Download Final Exam Review Math1316 Provide an appropriate response. 1

Final Exam Review Math1316
Provide an appropriate response.
1) Find the complement of an angle whose measure is 38°17′53′′ .
Convert the angle to decimal degrees and round to the nearest hundredth of a degree.
2) 148°59′42′′
Convert the angle to degrees, minutes, and seconds.
3) 13.59°
Find the angle of least positive measure coterminal with the given angle.
4) 867°
Solve the problem.
5) A wheel is rotating 720 times per minute. Through how many degrees does a point on the edge of the wheel
1
move in seconds?
3
Suppose that θ is in standard position and the given point is on the terminal side of θ. Give the exact value of the
indicated trig function for θ.
6) (-10, 24); Find sin θ.
Identify the quadrant for the angle θ satisfying the following conditions.
7) sin θ > 0 and cos θ < 0
Evaluate the function requested. Write your answer as a fraction in lowest terms.
8)
41
9
40
Find cos B.
1
Solve the problem.
9) Find the exact value of x in the figure.
24
Write the function in terms of its cofunction. Assume that any angle in which an unknown appears is an acute angle.
10) cos 59°
Solve the problem for the given information.
11) Find the equation of a line passing through the origin and making a 45° angle with the positive x-axis.
Find the reference angle for the given angle.
12) 236.7°
Evaluate.
13) 2 tan2 240° + 4 sin2 330° - cos2 0°
Find the sign of the following.
θ
14) tan , given that θ is in the interval (180°, 270°).
2
Find all values of θ, if θ is in the interval [0, 360°) and has the given function value.
3
15) sin θ =
2
Solve the problem.
16) A 34-foot ladder is leaning against the side of a building. If the ladder makes an angle of 23° 54′ with the side of
the building, how far up from the ground does the ladder make contact with the building? Round your answer
to the hundredths place when necessary.
2
Solve the right triangle. If two sides are given, give angles in degrees and minutes.
17)
a = 18.9 cm, b = 20.2 cm
Round the missing side length to one decimal place.
Convert the degree measure to radians. Leave answer as a multiple of π.
18) 510°
Convert the radian measure to degrees. Round to the nearest hundredth if necessary.
2π
19)
9
Solve the problem.
20) Through how many radians will the hour hand on a clock rotate in 24 hours?
Find the length of an arc intercepted by a central angle θ in a circle of radius r. Round your answer to 1 decimal place.
21) r = 87.74 in.; θ = 35°
Assume that the cities lie on the same north-south line and that the radius of the earth is 6400 km.
22) Find the distance between City A, 18° N and City B, 34° N. (Round to the nearest kilometer.)
Solve the problem.
23) A bicycle with a 24-inch wheel (diameter) travels a distance of 500 feet. How many revolutions does the wheel
make (to the nearest revolution)?
24) A pendulum swinging through a central angle of 74° completes an arc of length 15.4 cm. What is the length of
the pendulum? Round to the nearest hundredth.
25) Find the measure (in radians) of a central angle of a sector of area 56 square inches in a circle of radius 7 inches.
Round to the nearest hundredth.
Find the exact values of s in the given interval that satisfy the given condition.
1
26) [0, 2π); tan2 s =
3
Solve the problem.
27) The angle of elevation θ of the sun in the sky at any latitude L is calculated with the formula
sin θ = cos D cos L cos ω + sin D sin L. ω is the number of radians that the Earth has rotated through since noon
when ω = 0. D is the declination of the sun. Westville has latitude L = 37.4°. Find the angle of elevation θ in
degrees of the sun at 3 PM on February 29, 2012, where at that time D ≈ -0.1425 and ω ≈ 0.7854.
3
28) Find ω for the minute hand of a clock.
Match the function with its graph.
2) y = 3 cos x
29) 1) y = sin 3x
3) y = 3 sin x
4) y = cos 3x
A)
B)
3
-2
y
3
2
2
1
1
-

