Exam
Name___________________________________
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Assume that the cities lie on the same north-south line and that the radius of the earth is 6400 km.
1) Find the distance between City A, 62° N and City B, 29° N. (Round to the nearest kilometer.)
A) 3777 km
B) 3624 km
C) 3715 km
D) 3686 km
1)
Solve the problem.
2) Let angle POQ be designated . Angles PQR and VRQ are right angles. If
OQ accurate to four decimal places.
2)
A) 2.1445
B) 2.3662
= 65°, find the length of
C) 0.9063
D) 0.4226
Convert the radian measure to degrees. Round to the nearest hundredth if necessary.
3)
3)
4
A) 0.785°
B) 45°
C)
4
°
D) 45 °
Convert the degree measure to radians, correct to four decimal places. Use 3.1416 for .
4) 67.45°
A) 0.9772
B) 1.2772
C) 1.1772
D) 1.0772
4)
Convert the radian measure to degrees. Give answer using decimal degrees to the nearest hundredth. Use 3.1416 for .
5) 5
5)
A) 572.86°
B) 286.58°
C) 286.48°
D) 572.96°
Find the exact values of s in the given interval that satisfy the given condition.
6) [0, 2 ); 4 sin2 s = 3
5 7
11
,
,
A) ,
6 6 6
6
C)
3
,
2 4
5
,
,
B) ,
3 3 3
3
2
3
D)
1
6
,
5
6
6)
Find the exact value of s in the given interval that has the given circular function value.
3
, ; cos s = 7)
2
2
A) s =
Use the formula
8)
6
=
5
6
B) s =
t
2
3
C) s =
7)
D) s =
3
4
to find the value of the missing variable. Give an exact answer unless otherwise indicated.
= 7.4962 radians per min, = 12.25 radians (Round to four decimal places when necessary.)
A) 0.6119 min
B) 1.6342 min
C) 91.8285 min
D) 19.7462 min
Find the exact value without using a calculator.
2
9) cos
3
A) undefined
B) -
9)
1
2
C) -
3
2
D)
3
2
Solve the problem.
10) A circular pulley is rotating about its center. Through how many radians would it turn in 29
rotations?
A) 87
B) 116
C) 145
D) 58
Convert the degree measure to radians, correct to four decimal places. Use 3.1416 for .
11) -201°48
A) -3.4921
B) -3.5221
C) -3.5021
D) -3.5121
Find the exact value without using a calculator.
-5
12) cot
6
A) - 3
B) -
3
3
3
C)
Solve the problem.
14) Find for the minute hand of a clock.
C)
30
60
10)
11)
12)
D)
3
3
Find the value of s in the interval [0, /2] that makes the statement true. Round to four decimal places.
13) cos s = 0.6577
A) 0.7178
B) 0.8530
C) 5.4302
D) 0.9946
A)
8)
13)
14)
radians per min
B)
radians per min
D) 30 radians per min
2
6
radians per min
Convert the radian measure to degrees. Round to the nearest hundredth if necessary.
15) -
15)
4
A) -0.79°
B) -45 °
C) -
4
°
D) -45°
The figure shows an angle in standard position with its terminal side intersecting the unit circle. Evaluate the indicated
circular function value of .
16) Find cot .
16)
7
24
,25
25
A)
7
24
B)
24
7
C) -
7
24
D) -
24
7
Convert the radian measure to degrees. Give answer using decimal degrees to the nearest hundredth. Use 3.1416 for .
17) 1.6816
17)
A) 96.85°
B) 96.35°
C) 97.35°
D) 95.65°
Assume that the cities lie on the same north-south line and that the radius of the earth is 6400 km.
18) Find the latitude of Spokane, WA if Spokane and Jordan Valley, OR, 43.15° N, are 486 km apart.
A) 47.5 °N
B) 39.5 °N
C) 52.46 °N
D) 38.8 °N
Convert the degree measure to radians. Leave answer as a multiple of .
