Document 6530026

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Document 6530026

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© Wayne Hacker 2007. All rights reserved.
Math 182: Trigonometry
Sample Exam 1
Solutions
This practice test will give you an idea of the format and difficulty level of the actual test.
Like the actual test, it consists of 27 questions drawn from the problem sets and modified
slightly (e.g. by changing the numbers). Any problem on the problem sets can appear on the
actual test; it is not enough just to know how to solve the problems on this practice test.
No partial credit will be given for wrong answers. You will be graded on the best 25 of the 27
questions, so you can get up to two answers wrong without penalty. You cannot score more than
100% on the test.
Read the instructions carefully and follow them exactly. Your answers on the answer page
must exactly match the solutions, or they will be graded as wrong.
If the question asks that you round the answer to a certain number of decimal places, you must
do so. Your answer will be considered wrong if it is rounded incorrectly, or to the wrong
number of decimal places.
Unless you’re told to round your answer, it should be exact. Some things that can cause you
to lose credit for an exact answer are:
15
3
● Failure to simplify fractions.
is wrong; is correct.
20
4
2
2 3
● Failure to rationalize denominators.
is wrong;
is correct.
3
3
● Failure to simplify radicals.
18 is wrong; 3 2 is correct.
● Using decimal approximations instead of exact answers. 1.0472 is wrong;
π
is correct.
3
On problems where the answer involves units of measure (e.g. “23 ft”), you won’t lose credit
for not including the units. However, it’s a good idea to do so. Your answer must be in the
correct units: if, for example, the correct answer is “30°”, then “π/6” will be graded as wrong.
Your answers must appear in the correct format on the answer sheet. If no answer or a wrong
answer appears there, the grader will not check the page with the question to see if you’ve
answered it correctly there. That means it’s a good idea to double-check at the end of the test
and make sure that you’ve copied the answers correctly and in the right places on the answer
sheet.
Math 182: Trigonometry
Sample Exam 1
Name
Date
Score
Grader
This exam consists of 27 multiple-choice and short-answer questions. There will be no partial
credit for wrong answers. You will be graded on the best 25 of the 27, so you can get two
questions wrong without penalty. You cannot get a higher grade than 100% on this exam.
Write only your answers in the spaces provided on this page. Circle the correct answers for
multiple-choice questions. Do not do your work on this page.
Write your answers in exactly the format that the question asks for. If, for example, you round
to the wrong number of decimal places, or fail to rationalize denominators or simplify fractions
and radicals in exact solutions, your answer will be graded as wrong.
Unless otherwise indicated, your answers should be exact. Rationalize all denominators and
simplify all fractions and radicals.
Your answer must be in the correct units of measure. If, for example, the problem asks for an
angle in degrees, then an answer given in radians would be considered wrong.
Your grade will be based on the answers that you write on this page. If you have a wrong
answer or no answer on this page, the grader will not look at the page with the question to see if
the correct answer appears there. Illegible or ambiguous answers will be graded as wrong. You
are responsible for copying your answers clearly, correctly, and in the appropriate blanks.
You must show your work on the page with the question. Credit will not be given for lucky
guesses.
1. 4.5 cm
8. 2/3
15. 2.1790 rad
22. b
2. 66°
9. 2 53 53
16. 59.74°
23. e
3. 18.3503
10. 2π/9 rad
17. 1/2
24. e
4. arccos(5/7)
11. 13.5788
18. 0.4797 rad
25. a
5. b
12. c
19. 113 ft
26. a
6. 349 ft
13. 0.4447 rad
20. b
27. 1.9443 rad
7. III
14. d
21. a
1. The radius of a circle is 15 cm. A central angle measures 0.3 radians. What is the length of
the arc intercepted by the angle? Your answer should be exact.
Use the formula: s = rθ, where s is the arc length, r is the radius, and θ is the angle in radians.
Then s = (15)(0.3) = 4.5 cm.
2. What is the complement of 24°?
The complement of θ° is 90 – θ. In this case, it’s 90 – 24 = 66°.
3. In the figure at right, the triangle ABC has angles A = 0.61 rad and
B = 1.13 rad, and side c = 20. Find the length of side b. Round your
answer to four decimal places.
This is an ASA problem. Start by calculting C = π – (A + B) = 1.4016.
(Store the value of C in your calculator; do not write down “1.4016”
and re-enter it, or you risk round-off error.) Now use the law of sines:
20
b
20 sin(1.13)
; so b =
= 18.3503.
=
sin(1.13) sin C
sin C
4. For the right triangle at right, express θ as an arc function, e.g.
θ = arcsin(3/4). Do not calculate the actual value of θ.
cos θ = 5/7 (and θ is acute); so θ = arccos(5/7)
C
b
A
a
c
B
7
θ
5
5. An angle of 2π/5 radians is:
(a) straight
(b) acute
(c) laminar
(d) right
(e) obtuse
(f) haploid
2π/5 < π/2; so the angle is acute. The correct answer is (b).
6. You want to know the width of a river. You begin by
standing directly across from a conspicuous tree on the
opposite bank. You then walk 170 feet straight
downstream. From the new point, the tree is at an angle of
θ = 64° to the upstream direction. How wide is the river?
