Honors PreCalculus Final Exam Review
Mr. Serianni
Name___________________________________
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Convert the angle to decimal degrees and round to the nearest hundredth of a degree.
1) 295°6'24''
A) 295.11°
B) 295.07°
C) 295.12°
D) 295.17°
Convert the angle to degrees, minutes, and seconds.
2) 193.32°
A) 193°19'12''
B) 193°20'12''
D) 193°17'32''
C) 193°19'32''
Convert the degree measurement to radians. Express answer as multiple of π.
3) 480°
7π
7
8π
A)
B) π
C)
2
3
3
D)
9π
4
Convert the radian measure to degrees. (Round to the nearest hundredth when necessary)
43
4)
π
18
A) 860π°
B) 430°
C) 8°
D) 215°
Solve the problem.
5) For what numbers θ is f(θ) = cot θ defined?
A) all real numbers, except odd multiples of π (180°)
B) all real numbers, except integral multiples of π (180°)
π
C) all real numbers, except odd multiples of (90°)
2
D) all real numbers
6) What is the range of the tangent function?
A) all real numbers greater than or equal to 1 or less than or equal to -1
π
B) all real numbers, except odd multiples of (90)°
2
C) all real numbers
D) all real numbers from -1 to 1, inclusive
Use the fact that the trigonometric functions are periodic to find the exact value of the expression.
13π
7) tan
4
A) -1
B)
3
C)
3
3
D) 1
C)
3
3
D)
8) tan 390°
A)
3
B) -
3
1
3
2
Solve the problem.
9) If sin θ = -0.3, find the value of sin θ + sin (θ + 2π) + sin (θ + 4π).
A) -0.3
B) -0.9
C) -0.9 + 6π
D) 1.1
Name the quadrant in which the angle θ lies.
10) csc θ > 0 and sec θ > 0
A) Quadrant III
B) Quadrant IV
D) Quadrant I
C) Quadrant II
Use the given values of the sine and cosine to find the function value.
7
3
11) sin θ = , cos θ = . Find cot θ.
4
4
A)
7
3
B)
-4 7
7
C)
4
3
D)
-3 7
7
Find the exact value of the expression.
sin 55°
12) tan 55° cos 55°
A) 0
B) 1
C) Undefined
D) 55
Find the exact value of the requested trigonometric function of θ.
7
13) csc θ = - and θ in quadrant III
4
Find cot θ.
4 33
A) 33
14) sec θ =
B) -
7 33
33
B) -
17
8
C)
33
4
D) -
33
7
C) -
9
8
D) -
17
9
9
and θ in quadrant IV
8
Find tan θ.
A) -
17
Give the amplitude or period as requested.
15) Amplitude of y = -3 sin 5x
3
A)
B) 3
5
C)
π
3
D)
π
5
Determine the amplitude and period of the function without graphing.
3
2
16) y = - sin ( x)
4
5
A) amplitude =
3
; period = 5π
4
B) amplitude =
C) amplitude =
3
4π
; period =
4
5
D) amplitude = -
2
4
; period = 5
3
3
; period = 5π
4
Find an equation for the graph.
17)
y
5
4
3
2
1
-2π
-π
-1
π
2π
x
-2
-3
-4
-5
A) y = 2 cos (3x)
B) y = 3 cos
1
x
2
C) y = 3 cos (2x)
D) y = 2 cos
1
x
3
Write the equation of a sine function with the given characteristics.
5
18) Amplitude = ; Period = 5π
3
A) y =
5
sin (10πx)
3
B) y =
5
1
sin (
x)
3
10π
C) y =
5
2π
sin (
x)
3
5
D) y =
5
2
sin ( x)
3
5
Find the phase shift of the function.
π
19) y = -2 sin (4x - )
2
A) 2π units up
C) π/8 units to the right
B) π/2 units to the left
D) 4π units down
20) y = 3 sin (2πx - 5)
5
A) units to the left
2
B) 5 units to the left
C) 5 units to the right
D)
5
units to the right
2π
Solve the problem.
21) The temperature T of a patient during a 5-day illness is given in the following table.
Day, x
0
1
2
3
4
5
Temperature, T 102.5 104.8 105.2 102.1 99.5 98.9
Fit a sine function to the data in the table. From the graph of the sinusoidal function of best fit, estimate the
highest temperature reached during the 5 -day illness. Round answer to 1 decimal place.
A) 105.4°
B) 105.2°
C) 105.3°
D) 105.5°
Graph the function.
