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Math 140
Trigonometry 11th edition Lial, Hornsby, Schneider, and Daniels
Practice Final
(Ch. 1-8)
PART I: NO CALCULATOR (144 points)
(4.1, 4.2, 4.3, 4.4)
For the following functions:
a) Find the amplitude, the period, any vertical translation, and any phase shift.
If not applicable, write “none” in the blank.
b) Graph over the interval −2π ≤ x ≤ 2π . Identify and label any asymptotes.
1
3π 

1.
2.
y = 4sin x
y=
−2cos  x +

2
4 

amplitude:
amplitude:
period:
period:
vertical translation:
vertical translation:
phase shift:
phase shift:
−2π
:
4
4
2
2
−π
π
−2π
2π
−π
−2
−2
−4
−4
π

3. =
y tan  x − 
4

4.
amplitude:
period:
period:
vertical translation:
vertical translation:
phase shift:
phase shift:
4
4
2
2
−π
π
2π
2π
π
2π
y = csc x
amplitude:
−2π
π
−2π
−π
−2
−2
−4
−4
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Math 140 Practice Final (cont.)
(4.1, 4.2, 4.3, 4.4)
For the following functions:
a) Find the amplitude, the period, any vertical translation, and any phase shift.
If not applicable, write “none” in the blank.
b) Graph over the interval −2π ≤ x ≤ 2π . Identify and label any asymptotes.
1
5.
6.
y= 2 − sec x
y = cot x
2
amplitude:
amplitude:
period:
period:
vertical translation:
vertical translation:
phase shift:
phase shift:
−2π
4
4
2
2
−π
π
−2π
2π
−π
π
−2
−2
−4
−4
(6.1)
Give the exact radian measure of y if it exists.
(
7.=
y arctan − 3
)

2
=
y arcsin  −
10.

