Download Practice Problems Test 2 Fall 2013

Practice Problems Test 2 Fall 2013
Use a calculator to find the value of the expression
rounded to two decimal places.
7
15) csc-1 (- )
3
Find the exact value of the expression.
2
)
1) cos-1 (- 2
2) tan-1 (1)
5
16) cot-1 ( )
8
3) sin-1 (-0.5)
Simplify the trigonometric expression by following the
indicated direction.
3 cos2 θ + 4 cos θ + 1
17) Factor and simplify: cos2 θ - 1
3
4) tan-1 3
Find the exact value of the expression. Do not use a
calculator.
π
5) cos-1 [cos ( )]
10
Complete the identity.
*Note - you must practice verifying identities by
completing homework problems out of the book from
section 5.1
3π
6) cos-1 [cos (- )]
5
18)
Use a calculator to find the value of the expression
rounded to two decimal places.
6
7) cos-1 (- )
3
19) tan2 θ - 3 sin θ tan θ sec θ = ?
20) sec 4 θ - 2 sec 2 θ tan 2 θ + tan 4 θ = ?
Solve the problem.
8) Which one exists and why? sin [sin-1 (2.1)]
or sin-1 [sin (2.1)]
21)
9) State the domain and range of f(x) = tan-1 x.
csc θ cot θ
= ?
sec θ
Find the exact value of the trigonometric function.
22) tan 255°
Find the exact value of the expression.
3
10) cos (sin-1 )
5
11) cos-1 (cos sec θ sin θ
- 1 = ?
tan θ
23) sin 7π
)
6
11π
12
Find the exact value of the expression.
tan 20° + tan 10°
24)
1 - tan 20° tan 10°
7π
12) cos-1 (sin )
6
25)
1
13) tan (cos-1 )
3
tan 170° - tan 50°
1 + tan 170° tan 50°
26) cos 15° cos 45° - sin 15° sin 45°
14) cot-1 (-1)
1
Find the exact value under the given conditions.
4
3π
24 π
27) tan α = , π < α < ; cos β = - , < β
3
2
25 2
40) sec θ = - < π Find sin (α + β).
25 π
, < θ < π
24 2
θ
Find sin .
2
Use the Half-angle Formulas to find the exact value of the
trigonometric function.
41) cos 22.5°
Use the given information to find the exact value of the
expression.
20
28) Find cos (α + β).
sin α = , α lies in
29
42) sin 75°
4
quadrant I, and cos β = , β lies in quadrant I.
5
43) sin Complete the identity.
π
29) cos ( + θ) = ?
2
5π
12
Solve the right triangle using the information given.
Round answers to two decimal places, if necessary.
30) sin (π - θ) = ?
Find the exact value of the expression.
1
3
31) sin (cos-1 - sin-1
)
2
2
44) b = 8, α = 40°; find a, c, and β
Solve the triangle. Round any answers to 2 decimal
places.
β = 50°,
a = 3
45) α = 30°,
1
1
32) cos (sin-1 - tan-1 )
3
2
33) sin-1 [sin (
6π
)]
7
For #ʹs 3 - 5, 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. Round any answers to 2 decimal places.
46) a = 7, b = 9, β = 49°
34) csc-1 (-2)
2 3
35) sec-1 (- )
3
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).
36) cos θ = - , csc θ < 0
7
5
37) csc θ = - , tan θ > 0
2
Find cos (2θ).
4 3π
38) cos θ = , < θ < 2π
5
2
Find sin (2θ).
1
39) cos θ = , csc θ > 0
4
θ
Find sin .
2
47) a = 8, b = 6, β = 15°
48) a = 5,
b =69, α = 65°
An object is attached to a coiled spring. The object is
pulled down (negative direction from the rest position)
and then released. Write an equation for the distance of
the object from its rest position after t seconds.
49) amplitude = 5 cm; period = 4 seconds
50) amplitude = 11 in.; period = 8 seconds
An object moves in simple harmonic motion described by
the given equation, where t is measured in seconds and d
in meters . Find the maximum displacement, the
frequency, and the time required for one cycle.
51) d = 2 sin (5t) meters
2
52) d = -6 sin (3t) meters
Solve the problem.
53) An airplane is sighted at the same time by two
ground observers who are 4 miles apart and
both directly west of the airplane. They report
the angles of elevation as 13° and 21°. How
high is the airplane?
54) A ship sailing parallel to shore sights a
lighthouse at an angle of 13° from its direction
of travel. After traveling 5 miles farther, the
angle is 22°. At that time, how far is the ship
from the lighthouse?
55) A guy wire to the top of a tower makes an
angle of 53° with the level ground. At a point
32 feet farther from the base of the tower and
in line with the base of the wire, the angle of
elevation to the top of the tower is 25°. What is
the length of the guy wire?
3
Answer Key
Testname: 113REVIEWT2SP05
1)
3π
4
2)
π
4
3) - π
6
4)
π
6
5)
π
10
6)
3π
5
7) 2.53
8) The expression sin-1 [sin (2.1)] exists because 2.1 is in the domain of the sine function, but 2.1 is not in the range of the
sine function.
9) domain: all real numbers
range: all real numbers
4
10)
5
11)
5π
6
12)
2π
3
13) 2 2
3π
14)
4
15) -0.44
16) 1.01
3 cos θ + 1
17)
cos θ - 1
18) 0
19) -2 tan2 θ
20) 1
21) cot2 θ
3 + 2
2( 3 - 1)
4
22)
23)
24)
3
3
25) - 3
1
26)
2
27)
3
5
4
Answer Key
Testname: 113REVIEWT2SP05
28)
24
145
29) -sin θ
30) sin θ
31) 0
4 10 + 5
32)
15
33)
π
7
34) - 35)
5π
6
36) - 37)
π
6
47
49
17
25
38) - 24
25
39)
6
4
40)
7 2
10
41)
1
2
2 + 2
42)
1
2
2 + 3
43)
1
2
2 + 3
44) a = 6.71
c = 10.44
β = 50°
45) γ = 100°, b = 4.6, c = 5.91
46) one triangle
α = 35.94°, γ = 95.06°, c = 11.88
47) two triangles
α1 = 20.19°, γ1 = 144.81°, c1 = 13.36 or
α2 = 159.81°, γ2 = 5.19°, c2 = 2.1
48) one triangle
α = 35.94°, γ = 95.06°, c = 11.88
1
49) d = -5 cos πt
2
50) d = -11 cos 1
πt
4
5
Answer Key
Testname: 113REVIEWT2SP05
2
5
51) displacement = 2 meters; period = π seconds; f = oscillations/second
5
2π
3
2
oscillations/second
52) displacement = 6 meters; period = π seconds; f = 2π
3
53) 2.32 mi
54) 7.19 mi
55) 28.81 ft
6