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PRECALCULUS GT/H
REVIEW FOR FIRST SEMESTER FINAL
Work the following on notebook paper.
1. Find all of the zeros of f  x   6 x 4  x3  32 x 2  5x  10.
2. Find all of the zeros of f  x   21x 4  95x3  109 x 2  105x  200.
3. Find two positive numbers such that their product is 192 and the sum of the first plus
three times the second is a minimum.
4. A manufacturer wants to design an open box having a square base and a surface area of
108 sq. in. What dimensions produce a box with maximum volume?
5. Four feet of wire is to be used to form a square and a circle. Find the length of the side
of the square and the radius of the circle so that the enclosed total area is:
(a) a maximum (b) a minimum
6. Find all of the asymptotes of:
x2  9
2 x 2  3x  2
(a) y  2
(b) y 
x 1
x  16
________________________________________________________________________________
Solve.
7. x  6  x  14
8. x  1  1  x  6
9. x3  4x2  x  4
x2  x  6
10.
0
x 1
11. x  2  3x  1
12.Find two angles, one positive and one negative, that are coterminal with the given angle.
5
a. 35
b.
6
13. a. Convert 165 to radians. Give answers in terms of  .
b. Convert 208 to radians. Give the answer to the nearest hundredth of a radian.
14. a.
Convert

7
to degrees.
6
b. Convert 1.8 radians to nearest tenth of a degree.
15. Write an equation for a sine wave that has amplitude 3 and period 8.
2
16. If  is a third quadrant angle and cos   , find sin  .
3
17. Express each of the following in terms of a reference angle.
a. cos236 b. sin485 c. sin( 62 )
18. Give the exact value of the six trigonometric functions for each angle.


a.
b. 225 c. 
6
2
Let cot x 
19.
5
3
, where   x 
. Find the values of the other five trigonometric functions.
12
2
20. Without using your calculator or a table, find the value of each expression. Leave your
answers in terms of  whenever appropriate.
3
2
a. Tan 11
b. Sin 1
e. Sin 1 0
f. sin(Cos 1 )
3
)
2
5
g. sec(Tan 1 )
12
d. Tan 1 3
c. Cos 1 ( 
3
5
h. cos( Sin 11)
21. Solve the equation to the nearest hundredth of a radian. 2 sec x  5  0 for 0  x  2 .
22.
Evaluate without using a table or calculator.
5
a. sin120
b. sec
4
23. Evaluate without using tables or calculators.
F 3 I b. cos( Arc tan
G
H2 J
K
a. Cos 1 
3)
3 I
F
G
H2 J
K
c. cos 
3
1I
F
) d. Arcsin(sin )
G
J
H2 K
4
c. sin( Sin 1 
7 I
F
G
H6 J
K
d. tan 
1I
F
G
H2 J
K)
e. csc(Tan 1 
24. If sec x  5 and   x  2 , find exact value for the other five trigonometric functions of x.
25. Determine the sinusoidal function of sine with amplitude
1
3
, period  , and translation 2 units
up.
26. Determine the angle coterminal with 
27. A
4
9
7
2
that has a radian measure between 0 and 2 .
clockwise rotation would terminate in which quadrant and yield what angle measurement?
28. Express the function sin

8
in terms of its cofunction.
29. Find the value of cos if sin   
2
3
and angle  is in standard position with terminal side in
the fourth quadrant.
30. Sec and cot  have opposite signs in which quadrant?
31. Determine the following: sin( Cos 1
5
).
6
1  cos
 tan 2   sec 2 
2
sin 
32. Simplify the following .
33. Solve the equation: cos x sin x  sin2 x for 0  x  2 .
34. Match the function to the graph.
A. y  2 sin 2 x
B. y  3cos 3x
1
2
5
2
E. y  sin x
F. y  cos
3
2
C. y  cos
x
x
3
2x
H. y  cos
3
x
2
D. y  2 sin
G. y  2sin x
2
3
3
3
3
2
2
2
2
1
1
1
1
0
1
2
3
4
5
6
1
1
1
1
2
2
2
2
3
3
3
3
A. __________
B. _________
C. _________
D. _________
3
3
3
3
2
2
2
2
1
1
1
1
0
1
2
3
4
5
6
1
1
1
1
2
2
2
2
3
3
3
3
E. __________
F. _________
35. Find the exact value of sin 15.
36. Find the exact value of
cos
3
8
.
G. _________
H. _________
37. Find
sin 
     2 .