19) 570°
19
19
19
A)
B)
C)
6
12
5
19
D)
3
18)
19)
Solve the problem.
20) The temperature in Verlander is modeled by T(x) = 48 sin
365
(x - 102) + 41 where T(x) is the
temperature in degrees Fahrenheit on day x, with x = 1 representing January 1 and x = 365
representing December 31. Find the temperature on July 21.
A) -6°F
B) 88°F
C) 124°F
D) 86°F
3
20)
Use a table or a calculator to evaluate the function. Round to four decimal places.
21) cos 0.2116
A) 0.9777
B) 0.2100
C) 1.0228
D) 0.2148
Find the exact values of s in the given interval that satisfy the given condition.
3
22) [- , ); cos2 s =
4
21)
22)
A) -
5
,- ,
3
3 3
B) -
11
,- ,
6
6 6
C) -
5
,,,- , ,
3
3
3
3 3 3
D) -
11
7
5
,,,- , ,
6
6
6
6 6 6
The figure shows an angle in standard position with its terminal side intersecting the unit circle. Evaluate the indicated
circular function value of .
23) Find tan .
23)
7
24
,25
25
A) -
25
7
B) -
7
24
C) -
24
7
D)
25
24
Convert the radian measure to degrees. Round to the nearest hundredth if necessary.
5
24)
12
A) 75°
B) 144°
C) 150°
24)
D) 432 °
Solve the problem.
25) Find the radius of a circle in which a central angle of
meters. Round to the nearest hundredth.
A) 12.67 m
B) 25.33 m
7
radian determines a sector of area 72 square
C) 17.91 m
4
D) 320.86 m
25)
26) A circular sector has an area of 16 in2 and an arc length of 6 inches. What is the measure of the
central angle in degrees? Round to the nearest degree.
A) 32°
B) 129°
C) 11°
D) 64°
26)
27) A weight attached to a spring is pulled down 3 inches below the equilibrium position. Assuming
5
that the frequency of the system is
cycles per second, determine a trigonometric model that gives
27)
the position of the weight at time t seconds.
A) s(t) = -3cos 10t
B) s(t) = 3cos 10t
C) s(t) = 3cos 5t
D) s(t) = -3cos 5t
C) 2
2
D)
3
Give the amplitude or period as requested.
28) Period of y = sin 3x
A) 3
B) 1
Solve the problem.
29) The voltage E in an electrical circuit is given by E = 4.4 cos 60 t, where t is time measured in
seconds. Find the frequency of the function (that is, find the number of cycles or periods completed
in one second).
1
1
A) 30
B)
C)
D) 60
60
30
28)
29)
The function graphed is of the form y = a sin bx or y = a cos bx, where b > 0. Determine the equation of the graph.
30)
30)
A) y = 5 sin (3x)
B) y = 5 cos
1
x
3
C) y = 3 sin
Graph the function.
5
1
x
5
D) y = 5 cos (3x)
31) y = sin
1
x
2
31)
A)
B)
C)
D)
6
32) y = 2 - 3 sec x +
32)
4
A)
B)
C)
D)
7
The function graphed is of the form y = a sin bx or y = a cos bx, where b > 0. Determine the equation of the graph.
33)
33)
A) y = 5 sin
1
x
3
C) y = 5 cos
B) y = -5 cos (3x)
8
1
x
3
D) y = 5 cos (3x)
Match the function with its graph.
34) 1) y = - tan x 3) y = - cot x -
2
2
2) y = tan x +
4) y = cot x +
34)
2
2
A)
B)
C)
D)
A) 1C, 2B, 3D, 4A
B) 1A, 2D, 3B, 4C
C) 1A, 2B, 3C, 4D
9
D) 1D, 2A, 3C, 4B
The function graphed is of the form y = cos x + c, y = sin x + c, y = cos(x - d), or y = sin(x - d), where d is the least possible
positive value. Determine the equation of the graph.
35)
35)
A) y = cos (x - 4)
B) y = sin x + 4
C) y = cos x - 4
D) y = sin x - 4
Find the phase shift of the function.