3
1
22) y = sec ( x)
4
y
3
-2π
2π
4π
6π
8π x
-3
A)
B)
y
y
3
3
-2π
2π
4π
6π
8π x
-2π
-3
2π
4π
6π
8π x
2π
4π
6π
8π x
-3
C)
D)
y
y
3
-2π
3
2π
4π
6π
8π x
-2π
-3
-3
4
23) y = -3tan (x +
π
)
4
y
4
2
-π
2
-π
π
2
3π
2
π
2π
5π
2
3π
x
-2
-4
A)
B)
y
-π
y
4
4
2
2
-π
2
π
2
π
3π
2
2π
5π
2
3π
x
-π
-π
2
-2
-2
-4
-4
C)
π
2
π
3π
2
2π
5π
2
3π
π
2
π
3π
2
2π
5π
2
3π
D)
7π
3
x
D)
y
-π
y
4
4
2
2
-π
2
π
2
π
3π
2
2π
5π
2
3π
x
-π
-π
2
-2
-2
-4
-4
Find the value of the expression.
24) sin-1 -0.5
A) -
π
6
B)
π
3
C)
5
π
6
x
Find the exact value of each expression.
3
25) cos-1 ()
2
A)
5π
6
5π
6
B) -
C) -
π
6
D)
π
6
Complete the identity.
26) sec4 x - 2 sec2x tan2 x + tan4x = ?
A) 1
27) sin2 θ + tan2θ + cos2 θ = ?
A) cos3 θ
B) 2
C) sec2x + tan2 x
D) sec2 x (1 + tan2 x)
B) sec2 θ
C) tan2 θ
D) sin θ
C) -2 cot2 θ
D) 0
Simplify the expression as far as possible.
28) (1 + cot θ)(1 - cot θ) - csc2 θ
B) 2 cot2 θ
A) 2
Use trigonometric identities to find the exact value.
5π
π
5π
π
29) cos
cos
sin
+ sin
12
4
12
4
A) 1
B)
1
4
3
2
C)
D)
1
2
Find the exact value of the expression.
1 - tan 80° tan 70°
30)
tan 80° + tan 70°
A) -
3
3
B)
3
3
C) -
3
D)
3
Find the exact value by using a sum or difference identity.
π
31) sin
12
A) -
2( 3 - 1)
4
B)
2( 3 - 1)
C)
2( 3 - 1)
4
D) -
2( 3 - 1)
Find the exact value of the expression under the given conditions.
2 6
32) sin θ =
, tan θ < 0
5
Find sin (2θ).
23
A)
25
B)
-4 6
25
C)
6
4 6
25
D) -
23
25
33) sin θ =
20
π
, 0<θ<
29
2
Find cos (2θ).
41
A) 841
B)
840
841
C)
41
841
D)
42
841
Using the information given, find the exact value of the trigonometric function.
34) Find sin 2θ, given that tan θ = -2, sin θ < 0.
A)
4
5
B) -
4
5
Solve the equation for solutions in the interval 0 ≤ θ < 2π.
35) 5 csc θ - 2 = 3
3π
A) 2π
B)
2
C)
4
5
D) -
C)
π
2
D) π
4
5
Solve the equation. Give a general formula for all the solutions.
θ 2 3
36) csc =
3
3
A) θ =
π
+ 6kπ
2
B) θ =
π
+ 2kπ
9
C) θ =
π
+ 2kπ
18
D) θ = π + 6kπ
π 2π
,
3 3
C) x =
π 5π
,
6 6
D) x =
B) x =
π 9π
,
8 8
Solve the equation for the interval [0, 2π).
37) 2 sin 2x = sin x
A) x = 0, π,
π 5π
,
6 6
B) x =
Solve the equation for solutions in the interval 0 ≤ θ < 2π.
38) sin2 2x = 1
A) No solution
C) x =
π 3π 5π 7π
,
,
,
4 4
4 4
D) x = 0,
Solve the equation on the interval 0 ≤ θ < 2π.
39) sin θ = - 2 - cos θ
3π
π
A)
B)
2
2
C)
Find the missing parts of the triangle.
40) α = 44°30'
β = 24°10'
a = 16.30
A) γ = 112°20', b = 9.52, c = 21.66
C) γ = 112°20', b = 21.66, c = 9.52
π
4
2π
4π
, π,
3
3
D)
5π
4
B) γ = 111°20', b = 21.66, c = 9.52
D) γ = 111°20', b = 9.52, c = 21.66
7
π 3π π 2π
,
, ,
2 2 3 3
41) β = 11.6°
b = 5.69
a = 9.43
A) α = 19.47°, γ = 148.93°, c = 14.6;
α' = 160.53°, γ' = 7.87°, c' = 3.87
C) α = 160.53°, γ = 7.87°, c = 3.87
B) No solution
D) α = 19.47°, γ = 148.93°, c = 14.6
Solve the problem.
42) An airplane is sighted at the same time by two ground observers who are 2 miles apart and in line with the
airplane. They report the angles of elevation as 10° and 22°. How high is the airplane?