2  7
cos 
, given that
38. Graph with the correct trig function.
A. y  2  3cos x
C.
B. y  2 cos x  3
y  3sin x  2
D. y  2  3sin x
39. A right triangle has an acute angle  such that cot   15. Find cos .
40. Simplify:
1
1
.

sec x  1 sec x  1
41. Evaluate: sec( Arc tan 3) .
42. Evaluate:
FI.
HK
Arccos
1
2
43. Find the reference angle for   305 .
44. Find all solutions (exact value) in the interval
0, 2 , sec 3x  2 .
45. What is 1  tan 2   ?
46. What is the reference angle for 188  ?
47. Graph y  2 sec(2 x)  1.
48. Find the exact value of
cos
7
.
12
49. Express 315 degrees in radians.
50. Express 8420'40" in decimal degrees.
51 Express in degrees, minutes, and seconds : 38.405
52. Find the value of cos of the angle in standard position that passes through point (3, 4).
53. An angle with rotation of – 220 degrees terminates in which quadrant?
54. Find the domain of the function : g( x)  x  4 .
55. Graph y = 2tan x +1.
f ( x)  x 2 , g ( x)  3x  4 , evaluate
56. Given:
y  csc
57. Graph
1
x .
bf  ggbg
x 1
2
58. Simplify the following.
A. csc x  cos x cot x
B. tan2 x  sin2 x  cos2 x
C.
1
1  tan x
2

1
1  cot 2 x
59. Find the period, amplitude, vertical displacement, and phase displacement of the graph of
each equation.
A. y  3cos(2 x   )  1
B. y  0.5sin(.5x .5 )  3
C. y  3sin 2 x
60. State the type of symmetry: A. y  x2  1
B. y  x3  x
61. State the domain and range for:
A. y  sin x
C. y  x
B. y  cos x
D. y  sec x
62. State the domain and range for:
A. y  x


C. y  3sec 2  x    1
6



B. y  3sin 2  x    1
4

1
3 
D. y  cot  x 

3
4 
63. Find: csc 90  cot 0  sin180  cos 270 .
64. Graph
y  3 cos
1

 x   1.
2
3
65. Refer to the triangle to find exact value for the following:
A. sin T =
B. cos T =
C. tan R =
D. sec T =
E. csc R =
F. csc T =
R
4
x
9
S0
6
G. cot R
66. Given:
cos A  
12
, csc B  
13
A.
cos  A  B 
B.
sin
1
5
.
Neither A nor B is in Quadrant III. Find:
3
B
2
C. tan 2A
67. Describe the transformations of each equation.
A.
B.
C.
D.
f ( x)  2 cos x
f ( x )  cos 21 x
f ( x)  cos( x  2 )
f ( x)  cos x  2
68. Write an equation for the graph of the cosine function with the following characteristics:
period of  , phase shift left 2 , vertical stretch of 3
69. If f ( x)  2 x  4 and g( x)  3x 2  1 , find:
A. f g x
B.
bg
g f bg
x
C. f 2a
D. g 5a
b g
70. If g( x)  x 2  1 and f ( x)  x  2 , find f ( g( x)).
T
71. Which graph is one-to-one?
A.
B.
C.
D.
72. Describe the graph as odd, even, both, or neither.
A) ____________
B) ___________
C) ____________
73. Name the asymptotes for y  2 sec x  3 .
74. Evaluate: csc14  to 4 decimal places.
.
. (in radians)
75. Find an acute angle such that tan   12617
Given: sin A 
8
17
76. sin  A  B 
1
77. cos B
2
,

2
 A   , tan B  
7
24
,
3
2
 B  2 .
Find:
78. tan 2A
Solve, 0  x  2 .
1
2
79. cos 2x  sin x  1
80. sin 3 x  
81. 2sin 2 x  cos x  1  0
82. tan 2 x  sec x  1
83. sin 2x  2cos x
________________________________________________________________________________
Solve for all values of x (general solution).
84. sin 2x  sin x
85. cos 2x cos x  sin 2x sin x  0
________________________________________________________________________________
86. A paddlewheel on a steamboat makes one complete revolution every 18 seconds. The diameter
of the wheel
is 10 feet. A point is at the bottom of the wheel, 2 feet below the surface of the water, at t = 0.
Write an
equation to represent the point's distance from the surface of the water.
87. A gear makes 6.2 rotations about its axis. what is the angular displacement in radians of a point
on the gear?
88. What is the angular velocity in radians per minute of a notch on a wheel that makes 24 rotations
per second
about its axis?
89. The minute hand of a watch is 1.3 cm long. What is the linear velocity of the tip of the hand?
Be sure to study: the graphs of the parent functions and trig functions with their domains and
ranges
the domain and range for all inverse trig functions
the unit circle
all of the trig identities