36) y = 4 - 3 sin 4x +
A)
C)
6
2
36)
2
units to the right
B)
units to the left
D)
8
8
units to the left
units to the right
Solve the problem.
37) The position of a weight attached to a spring is s(t) = -7 cos 16 t inches after t seconds. What is the
maximum height that the weight reaches above the equilibrium position and when does it first
reach the maximum height?
A) The maximum height of 7 inches is first reached after 8 seconds.
B) The maximum height of 14 inches is first reached after 4 seconds.
C) The maximum height of 14 inches is first reached after 8 seconds.
D) The maximum height of 7 inches is first reached after 0.06 seconds.
10
37)
Graph the function over a one-period interval.
38) y = 1 + sin(2x- )
38)
A)
B)
C)
D)
11
Match the function with its graph.
2) y = 3 cos x
39) 1) y = sin 3x
3) y = 3 sin x
4) y = cos 3x
39)
A)
B)
C)
D)
A) 1A, 2B, 3C, 4D
B) 1A, 2C, 3D, 4B
C) 1B, 2D, 3C, 4A
D) 1A, 2D, 3C, 4B
Solve the problem.
40) A generator produces an alternating current according to the equation I = 56 sin 127 t, where t is
time in seconds and I is the current in amperes. What is the smallest time t such that I = 28?
1
1
1
1
sec
sec
sec
sec
A)
B)
C)
D)
762
508
254
381
41) A rotating beacon is located 14 ft from a wall. The distance from the beacon to the point on the wall
where the beacon is aimed is given by
a = 14 sec 2 t ,
where t is time measured in seconds since the beacon started rotating. Find a for t = 0.37 seconds.
Round your answer to the nearest hundredth.
A) 27.85 ft
B) -20.45 ft
C) 20.45 ft
D) 35.25 ft
12
40)
41)
Graph the function.
42) y = -1 + 2 cos x
42)
A)
B)
C)
D)
Graph the function over a one-period interval.
13
43) y =
1
2
+ cos 2x 2
3
43)
A)
B)
C)
D)
Solve the problem.
44) Suppose that the average monthly low temperatures for a small town are shown in the table.
Month
1 2 3 4 5 6 7 8 9 10 11 12
Temperature (°F) 19 27 38 45 57 62 65 58 51 41 33 25
Model this data using f(x) = a sin(b(x - c)) + d.
A) f(x) = 23 sin
C) f(x) = 23 sin
6
(x - 7) + 42
12
B) f(x) = 42 sin
(x - 4) + 42
D) f(x) = 23 sin
14
6
6
(x - 4) + 23
(x - 4) + 42
44)
Find the exact value of the expression using the provided information.
3
5
, with s in quadrant IV, and sin t = , with t in quadrant
45) Find cos(s + t) given that sin s = 3
6
45)
IV.
A)
186 +
18
15
B)
186 18
15
C)
55 + 15
18
D)
55 - 15
18
Complete the sentence so the result is an identity. Let x be any real number.
46) cos x = (cot x)( )
A) tan x
B) sin x
C) csc x
D) sec x
Use a sum or difference identity to find the exact value.
47) tan 75°
A) 3 + 2
B) - 3 + 2
D) -
46)
47)
C)
3-2
3-2
Solve the problem.
48) The power dissipated in an electric circuit is given by the expression P = RI2 , where R is the
resistance of the circuit and I is the current through the circuit. For a sinusoidal alternating current,
the current might be represented by the relation I = A sin(2 ft), where A is the amplitude, f is the
frequency, and t is time. Write an expression for P involving the sine function, and use a
fundamental identity to write P in terms of the cosine function.
A) P = RA sin2 (2 ft) ; P = RA - RA cos2 (2 ft)
48)
B) P = RA2 sin2 (2 ft) ; P = RA2 - RA2 cos2 (2 ft)
C) P = RA sin2 (2 ft) ; P = RA - cos2 (2 ft)
D) P = RA2 sin2 (2 ft) ; P = - RA2 cos2 (2 ft)
Use a sum or difference identity to find the exact value.