Round your answer to the nearest foot.
y
Let y be the width of the river. Then tan 64° =
, so
170
y = 170 tan 64° = 349 ft
θ
7. In which quadrant would you find 213°? Circle the correct answer on the answer page.
180° < 213° < 270°; so 213° is in the third quadrant. The correct answer is III.
8. Fill in the blank to make the statement true. Assume θ is in the first quadrant.
cos θ = _________ if and only if θ = arccos(2/3)
2/3; this comes straight from the definition of an arc cosine
9. In the right triangle at right, what is sin A? Your answer should be
exact.
2
A
Use Pythagoras to find the hypotenuse: r = 7 2 + 2 2 = 53 . Then
7
2 53
. Don’t forget to rationalize the denominator: otherwise, your answer will be
53
graded as wrong.
sin A =
10. An angle measures 40°. What is its measure in radians? Your answer should be exact.
180° = π radians; so to convert from degrees to radians, multiply by
π
180
. Thus
40π 2π
radians. Your answer will be counted as wrong if you don’t simplify the
=
180
9
fraction, or if you use a decimal approximation like 0.698132.
40° =
11. In the figure at right, the triangle ABC has angle A = 53°
and sides b = 17 and c = 10. Find the length of side a.
Round your answer to four decimal places.
This is an SAS situation. By the law of cosines,
a 2 = 17 2 + 10 2 − 4(17)(10) cos 53° ; so a = 13.5788.
B
a
c
b
A
C
12. Which function does the graph below represent? Angles are measured in degrees.
(a) y = sin x
(b) y = cot x
(b) y = cos x
(e) y = sec x
(c) y = tan x
(f) y = csc x
10
8
6
4
2
0
-120
-60
0
60
120
180
240
300
360
420
480
540
600
660
720
780
840
-2
-4
-6
-8
-10
The graphs of sin x and cos x oscillate back and forth between -1 and 1, so this isn’t one of those.
The values of sec x and cos x can never be zero; they must always be in the range [1,∞) or
(-∞,-1]. Thus this graph is either tan x or cot x. It goes through the point (0,0). That’s consistent
with y = tan x, since tan(0) = 0; but inconsistent with y = cot x, since cot(0) is undefined. Hence
the correct answer is (c).
13. Your friend’s house is 8520 feet away from yours. An airplane
flies over your friend’s house at an elevation of 4060 feet. What
is the angle of elevation of the airplane, as seen from your house?
Round your answer to the nearest 0.0001 radians.
4060
Let θ be the angle of elevation. Then tan θ =
. Since θ is an
8520
 4060 
acute angle, θ = tan −1 
 = 0.4447 rad.
 8520 
θ
14. If cos θ > 0, which quadrant could θ be in?
(a) I or II
(b) I or III
(c) I or IV
(d) II or III (e) II or IV (f) III or IV
cos θ is negative when θ is in the II or III quadrant; so the correct answer is (d).
15. In the figure at right, the triangle ABC has sides a = 12, b = 17,
and c = 7. What is the measure of angle B? Round your answer
to the nearest 0.0001 radian.
This is an SSS situation. By the law of cosines,
12 2 + 7 2 − 17 2
. Thus B = 2.1790 rad.
cos B =
2(12)(7)
B
a
c
A
16. In the right triangle at right, calculate the angle θ. Round your answer to the
nearest 0.01°.
180
 12 
tan −1   = 59.74°.
tan θ = 12/7; so θ =
π
7
C
b
12
θ
7
17. If θ = 5π/3 radians, what is cos θ? Do not use a calculator. Your answer should be exact.
θ is in the fourth quadrant, so the reference angle is 2π – θ = π/3 rad. The cosine of π/3 is 1/2.
Since θ is in the fourth quadrant, its cosine is positive; so cos θ = 1/2.
18. In the right triangle at right, calculate the angle A. Round your
answer to the nearest 0.0001 rad.
sin A = 6/13; so A = arcsin(6/13) = 0.4797 rad.
B
13
6
C
A
19. You are flying a kite on a windy day. The string is 180 feet long,
and the wind is strong enough to stretch it straight. A friend who’s
standing 140 feet from you tells you that the kite is directly over him.
How high above the ground is the kite? Round your answer to the
nearest foot.
You, the kite, and the friend form a right triangle with hypotenuse r = 180 and with side x = 140.
You want the third side: the height h of the kite. By Pythagoras, h = 180 2 − 140 2 = 113 ft.
20. Find the distance between the points (-1,4) and (3,2).
(a) 2 2
(d) 10
(b) 2 5
(e) 6
(c) 26
(f) 4
The distance between points ( x1 , y1 ) and ( x2 , y 2 ) is: d =
(x2 − x1 )2 + ( y 2 − y1 )2 .
our numbers gives d = 4 2 + 2 2 = 20 = 2 5 . Hence the correct answer is (b).
Plugging in
21. What is the amplitude of the function: y = 6 cos (5x − 7)? Angles are measured in radians.
(a) 6
(b) 5
(c) 7
(d) 12
(e) 10
(f) 14
In a function of the form y = a cos(bx − c ) + d , the amplitude is a. In this problem, it’s 6.
Hence the correct answer is (a).
22. A boat is bobbing up and down on the waves. The depth of water z in meters below the boat
as a function of time t in seconds is: z = 0.8 sin (bt ) + 8.5 , where angles are measured in degrees.
If two consecutive high points are separated by 10 sec, what is b?
(a) 6
(b) 36
(c) 10
(d) 1/6
(e) 1/36
(f) 1/10
In a function of the form y = a sin(bx − c) + d , with angles measured in degrees, the period is
360
. Two consecutive maxima of the function are separated by one period; so in this
T=
b
360
problem, T = 10 sec. Hence b =
= 36. The correct answer is (b).
10
23. If θ is in the fourth quadrant, and cosθ =
(a)
7
4
(d) −
(b)
7
4
11
, what is sin θ?
4
5
4
(e) −
5
4
We can use the identity sin 2 θ + cos 2 θ = 1 ; so sin 2 θ = 1 −
quadrant, sin θ is negative; so sin θ = −
(c)
3
4
(f) −
3
4
11 5
= . Since θ is in the fourth
16 16
5
. The correct answer is (e).
4
24. What function does the graph below show? x is in degrees.
(a) y = sin(3 x − 90)
(b) y = sin( 2 x) − 5
(c) y = 3 sin(2 x) − 5