 2 

3
y cos −1  −
8. =

 2 
11.
 2
y = csc −1 

 2 
9.
y = sec −1 ( 2 )
y cot −1 ( −1)
12. =
Write the following trigonometric expression as an algebraic expression in u, for u > 0 .
13. cot sec −1 u
14. cos ( arcsin u )
(
)
2
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2π
Math 140 Practice Final (cont.)
PART II: YOU MAY USE A CALCULATOR (256 points)
DOUBLE-ANGLE IDENTITIES
sin 2 A = 2sin A cos A
cos
=
2 A cos 2 A − sin 2 A
tan 2 A =
2 tan A
1 − tan 2 A
=
cos 2 A 2cos 2 A − 1
cos 2 A = 1 − 2sin 2 A
SUM AND DIFFERENCE IDENTITIES
HALF-ANGLE IDENTITIES
sin( A
=
+ B) sin A cos B + cos A sin B
sin( A − B) = sin A cos B − cos A sin B
sin
A
1 − cos A
= ±
2
2
cos
A
1 + cos A
= ±
2
2
cos( A + B) = cos A cos B − sin A sin B
cos( A − B) = cos A cos B + sin A sin B
tan A + tan B
1 − tan A tan B
tan A − tan B
tan( A − B) =
1 + tan A tan B
tan( A + B) =
tan
A
sin A
=
2 1 + cos A
tan
A 1 − cos A
=
2
sin A
tan
LAW OF COSINES
a 2 = b 2 + c 2 − 2bc cos A
A
1 − cos A
= ±
2
1 + cos A
b 2 = a 2 + c 2 − 2ac cos B
DE MOIVRE’S THEOREM
c 2 = a 2 + b 2 − 2ab cos C
 r ( cos θ + i sin θ )  = r n ( cos nθ + i sin nθ )
where r ( cos θ + i sin θ ) is a complex
number and n is any real number.
n
LAW OF SINES
a
b
c
= =
sin A sin B sin C
sin A sin B sin C
= =
a
b
c
3
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Math 140 Practice Final (cont.)
(1.1)
1.
Convert the following angles to decimal degrees. If applicable, round to the nearest hundredth of
a degree.
a) 76  48'
b) 34  51'35''
c) 249 15'
(1.4)
2.
Identify the quadrant satisfying the given conditions:
a) cos θ < 0 and cot θ > 0
b) tan θ < 0 and cscθ > 0
(2.1, 2.2)
3.
Find the exact values of the six trigonometric functions for the given angles.
Rationalize denominators when applicable.
a) 135 
b) 210 
c) 270 
d) 300 
(5.5)
4.
Given cos 2 x = −
5
and 90° < x < 180° , find the exact values of the following.
12
sin x =
cos x =
tan x =
5.
Given csc x = −
7 5
and cos x > 0 , find the exact values of the following.
5
sin 2x =
cos 2x =
tan 2x =
(2.5)
6.
Find h as indicated in the figure.
h
41.2 
52.5 
168 m
4
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Math 140 Practice Final (cont.)
(7.3)
Solve the following problem. Include a labeled sketch in your work.
7.
Starting at point A, a ship sails 18.2 km on a bearing of 190 , then turns and sails 47.4 km on a
bearing of 319 . Find the distance of the ship from point A to the nearest tenth of a kilometer.
(7.1)
Solve the following problem. Include a labeled sketch in your work.
8.
Francesco is flying in a hot air balloon directly above a straight road 3.4 mi long that joins two
towns. He finds that the town closer to him is at an angle of depression of 52.9 , and the farther
town is at an angle of depression of 28.5 . How high above the ground is the balloon?
Round your answer to the nearest hundredth of a mile.
(5.6)
9.
Use the appropriate half-angle identity to find the exact value of the following.
Work demonstrating use of appropriate identity must be shown.
a) sin 202.5°
b) tan 75°
(5.3, 5.4)
10. Use the appropriate sum or difference identity to find the exact value of the following.
Work demonstrating use of appropriate identity must be shown.
13π
5π
π
a) cos
b) tan
c) sin
12
12
12
(3.1)
11. Convert the following angles to radians. Leave answers as multiples of π .
a) 110 
b) 216 
(3.1)
12. Convert the following angles to degrees.
4π
a) −
15
b)
8π
5
5
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Math 140 Practice Final (cont.)
(5.1, 5.2, 5.5)
Verify that each equation is an identity.
sinθ + tan θ
13.
= tan θ
1 + cos θ
15.
cos 2θ
cot θ − tan θ =
sin θ cos θ
14.
2
sec 2 θ csc
=
θ sec 2 θ + csc 2 θ
16.
tan 8θ − tan 8θ tan 2 4θ =
2 tan 4θ
(6.2)
17. Solve the following equations for all exact solutions in radians.
Write answers using the least possible nonnegative angle measures.
0
a) 2sin x − 3 =
b) cos x + 1 =2sin 2 x
(6.3)
18. Solve the following for all solutions in degrees. Use exact values for x whenever possible.
If necessary, approximate answers to the nearest tenth of a degree.
Write answers using the least possible nonnegative angle measures.
a) 3cot 3 x = 3
b) 2 − sin 2 x =
4sin 2 x
(6.2, 6.3)
19. Solve the following equation over the interval 0  , 360  Use exact values for x whenever
possible. If necessary, approximate answers to the nearest tenth of a degree.
40
a) 5 tan 2 x + 16 tan x =
b) 5sec 2 x= 3 + 3sec x
c) 2sin 2 x = 1
)
(8.4)
20.
(
)
6
Use DeMoivre’s Theorem to find 2 − 2i 3 . Write your answer in rectangular form.
(8.5)
21. Convert the following to polar coordinates with 0° ≤ θ < 360° and r > 0 .
a)
( −1, 3 )
b)
(
2, − 2
)
6
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Math 140 Practice Final (cont.)
(8.6)
22. A golf ball is hit from the ground with initial velocity of 150 feet per second at an
angle of 60° with the ground.
x = 75 t
The parametric equations that model the path of the rocket are given by
−16t 2 + 75 3 t
y=
Determine a rectangular equation that models the path of the projectile.
Use exact values for any numbers in the equation.
7
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Math 140 Practice Final (cont.)
Part I Answers:
1)
a) amplitude: 4
period:
4π
vertical translation: none
phase shift: none
b)
2)
a) amplitude: 2
period:
2π
vertical translation: none
phase shift:
3π
left
4
b)
8
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Math 140 Practice Final (cont.)
Part I Answers:
3)
a) amplitude: none
period:
vertical translation: none
π
phase shift:
π
4
right
b)
asymptotes:
x= −
5π
4
x= −
π
4
x=
3π
4
x=
7π
4
4)
a) amplitude: none
period:
2π
vertical translation: none
phase shift: none
b)
asymptotes:
x = −2π
x = −π
x =π
x=0
x = 2π
9
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Math 140 Practice Final (cont.)
Part I Answers:
5)
vertical translation: 2 up
a) amplitude: none
period:
2π
phase shift: none
b)
asymptotes:
x= −
3π
2
x= −
π
2
x=
π
2
x=
3π
2
6)
a) amplitude: none
period:
2π
vertical translation: none
phase shift: none
b)
asymptotes:
x = −2π
x = 2π
x=0
10
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Math 140 Practice Final (cont.)
Part I Answers:
π
7)
−
8)
5π
6
9)
does not exist
10)
−
11)
does not exist
12)
−
3
π
4
π
4
13)
u2 −1
u2 −1
14)
1− u2
11
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Math 140 Practice Final (cont.)
Part II Answers:
1)
a) 78.8 
b) 34.86 
2)
a) III
b) II
3a)
sin135 =
2
2
c) 249.25 
cos135 = −
csc135 = 2
2
2
sec135 = − 2
1
2
cos 210 = −
3
2
csc 210 = −2
sec 210 = −
2 3
3
3c)
sin 270 = −1
cos 270 = 0
3d)
sin 300 = −
3
2
cos 300 =
csc 300 = −
2 3
3
sec 300 = 2
3b)
4)
sin 210 = −
sin x =
102
12
cos x = −
1
2
42
12
12
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tan135 = −1
cot135 = 1
tan 210 =
3
3
cot 210 = 3
tan 300 = − 3
cot 300 = −
tan x =
119
7
3
3
Math 140 Practice Final (cont.)
Part II Answers:
5)
sin 2 x = −
6)
448 m
7)
38.6 km
8)
1.31 miles
9)
a) −
10)
a)
11)
a)
12)
a) −48
4 55
49
2− 2
2
6− 2
4
11π
18
cos 2 x =
39
49
tan 2 x =
−4 55
39
b) −2 − 3
b) 2 − 3
b)
6π
5
b) 288
13
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c)
6− 2
4
Math 140 Practice Final (cont.)
Part II Answers:
13)
One possible way of verifying
sin θ + tan θ
= tan θ :
1 + cos θ
sin θ + tan θ
1 + cos θ
sin θ
sin θ +
cos θ
=
1 + cos θ
1 