49) sin 205° cos 85° - cos 205° sin 85°
3
1
A) B) 2
2
50) tan 345°
A) - 3 - 2
49)
3
C)
2
41
D)
12
50)
B)
3-2
C) -
15
3+2
D)
3+2
For the graph of a circular function y = f(x), determine whether f(-x) = f(x) or f(-x) = -f(x) is true.
51)
A) f(-x) = f(x)
B) f(-x) = -f(x)
Decide whether the expression is or is not an identity.
52) sin2x = sin x
A) Identity
52)
B) Not an identity
Factor the trigonometric expression and simplify.
53) sec4 x + sec2 x tan2 x - 2 tan4 x
A) tan2 x - 1
51)
B) sec2 x + 2
C) 3 sec2 x - 2
53)
D) 4 sec2 x
Graph the expression on each side of the equals symbol to determine whether the equation might be an identity.
sin + 1
= tan
54)
54)
cos + cot
A) Not an Identity
B) Identity
Find the exact value by using a half-angle identity.
55) sin 165°
1
1
2- 3
2+ 3
A) B) 2
2
1
C)
2
2+
3
1
D)
2
55)
2-
Use an identity to write the expression as a single trigonometric function or as a single number.
sin 72°
56)
1 - cos 72°
A) sin 36°
B) tan 36°
C) cot 36°
3
56)
D) cos 36°
Use the formula s = r t to find the value of the missing variable. Give an exact answer.
57) s =
11
m, r = 5 m, t = 2 sec
55
2
57)
radians per sec
B)
C) 110 radians per sec
D)
A)
16
110
radian per sec
2
radian per sec
55
Solve the problem.
58) Let angle POQ be designated . Angles PQR and VRQ are right angles. If
length of OU.
A) 1
B)
2
2
= 45°, find the exact
C) 0
D)
2
Convert the degree measure to radians, correct to four decimal places. Use 3.1416 for .
59) 23.2170°
A) 0.6052
B) 0.4052
C) 0.3052
D) 0.5052
Convert the degree measure to radians. Leave answer as a multiple of .
60) 810°
9
9
A) 9
B) C)
2
2
D) -
9
4
Convert the radian measure to degrees. Round to the nearest hundredth if necessary.
28
61)
9
A) 280°
B) 9.77°
C) 1120 °
58)
59)
60)
61)
D) 560°
Solve the problem.
62) A pendulum swinging through a central angle of 133° completes an arc of length 11.3 cm. What is
the length of the pendulum? Round to the nearest hundredth.
A) 4.67 cm
B) 4.97 cm
C) 4.87 cm
D) 4.77 cm
62)
63) Electrical wire is being wound around a drum with radius of 0.95 meters. How much line (to the
nearest hundredth of a meter) would be wound around the drum if it is rotated through an angle of
340.8°?
A) 5.55 m
B) 5.45 m
C) 5.75 m
D) 5.65 m
63)
64) Suppose the tip of the minute hand of a clock is 8 inches from the center of the clock. Determine the
distance traveled by the tip of the minute hand in 30 minutes. Give an exact answer.
A) 64 in.
B) 16 in.
C) 4 in.
D) 8 in.
64)
Graph the function.
17
65) y =
2
sin x +
3
3
65)
A)
B)
C)
D)
18
The function graphed is of the form y = a sin bx or y = a cos bx, where b > 0. Determine the equation of the graph.
66)
66)
A) y = -4 cos (3x)
C) y = 4 cos
B) y = 4 sin (3x)
1
x
3
D) y = -4 cos
1
x
3
Graph the function.
67) y = sec x -
67)
6
A)
B)
19
C)
68) y =
D)
2
2
tan x +
5
3
3
68)
A)
B)
20
C)
D)
69) y = 1.5 sin x
69)
A)
B)
21
C)
D)
Use a sum or difference identity to find the exact value.