2
x
-2
-
-1
-1
-2
-2
-3
-3
C)
y

2

2
x
D)
y
-2
-
y
3
3
2
2
1
1

-1
2
x
-2
-
-1
-2
-2
-3
-3
Graph the function.
1
30) y = sin x
2
y
2
1
-2
2
x
-1
-2
4
x
Give the amplitude or period as requested.
1
31) Period of y = -3 cos x
4
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.
32)
6
y
5
4
3
2
1
-1

2

3
2
2
x
-2
-3
-4
-5
Solve the problem.
33) The voltage E in an electrical circuit is given by E = 4.9 cos 30πt, where t is time measured in seconds. Find the
amplitude.
Find the specified quantity.
34) Find the period of y = 3 cos
1
π
x+
.
4
3
Find the phase shift of the function.
π
35) y = -2 sin 4x 2
5
Match the function with its graph.
36) 1) y = tan x
2) y = cot x
3) y = -tan x
4) y = -cot x
A)
B)
3
-2
y
3
2
2
1
1
-

2
x
-2
-
-1
-1
-2
-2
-3
-3
C)
y

2

2
x
D)
3
-2
y
3
2
2
1
1
-

2
x
-2
-
-1
-1
-2
-2
-3
-3
6
y
x
37) 1) y = sec x
3) y = -sec x
2) y = csc x
4) y = -csc x
A)
B)
3
-2
y
3
2
2
1
1
-