sin θ 1 +

cos θ 

=
1 + cos θ
1 
 cos θ
+
sin θ 

 cos θ cos θ 
=
1 + cos θ
 cos θ + 1 
sin θ 

 cos θ 
=
1 + cos θ
sin θ
( cos θ + 1)
θ
cos
=
1 + cos θ
= tan θ
14)
One possible way of verifying
2
sec 2 θ csc
=
θ sec 2 θ + csc 2 θ :
sec 2 θ + csc 2 θ
=
1
1
+ 2
2
cos θ sin θ
sin 2 θ + cos 2 θ
=
sin 2 θ cos 2 θ
1
=
2
sin θ cos 2 θ
= sec 2 θ csc 2 θ
14
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Math 140 Practice Final (cont.)
Part II Answers:
15) One possible way of verifying
cos 2θ
:
cot θ − tan θ =
sin θ cos θ
cos 2θ
sin θ cos θ
cos 2 θ − sin 2 θ
=
sin θ cos θ
cos 2 θ
sin 2 θ
=
−
sin θ cos θ sin θ cos θ
cos θ ⋅ cos θ sin θ ⋅ sin θ
=
−
sin θ cos θ
sin θ cos θ
= cot θ − tan θ
16) One possible way of verifying
tan 8θ − tan 8θ tan 2 4θ =
2 tan 4θ :
tan 8θ − tan 8θ tan 2 4θ
= tan 8θ (1 − tan 2 4θ )
= tan 2 ( 4θ ) (1 − tan 2 4θ )
=
2 tan 4θ
(1 − tan
1 − tan 2 4θ
= 2 tan 4θ
2
4θ )
1
15
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Math 140 Practice Final (cont.)
Part II Answers:
π
π + 2 kπ
π
b) + 2kπ
+ 2 kπ
3
2π
+ 2kπ , where k is an integer
3
17)
a)
18)
a) 20 + 60 k , where k is an integer
19)
a) 58.8 , 101.7 , 238.8 , 281.7
3
5π
+ 2kπ , where k is an integer
3
b)
11.8 + 180 k
78.2 + 180 k , where k is an integer
b) 27.8 , 332.2
c) 45 , 135 , 225 , 315
20)
4096 or 4096 + 0i
21)
a)
22)
16 2
y=
−
x + 3x
5625
( 2, 120 )

b)
( 2, 315 )

16
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