5
70) sin 12
6+
4
A)
71) tan
2
B)
-
64
70)
2
C)
-
6+
4
2
64
D)
2
71)
12
A) 2 -
3
B) 2 +
3
C) - 2 +
3
D) - 2 -
3
Perform the transformation.
72) Write sec x in terms of sin x.
72)
A) ±
1 - sin2 x
B)
±
1 - sin2 x
sin x
D)
C)
Use the formula v = r
73) r = 2 cm,
± sin x 1 - sin2 x
1 - sin2 x
±
1 - sin2 x
1 - sin2 x
to find the value of the missing variable. Give an exact answer unless otherwise indicated.
=
3
radian per sec
A) 6 cm per sec
B)
73)
2
cm per sec
3
C)
3
cm per sec
2
D)
6
cm per sec
Convert the radian measure to degrees. Give answer using decimal degrees to the nearest hundredth. Use 3.1416 for .
74) -3.7
74)
A) -211.99°
B) -423.98°
C) -423.48°
D) -212.49°
Solve the problem.
75) Suppose that a weight on a spring has an initial position of s(0) = -2 inches and a period of
P = 2.5 seconds. Find a function s(t) = a cos(2 Ft) that models the displacement of the weight.
A) s(t) = -2 cos (2 (2.5)t)
B) s(t) = -2 cos (2 (0.4)t)
C) s(t) = -4 cos (2 (0.4)t)
D) s(t) = -4 cos (2 (2.5)t)
22
75)
Match the function with its graph.
76) 1) y = -csc x
2) y = -sec x
3) y = -tan x
4) y = -cot x
76)
A)
B)
C)
D)
A) 1C, 2A, 3B, 4D
B) 1B, 2D, 3C, 4A
C) 1A, 2D, 3C, 4B
D) 1A, 2B, 3C, 4D
Solve the problem.
77) Suppose that for an electrical appliance, voltage is given by V = 189 sin 120 t and amperage by
I = 1.07 sin 120 t, where t is time in seconds. Use identities to write the wattage W = VI in the form
W = a cos t + c.
A) W = - 101.115 cos 240 t + 101.115
B) W = 101.115 cos 240 t - 101.115
C) W = - 101.115 cos 120 t + 101.115
D) W = - 202.23 cos 240 t+ 202.23
Convert the radian measure to degrees. Round to the nearest hundredth if necessary.
78) 13
A) 1170°
B) 2340°
C) 2520°
23
D) 4680°
77)
78)
The function graphed is of the form y = a tan bx or y = a cot bx, where b > 0. Determine the equation of the graph.
79)
79)
A) y = cot 2x
B) y = 2 cot x
C) y = tan 2x
D) y = 2 tan x
Use a graphing calculator to make a conjecture as to whether each equation is an identity.
80) cos(x + y) = cos x + cos y
A) Identity
B) Not an Identity
24
80)
Answer Key
Testname: PRACATICETEST2
1)
2)
3)
4)
5)
6)
7)
8)
9)
10)
11)
12)
13)
14)
15)
16)
17)
18)
19)
20)
21)
22)
23)
24)
25)
26)
27)
28)
29)
30)
31)
32)
33)
34)
35)
36)
37)
38)
39)
40)
41)
42)
43)
44)
45)
46)
47)
48)
49)
50)
D
D
B
C
C
B
B
B
B
D
B
C
B
A
D
C
B
A
A
B
A
D
C
A
C
D
A
D
A
D
B
D
C
D
C
B
D
C
C
A
C
C
D
D
B
B
A
B
C
B
25
Answer Key
Testname: PRACATICETEST2
51)
52)
53)
54)
55)
56)
57)
58)
59)
60)
61)
62)
63)
64)
65)
66)
67)
68)
69)
70)
71)
72)
73)
74)
75)
76)
77)
78)
79)
80)
B
B
C
B
D
C
B
D
B
C
D
C
D
D
D
D
A
A
C
B
A
D
B
A
B
B
A
B
A
B
26