2
x
-2
-
-1
-1
-2
-2
-3
-3
C)
y

2

2
x
D)
3
-2
y
3
2
2
1
1
-

2
x
-2
-
-1
-1
-2
-2
-3
-3
y
x
Solve the problem.
38) A rotating beacon is located 15 ft from a wall. The distance from the beacon to the point on the wall where the
beacon is aimed is given by
a = 15 sec 2πt ,
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.
Use the fundamental identities to simplify the expression.
39) cos θ - cos θ sin2 θ
Use Identities to find the exact value.
7π
5π
7π
5π
40) cos
cos
sin
+ sin
12
12
12
12
Use a sum or difference identity to find the exact value.
41) tan 75°
42) sin 105° cos 45° - cos 105° sin 45°
7
43)
tan 25° + tan 5°
1 - tan 25° tan 5°
Find the exact value of the expression using the provided information.
1
1
44) Find sin(s - t) given that cos s = , with s in quadrant I, and sin t = - , with t in quadrant IV.
3
2
Using a sum or difference identity, write the following as an expression involving functions of x.
π
45) sin
-x
4
Use an identity to write the expression as a single trigonometric function or as a single number.
46) 2 cos2 75° - 1
47) sin 8x cos 8x
Use identities to find the indicated value for each angle measure.
3
48) cos θ = , sin θ < 0
Find sin(2θ).
5
Find the exact value by using a half-angle identity.
49) cos 22.5°
Determine all solutions of the equation in radians.
x
1
π
50) Find cos , given that cos x = and x terminates in 0 < x < .
2
4
2
Use an identity to write the expression as a single trigonometric function or as a single number.
1 + cos 84°
51)
2
Give the exact value of the expression.
1
52) cos arcsin
4
Write the following as an algebraic expression in u, u > 0.
53) cos(arctan u)
8
Solve the problem.
54) A painting 1 meter high and 3 meters from the floor will cut off an angle θ to an observer, where
x
θ = tan-1
, assuming that the observer is x feet from the wall where the painting is displayed and that
2
x + 1.8
the eyes of the observer are 1.8 meters above the ground (see the figure). Find the value of θ for x = 3. Round to
the nearest tenth of a degree.
1.2
1.8
Solve the equation for exact solutions over the interval [0, 2π).
55) sin2 x + sin x = 0
Solve the equation in the interval [0°, 360°). Give solutions to the nearest tenth, if necessary.
56) 2 cos2θ + 7 sin θ = 5
Solve the equation (x in radians and θ in degrees) for all exact solutions where appropriate. Round approximate answers
in radians to four decimal places and approximate answers in degrees to the nearest tenth.
57) 4 sin2 x - 1 = 0
58) 3 cos2 θ + 2 cos θ = 1
Solve the equation for solutions in the interval [0, 2π).
59) sin 2x + sin x = 0
Solve the equation for solutions in the interval [0°, 360°). Round to the nearest degree.
θ
60) cot = 1
3
Solve the equation for x, where x is restricted to the given interval.
π
61) y = 8 cos 3x, for x in 0,
3
Solve the equation for exact solutions.
62) 4 cos-1 x = π
9
Perform the transformation.
63) Write cot x in terms of sin x.
Write the expression in terms of sine and cosine, and simplify so that no quotients appear in the final expression.
64) tan x(cot x - cos x)
Use the fundamental identities to simplify the expression.
65) cot θ sec θ sin θ
Verify that each equation is an identity.
cos β
66) sec β + tan β =
1 - sin β
Use identities to write each expression as a function of θ.
67) cos (θ - π)
Use a calculator to give the real number value. Round the answer to 7 decimal places.
68) y = arcsec (2.8842912)
Solve the equation for solutions in the interval [0, 2π).
69) 2 cos 2x = 1
Solve the triangle for a Round to the nearest tenth when necessary
70)
26 m
Find the area of triangle ABC with the given parts. Round to the nearest tenth when necessary.
71) A = 39°50'
b = 16.9 m
c = 5.9 m
Solve the problem.
72) An airplane is sighted at the same time by two ground observers who are 2 miles apart and both directly west of
the airplane. They report the angles of elevation as 15° and 20°. How high is the airplane? Round to the nearest
hundredth of a mile.
Find the missing parts of the triangle.
73) A = 70°
a = 33 yd
b = 67 yd
If necessary, round angles to the nearest degree and side lengths to the nearest yard.
10
Find the indicated angle or side. Give an exact answer.
74) Find the exact length of side a.
5 3
Find the smallest angle of the triangle. Round to the nearest tenth
75) a = 7.1 in.
b = 13.9 in.
c = 15.3 in.
Find the area of triangle ABC with the given parts. Round to the nearest tenth when necessary.
76) a = 16.5 cm
