos x)2
= 1
1 + 2 sin x cos x
Write the expression as the cosine of an angle, knowing
that the expression is the right side of the formula for cos
(α - β) with particular values for α and β.
20) cos (155°) cos (35°) + sin (155°) sin (35°)
1
Find the exact value of the expression.
2π π
21) cos (
- )
9
18
31) tan α = < π Find tan (α + β).
Use the given information to find the exact value of the
expression.
4
22) Find cos (α - β). sin α = , α lies in quadrant
5
Use the figure to find the exact value of the trigonometric
function.
32)
3
2
II, and cos β = , β lies in quadrant I.
5
Find the exact value by using a sum or difference identity.
23) sin 15°
Find sin 2θ.
Use the given information to find the exact value of the
expression.
4
33) Find tan 2θ. sin θ = , θ lies in quadrant II.
5
Find the exact value of the expression.
25) sin 255° cos 15° - cos 255° sin 15°
Use the given information to find the exact value of the
expression.
4
26) Find cos (α + β).
sin α = , α lies in
5
34) Find sin 2θ. cos θ = Use a half-angle formula to find the exact value of the
expression.
36) cos 112.5°
4
sin α = - , α lies in
5
quadrant IV, and cos β = - 7
, θ lies in quadrant IV.
25
Write the expression as the sine, cosine, or tangent of a
double angle. Then find the exact value of the expression.
35) cos2 120° - sin2 120°
12
, β lies in quadrant
13
I.
27) Find sin (α - β).
5
4
24) cos (135° + 60°)
quadrant I, and cos β = 24
3π
20 π
, π < α < ; cos β = - , < β
7
2
29 2
21
, β lies in
5
Use a half-angle formula to find the exact value of the
expression.
5π
37) sin 12
quadrant III.
Find the exact value by using a difference identity.
28) tan 105°
Use the given information given to find the exact value of
the trigonometric function.
1
38) sin θ = ,
θ lies in quadrant I
Find sin
4
Use trigonometric identities to find the exact value.
tan 50° + tan 100°
29)
1 - tan 50° tan 100°
θ
.
2
Find the exact value under the given conditions.
7 π
21
30) cos α = - , < α < π; sin β = - , π <
25 2
5
5
39) sec θ = - ,
4
3π
Find tan (α + β).
β < 2
θ
sin .
2
2
θ lies in quadrant II
Find
Find all solutions of the equation.
40) 2 cos x + 2 = 0
41) tan x sec x = -2 tan x
42) 8 sin x + 6 2 = 6 sin x + 5 2
Solve the equation on the interval [0, 2 π).
3
43) sin 4x = 2
44) cos 2x = 3
2
45) cos2 x + 2 cos x + 1 = 0
46) cos x = sin x
47) sec 2 x - 2 = tan 2 x
48) tan x + sec x = 1
49) sin2 x + sin x = 0
Solve the equation on the interval [0, 2 π).
50) sin x - 2 sin x cos x = 0
51) tan2 x sin x = tan2 x
Solve the equation on the interval [0, 2 π).
52) 2 cos2 x + sin x - 2 = 0
53) sin 2x + sin x = 0
Use a calculator to solve the equation on the interval [0, 2
π). Round the answer to two decimal places.
54) sin x = 0.38
55) sin x = -0.29
56) 3 cos2 x + 2 cos x = 1
57) sin 3x = - sin x
3
Answer Key
Testname: MATH 113 TEST II PRACTICE PROBLEMS SPRING 2011
1)
π
3
30)
-48 + 7 21
14 + 24 21
2)
3π
4
31)
333
644
3)
π
6
32)
24
25
33)
24
7
4) -1.06
5) 0.41
6) 2.06
π
7)
7
8)
π
3
9) - 34) - 336
625
35) - 1
2
1
36) - 2 - 2
2
π
11
10)
4
5
37)
2 + 3 1
= 2
2
11)
8 89
89
38)
8 - 2 15
4
12) B = 50°, a = 39.1, c = 60.8
13) A = 74.4°, B = 15.6°, a = 395
14) B = 38.1°, a = 40.3, b = 31.6
15) 338 feet
16) 70.67°
17) 78.2 miles north and 38.1 miles east
18) Work on one side until you get it like the other side.
19) same as #1
20) cos (120°)
3
21)
2
22)
39)
3π
5π
+ 2nπ or x = + 2nπ
4
4
41) x = 4π
π
2π
+ 2nπ or x = + 2nπ or + nπ
3
2
3
42) 8 sin x - 6sinx= 5 2 - 6 2
2 sin x = - 2
2
sin x =- 2
x = 2( 3 - 1)
4
24) 25) - 2( 3 + 1)
4
16
65
27)
6 + 4 21
25
5π
7π
+ 2nπ or x = + 2nπ
4
4
43)
π π 2π 7π 7π 13π 5π 19π
, , , , , , , 12
3
12
12 6 3 12 6
44)
π 11π 13π 23π
, , , 12 12
12
12
45) π
π 5π
46) , 4 4
3
2
26)
3 10
10
40) x = -6 + 4 21
25
23)
2 + 3
47) no solution
48) 0
49) Factor out the GCF first.
3π
0, π, 2
28) -2 - 3
3
29) - 3
4
Answer Key
Testname: MATH 113 TEST II PRACTICE PROBLEMS SPRING 2011
50) 0, π
5π
, π, 3
3
π
51) 0, , π
2
52) 0, π, 53) 0, π 5π
, 6 6
2π
4π
, π, 3
3
54) 0.39, 2.75
55) 3.44, 5.99
56) 1.23, 3.14, 5.05
57) 0, 1.57, 3.14, 4.71
5