Pre-Calc Chapter 4 Sample Test

Pre-Calc Chapter 4 Sample Test
1. Determine the quadrant in which the angle lies. (The angle measure is given in radians.)
π
8
A) I B) II C) III D) IV E) The angle lies on a coordinate axis.
2. Sketch the angle in standard position 11π/12
A)
B)
C)
D)
E) none of these
3. Determine two coterminal angles (one positive and one negative) for the given angle. Give your answer
in radians: π/4
A) 9π/4, -7π/4
4. Find (if possible) the complement and supplement of the given angle: 1.2
A)
C)
B)
D)
E)
5. Convert the angle measure to decimal degree form: –115°26 '
A) –114.567° B) –115.026° C) –115.433° D) –2.008° E) –6590.504°
6. Convert the angle measure to D°M ′S ′′ form: 130.6225°
A) 130° 37' B) 130° 37' 21" C) 130° 6225' D) 130° 21' 37" E) 130° 21'
7. Find the angle in radians.
A)
B)
C)
D)
E)
8. Find the radian measure of the central angle of a circle of radius r that intercepts an arc of length s.
radius: r = 9 centimeters arc length: s = 43 centimeters
9
9π
43π
43
43
A)
C)
D)
B)
E)
43
43
9
18
9
9. Find the length of the arc on a circle of radius r intercepted by a central angle θ .
7π
radius: r = 11 meters central arc: θ =
4
7π
77π
847π
77π
77
A)
meters B)
meters C)
meters D)
meters E)
meters
4
8
4
4
4
10. Find the area of the sector of the circle with radius r and central angle θ .
2π
radius: r = 5 miles central arc: θ =
3
2π
10π
A)
D)
square miles
square miles
3
3
25π
10
E)
square miles
square miles
B)
3
3
50π
C)
square miles
3
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11. Determine the exact value of tan θ .
θ
⎛ 24 7 ⎞
⎜ ,− ⎟
⎝ 25 25 ⎠
A) –
7
24
B)
7
24
C) –
24
7
D)
24
7
E) –
25
7
12. Find the point (x, y) on the unit circle that corresponds to the real number t:
A)
B)
C)
D)
13. Evaluate the trigonometric function using its period as an aid.
⎛ 5π ⎞
cos ⎜ –
⎟
⎝ 3 ⎠
1
1
3
3
2 3
B) –
C)
D) −
A)
E)
2
2
2
2
3
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E)
14. Use the figure and a straightedge to approximate the value of cos 3.25
A) 1.00
B) –0.11
C) –0.99
D) 0.11
E) –1.01
15. Find the exact value of the given trigonometric function of the angle θ shown in the figure. (Use the
Pythagorean Theorem to find the third side of the triangle.) Find: csc θ
c
A) 4 2
B)
a
2
C)
2
2
D) 1 E)
2
4
θ
a
a=4
4
, find csc θ .
7
[Hint: Sketch a right triangle corresponding to the trigonometric function of the acute angle θ , then use
the Pythagorean Theorem to determine the third side.]
33
7
4
7
D) 28 33 E)
A)
B)
C)
7
4
33
33
16. Given that cos θ =
17. Use the given function values and the trigonometric identities (including the cofunction identities), to
find the indicated trigonometric function.
34
3 34
csc θ =
, cos θ =
; find sin ( 90° − θ )
5
34
5
3
5 34
3 34
3 34
A)
B)
C)
D)
E)
3
5
34
68
34
18. Use a calculator to evaluate the function. Round your answers to four decimal places. (Be sure the
calculator is in the correct angle mode.)
sec 47.7°
A) –1.1925 B) 0.6730 C) 1.4859 D) 0.9844 E) 1.0990
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19. Solve for r.
A) r =
21 3
2
D) r =
3
42
r
45°
21
B) r =
21 2
2
C) r =
21
3
E) r = 21 2
20. The point(3,4)is on the terminal side of an angle in standard position. Determine the exact value of
tan θ .
A)
B)
C)
D)
E)
fracnum(q, p)
fracden(q, p)
fracnum(q, cafac)
fracsign(q, cafac)
leadcoeff(fracden(q, cafac)) carad
fracden(q, p)
fracsign(q, p)
fracnum(q, p)
fracnum(q, p+ q)
fracsign(q, p+ q)
fracden(q, p+ q)
fracnum(q, dafac)
fracsign(q, dafac)
leadcoeff(fracden(q, dafac)) darad
4/3
21. State the quadrant in which θ lies: cot(θ ) < 0 and sec(θ ) < 0
A) Quadrant IV
D) Quadrant I
B) Quadrant III
E) Quadrant II or Quadrant IV
C) Quadrant II
22. Use the function value and constraint below to evaluate the given trigonometric function.
Function Value
Constraint
Evaluate:
sec θ = –4
tan θ < 0
cot θ
1
1
A) − 15 B) 15 C) −
D) −
E) undefined
4
15
23. Find the reference angle θ ′ for the given angle θ : θ = –306°
B) 36°
A) 144°
C) 64°
D) 54°
E) 44°
24. Find the indicated trigonometric value in the specified quadrant.
Function
13
III
9
2 22
13
B)
C)
13
2 22
csc θ = −
A)
9
2 22
Quadrant Trigonometric Value
tan θ
D)
9
13
E) undefined
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⎛ x π⎞
25. Determine the period and amplitude of y = 4 cos ⎜ + ⎟ .
⎝ 11 8 ⎠
2π
π
; amplitude: 4
D) period: ; amplitude: 4
A) period:
11
11
2π
B) period: 22π ; amplitude: 4
E) period: –
; amplitude: –4
11
C) period: 11π ; amplitude: 8
26. Describe the relationship between f ( x) = cos( x) and g ( x) = cos 3 x – 5 . Consider amplitude, period, and
shifts.