 x
x
(d) y = 5 sin  + 60 
(e) y = 3 sin   − 2
(f) y = −6 sin(3 x) + 1

 2
2
1.5
1
0.5
-120
0
-60-0.5 0
60
120
180
240
300
360
420
480
540
600
660
720
780
840
-1
-1.5
-2
-2.5
-3
-3.5
-4
-4.5
-5
-5.5
We want to find a function of the form y = a sin(bx − c) + d that matches this graph.
The easiest thing to check is the amplitude a. The graph ranges between a maximum of 1 and
a minimum of -5; its amplitude is half the difference between those, so a = 3. That immediately
rules out all of the choices except (c) and (e).
Next, look at the vertical translation d. The midpoint between the maximum and minimum is
−2; so the graph has been translated down by 2 units, and d = -2. That excludes (c), so (e) is the
only choice left.
Just to be safe, we’ll check the period and the horizontal translation. The graph rises through
the midpoint at (0,-2) and again at (720,-2), so the period is T = 720. When x is measured in
degrees, bT = 360; so b = 1/2. That’s consistent with (e) but not with (c).
The graph rises through the midpoint at (0,-2), which suggests that c = 0. The graph of (a)
would rise through the midline at x = 30; the graph of (d) would rise through the midline at
x = 120.
The correct answer is (e).
25. tan θ = ?
sin θ
sin θ
sin θ
(a)
(b)
(c)
cosθ
cscθ
secθ
cosθ
cosθ
cosθ
(d)
(e)
(f)
sin θ
cscθ
secθ
This is a basic trig identity that you should know by memory. It’s (a).
26. If angles are measured in radians, sin θ = ?
π

(a) cos − θ 
(b) cos(π − θ )
(c) 1 − cos θ
2

π

(d) − cos − θ 
(e) − cos(π − θ )
(f) 1 − cos 2 θ
2

The sine of an angle is the cosine of its complement. Hence: (a).
27. In the triangle ABC shown at right, A = 0.65 rad; a = 13; c = 20;
and C is obtuse. Calculate the value of C. Round your answer to
the nearest 0.0001 radian.
This is an SSA situation. From the law of sines,
sin(0.65) sin C
20 sin(0.65)
; so sin C =
. We want C to be obtuse;
=
13
20
13
 20 sin 0.65 
so if we let θ = sin −1 
 , then C = π − θ = 1.9443 rad.
13


B
c
a
A
b
C

Document 6530026

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