A) 0.63 miles
B) 0.35 miles
C) 0.75 miles
D) 1.35 miles
43) Given a triangle with b = 6, c = 7, and α = 146°, what is the length of a? Round the answer to two decimal places.
A) a = 3.92
B) a = 10.95
C) a = 4.65
D) a = 12.44
44) Solve the triangle given that a = 19, b = 16, c = 11.
A) α = 57.27°, β = 87.39°, γ = 35.33°
C) α = 87.39°, β = 35.33°, γ = 57.27°
B) α = 87.39°, β = 57.27°, γ = 35.33°
D) α = 35.33°, β = 57.27°, γ = 87.39°
45) 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 46 feet from point A and 65 feet from point B. The angle ACB is 53°. How far apart are points A and
B?
A) 90.2 feet
B) 67.4 feet
C) 52.4 feet
D) 99.7 feet
Find the area of the triangle with the given parts.
46) α = 34.3°
b = 13.0 in.
c = 4.0 in.
A) 13 in.2
B) 15 in.2
C) 23 in.2
D) 21 in.2
C) 57 cm2
D) 66 cm2
47) a = 11.3 cm
b = 10.3 cm
c = 16.8 cm
A) 60 cm2
B) 63 cm2
Find the rectangular coordinates for the point.
48) (9, 120°)
9 9 3
9 -9 3
A) - ,
B) ,
2 2
2
2
49) 5, -
C) -
9 -9 3
,
2
2
D)
9 9 3
,
2 2
C) -
5 5 3
,
2
2
D)
5 3 5
,
2 2
4π
3
A) -
5 3
5
,2
2
B)
5
5 3
,2
2
Find the polar coordinates for the point.
50) (-1, 2.6)
A) (2.79, 111°)
B) (-2.79, 21.04°)
C) (2.79, -21.04°)
8
D) (2.79, 21.04°)
Perform the indicated operation.
51) u = -10i - 4j, v = -3i + 8j; Find u + v.
A) -14i + 4j
B) -7i - 14j
C) 7i + 4j
D) -13i + 4j
Find the magnitude v of the vector.
52) v = 6i + 8j
A) 14
C) 10
D)
B) 100
Find the unit vector having the same direction as v.
53) v = -5i - 12j
13
13
5
12
A) u = ij
B) u = ij
5
12
13
13
C) u =
12
5
i+
j
13
13
10
D) u = -65i - 156j
Write the vector in the form ai + bj given its magnitude and the angle it makes with the positive x-axis.
54) v = 7, α = 270°
A) v = -7j
B) v = -7i
C) v = -7i - 7j
D) v = 7(
2
2
)
2
2
For the given vectors u and v, find their dot product u · v. Round to two decimal places, if necessary.
55) u = 3.45i + 6.72j, v = - 2.15i +5.60j.
A) 45.05
B) 13.62
C) 12.39
D) 30.21
Find the angle between the two vectors.
56) v = -5i + 7j, w = -6i - 4j.
A) 90°
B) 88.15°
C) 20.7°
Determine whether the vectors are parallel, orthogonal, or neither.
57) v = 2i - j, w = 4i - 2j
A) parallel
B) neither
58) v = i + 3j, w = i - 2j
A) orthogonal
D) 110.78°
C) orthogonal
B) parallel
C) neither
Decompose v into two vectors v1 and v2 , where v1 is parallel to w and v2 is orthogonal to w.
59) v = -3i + 5j, w = 2i + j
1
14
18
A) v1 = - (2i + j), v2 = i+
j
5
5
5
B) v1 = -
1
7
36
(2i + j), v2 = i +
j
5
5
5
1
13
26
(2i + j), v2 = i+
j
5
5
5
D) v1 = -
1
5
21
(2i + j), v2 = - i +
j
4
2
4
C) v1 = -
Solve the problem.
60) Find the work done by a force of 8 pounds acting in the direction of 44° to the horizontal in moving an object 7
feet from (0, 0) to (7, 0). Round answer to the nearest tenth of a foot-pound.
A) 40.3 foot-pounds
B) 80.6 foot-pounds
C) 38.9 foot-pounds
D) 43.5 foot-pounds
9
Answer Key
Testname: PC_SEM2REVIEW
1) A
2) A
3) C
4) B
5) B
6) C
7) D
8) C
9) B
10) D
11) D
12) A
13) C
14) B
15) B
16) A
17) D
18) D
19) C
20) D
21) A
22) C
23) A
24) A
25) A
26) A
27) B
28) C
29) C
30) C
31) C
32) B
33) C
34) D
35) C
36) D
37) A
38) C
39) D
40) D
41) A
42) A
43) D
44) B
45) C
46) B
47) C
48) A
49) C
50) A
10
Answer Key
Testname: PC_SEM2REVIEW
51) D
52) C
53) B
54) A
55) D
56) B
57) A
58) C
59) C
60) A
11