A) The period of g(x) is three times shorter than the period of f(x).
Graph of g(x) is shifted downward 5 unit(s) relative to the graph of f(x).
B) The amplitude of g(x) is three times the amplitude of f(x).
Graph of g(x) is shifted downward 5 unit(s) relative to the graph of f(x).
C) The period of g(x) is three times the period of f(x).
Graph of g(x) is shifted upward 5 unit(s) relative to the graph of f(x).
D) The amplitude of g(x) is three times the amplitude of f(x).
Graph of g(x) is shifted upward 5 unit(s) relative to the graph of f(x).
E) The period of g(x) is five times the period of f(x).
Graph of g(x) is shifted downward 3 unit(s) relative to the graph of f(x).
27. Find a and d for the function f ( x ) = a sin x + d such that the graph of f ( x) matches the graph below.
A) a = 2; d = -1
B) a = 4; d = 1
C) a = -2; d = 1
D) a = 2; d = 2
E) a = 4; d = -3
28. Use a calculator to evaluate the trigonometric function. Round your answer to four decimal places. (Be
sure the calculator is set in the correct angle mode.) cot
π
3
A) 0.0183 B) 1.7321 C) 0.9998 D) 0.5774 E) 0.8660
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29. Given the graph of f(x) below, sketch the graph of:
A)
D)
B)
.
C)
E)
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30. Sketch the graph of the function below, being sure to include at least two full periods:
A)
D)
B)
C)
E)
Page 8
31. Use a calculator to evaluate the function. Round your answers to four decimal places: csc8°28'
A) 0.1472 B) 1.2223 C) 6.7919 D) 1.011 E) 1.0381
32. Find a and d for the function f ( x) = a sin x + d such that the graph of f ( x) matches the graph below.
A)
B)
C)
D)
E)
33. Use the properties of inverse trigonometric functions to evaluate sin ⎡arcsin ( –0.46 ) ⎤ .
⎣
⎦
A) –0.9 B) –0.42 C) –0.76 D) –0.23 E) –0.46
34.
x⎞
⎛
Write an algebraic expression that is equivalent to sin ⎜ arctan ⎟ .
9⎠
⎝
9
A)
x 2 + 81
B)
9
x
C)
x 2 + 81
x
D)
x 2 + 81
9
E)
x
x 2 + 81
35. Find the altitude of the isosceles triangle shown below if θ = 30 ° and b = 5 centimeters . Round answer
to two decimal places.
θ
θ
A) 2.89 centimeters
C) 1.25 centimeters
E) 4.33 centimeters
B) 0.67 centimeters
D) 1.44 centimeters
b
36. A sign next to the highway at the top of Saura Mountain states that, for the next 6 miles, the grade is
11%. Determine the change in elevation (in feet) over the 6 miles for a vehicle descending the
mountain. Round answer to nearest foot.
A) –3464 feet B) –3485 feet C) –3474 feet D) –2978 feet E) –3226 feet
37. Use a graphing utility to graph the damping factor and the function in the same viewing window.
Describe the behavior of the function as x increases without bound.
Version 2 Page 9
B)
A)
C)
D)
E)
38. Two lifeguards, Tony and Sharon, are 24 kilometers apart and Tony is directly due south of Sharon on
the beach. A stranded boat offshore is spotted by both lifeguards, and the bearings from Tony and
Sharon, respectively, are N 21° E and S 12° E . Determine the distance the stranded boat is from the
beach. Round answer to nearest tenth of a kilometer.
A) 2.3 kilometers B) 3.3 kilometers C) 4.5 kilometers D) 5.2 kilometers E) 6.2 kilometers
39. While traveling across the flat terrain of Nevada, you notice a mountain directly in front of you. You
calculate that the angle of elevation to the peak is 4.5 ° , and after you drive 5 miles closer to the
mountain it is 7 ° . Approximate the height of the mountain peak above your position. Round your
answer to the nearest foot.
A) 4553 feet B) 5248 feet C) 5787 feet D) 6569 feet E) 8243 feet
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40. A land developer wants to find the distance across a small lake in the middle of his proposed
development. The bearing from A to B is N 33 ° W . The developer leaves point A and travels 53
meters perpendicular to AB to point C. The bearing from C to point B is N 57 ° W . Determine the
distance, AB , across the small lake. Round distance to nearest meter.
B
A) 94 meters
B) 101 meters
C) 119 meters
D) 134 meters
E) 149 meters
C
A
Answer Key
31.
32.
33.
34.
35.
36.
37.
38.
39.
40.
C
B
E
E
D
A
D
B
C
C
Answer Key
1. A 23. D
2. A 24. A
3. A 25. B
4. C 26. A
5. C 27. B
6. B 28. D
7. A 29. D
8. E 30. E
9. D
10. B
11. A
12. E
13. A
14. C
15. B
16. E
17. E
18. C
19. E
20. A
21. C
22. C
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