b = 17.0 cm
c = 15.3 cm
Solve the problem.
77) Two airplanes leave an airport at the same time, one going northwest (bearing 135°) at 424 mph and the other
going east at 326 mph. How far apart are the planes after 3 hours (to the nearest mile)?
Draw a sketch to represent the vector. Refer to the vectors pictured here.
78) b - c
Use the figure to find the specified vector.
79) Find -a.
11
Find the component form of the indicated vector.
80) Let u = -1, -6 , v = -3, -3 . Find -4u + 5v.
Find the magnitude and direction angle (to the nearest tenth) for each vector. Give the measure of the direction angle as
an angle in [0,360°].
81) -12, 5
Vector v has the given magnitude and direction. Find the magnitude of the indicated component of v rounded to the
nearest tenth when necessary.
82) α = 26°, ∣v∣ = 337; Find the vertical component of v.
Write the vector in the form <a, b>. If necessary, round values to the nearest hundredth.
83)
Two forces act at a point in the plane. The angle between the two forces is given. Find the magnitude of the resultant
force.
84) forces of 40.5 and 23.6 lb, forming an angle of 39.4°
(round to the nearest pound)
Write the vector in the form ai + bj.
85) 2, 4
Find the dot product for the pair of vectors.
86) -5, 10 , 12, 10
Determine whether the pair of vectors is orthogonal.
87) 7, 3 , -2 3, 14
Write the complex number in rectangular form.
88) 5(cos 3° + i sin 3°)
Write the complex number in trigonometric form r(cos θ + i sin θ), with θ in the interval [0°, 360°).
89) 5 3 + 5i
12
Find the product. Write the product in rectangular form, using exact values.
90) [7(cos 60° + i sin 60°)] [2(cos 90° + i sin 90°)]
Find the quotient and write in rectangular form. First convert the numerator and denominator to trigonometric form.
15cis 337°
91)
3cis 67°
Find the given power. Write answer in rectangular form.
92) 4 cis 15° 4
Find all cube roots of the complex number. Leave answers in trigonometric form.
93) 64(cos 294° + i sin 294°)
Determine the number of triangles ABC possible with the given parts.
94) a = 35, b = 53, A = 20°
Plot the point.
95) -4,
5π
4
5
-5
5
-5
Determine two pairs of polar coordinates for the point with 0° ≤ θ < 360°.
96) (5, -5)
For the given rectangular equation, give its equivalent polar equation.
97) 2x + 3y = 6
Find an equivalent equation in rectangular coordinates.
5
98) r =
1 + cos θ
99) r sin θ = 10
100) r = 10 sin θ
13
The graph of a polar equation is given. Select the polar equation for the graph.
101)
5
4
3
2
1
-5 -4 -3 -2 -1
-1
1
2
3
4
5
r
-2
-3
-4
-5
A) r = 3 sin(3θ)
B) r = 3 + cos(3θ)
C) r = 3
14
D) r = 3 cos(3θ)
Answer Key
Testname: 1316 FINAL EXAM REVIEW
1)
2)
3)
4)
5)
51°42′7′′
149.00°
13°35′24′′
147°
1440°
12
6)
13
7) Quadrant II
40
8) cos B =
41
9) 12 3
10) sin 31°
11) y = x
12) 56.7°
13) 6
14) negative
15) 60° and 120°
16) 31.08 ft
17) A = 43°6'; B = 46°54'; c = 27.7 cm
17π
18)
6
19) 40°
20) 4π
21) 53.6 in.
22) 1787 km
23) 80 revolutions
24) 11.92 cm
25) 2.29 radians
π 5π 7π 11π
26) ,
,
,
6 6
6
6
27) 28°
π
28)
radians per min
30
29) 1B, 2D, 3C, 4A
30)
y
2
1
-2
2
x
-1
-2
15
Answer Key
Testname: 1316 FINAL EXAM REVIEW
31) 8π
1
x
3
32) y = -4 sin
33) 4.9
34) 8π
π
35)
units to the right
8
36) 1C, 2A, 3B, 4D
37) 1C, 2A, 3B, 4D
38) 21.91 ft
39) cos3 θ
3
2
40)
3+2
3
42)
2
41)
3
3
43)
44)
2 6+1
6
2
2
cos x sin x
2
2
45)
3
2
46) 47)
1
sin 16x
2
24
25
48) 49)
50)
1
2
2+
2
10
4
51) cos 42°
15
52)
4
53)
u2 + 1
u2 + 1
54) 15.5°
55) 0, π,
3π
2
56) {90°, 48.6°, 131.4°}
π
5π
57)
+ nπ,
+ nπ
6
6
58) {70.5° + 360°n, 180° + 360°n, 289.5° + 360°n}
16
Answer Key
Testname: 1316 FINAL EXAM REVIEW
59) 0,
2π
4π
, π,
3
3
60) {135°}
1
y
61) x = arccos
3
8
2
2
62)
63)
±
1 - sin2 x
sin x
64) 1 - sin x
65) 1
66) sec β + tan β =
1
sin β 1 + sin β 1 + sin β 1 - sin β
1 - sin2 β
cos2 β
cos β
+
=
=
∙
=
=
=
cos β cos β
cos β
cos β
1 - sin β cos β(1 - sin β) cos β(1 - sin β)
1 - sin β
67) -cos θ
68) 1.2167397
π 9π 7π 15π
69)
,
,
,
8 8 8
8
70) a = 11.7 m
71) 31.9 m2
72) 2.03 mi
73) no such triangle
74) 7
75) A = 27.6°,
76) 114 cm2
77) 2082 mi
78)
79) -3, -7
80) -11, 9
81) 13; 157.4°
82) 147.7
83) 4.7, 1.71
84) 61 lb
85) 2i + 4j
86) 40
87) Yes
88) 5 + 0.3i
89) 10(cos 30° + i sin 30°)
90) -7 3 + 7 i
91) -5i
92) 128 + 128 i 3
93) 4 cis 98°, 4 cis 218°, 4 cis 338°
94) 2
17
Answer Key
Testname: 1316 FINAL EXAM REVIEW
95)
5
-5
5
-5
96) (5 2, 315°), (-5 2, 135°)
6
97) r =
2 cos θ + 3 sin θ
98) y2 = 25 - 10x
99) y = 10
100) x2 + y2 = 10y
101) D
18