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Math 104 Final Review
1.
Find all six trigonometric functions of  if ( 3, 7 ) is on the terminal side of  .
2.
Find cos and sin  if the terminal side of  lies along the line y  2 x in
quadrant IV.
3.
Find the remaining trigonometric functions of  if csc  3 and  terminates
in quadrant II.
4.
Use identity substitutions to simplify: a) csc  cot  cos
5.
Show that cos (sec  tan  )  1  sin  by transforming the left side into the right side.
6.
Simplify the expression
7.
If sin  
8.
In which quadrant will  lie if csc  0 and cos  0 ?
9.
In ABC , C = 90 , c= 4.79 cm, and b = 3.68 cm. Draw the triangle then find each of the
following:
a) Side a
b) Angle A
c) Angle B
10.
In ABC C = 90 , A = 60 , and side a = 12 cm. Find exact answers for each of the
following.
a) Side c
b) Side b
11.
Use a calculator to find: a) tan 63 50
12.
A CB antenna is located on the top of a garage that is 16 feet tall. From a point on level
ground that is 100 feet from a point directly below the antenna, the antenna subtends an
angle of 12 . Approximate the length of the antenna.
13.
A pilot, flying at an altitude of 5000 feet, wishes to approach a landing point on a runway
at an angle of 10 (angle of depression). Approximate, to the nearest 100 feet, the
distance from the airplane to the landing point at the beginning of the descent.
b) (1  cos )(1  cos  )
9  x2 as much as possible after substituting 3sin for x.
1
with  in quadrant I, find cos , csc , and cot  .
a
b)  if  is acute and sec  1.923
2
3
 3 
b) 4 cos   
 4 
Give the exact values of: a) sin
15.
Find the exact values of: a) sec 45
16.
Show that cotangent is an odd function.
17.
Convert to radians: a) 120
18.
Convert to degrees: a)
19.
Draw the following angles in standard position and find the reference angle:
13
a) 280 20
b) 225
c)
12
20.
Use a calculator to find  if  is between 0 and 360 and
a) cos  .4772 with  in quadrant III
b) sec  1.545 with  in quadrant IV
21.
If  
4
3
b) csc 60
c) csc
5
6
14.
c) sin 2 60  cos 2 45 (Simplify)
b) 250
b)
7
12
2

is a central angle that subtends an arc length of s  , find the radius of the
3
4
circle.
22.
Find the area of the sector formed by central angle   2.4 in a circle of radius r  3 cm.
23.
A conical paper cup is constructed by removing a sector from a circle of radius 5 inches
and attaching edge OA to OB (see figure). Find angle AOB so that the cup has a depth of
4 inches.
24.
Find the amplitude, period, and phase shift of:
a.


y  3 cos  x  
2
4
b.

1
y  5sin  x  
6
3
25.
26.
Sketch one period of:
1

cos x
2
2
a.
y
b.

1
y  sin  x  
2
2
c.
y  tan 3 x
d.
1
y  3sec x
2
Find the equations of each of the following:
a.
b.
c.
27.


Find an interval over which the graph of y  cos   x   completes one complete cycle.
2

28.
1
Find the range of y  3cos x  (3  2) .
4
29.
Find the exact value for each of the following:
a.
30.
b.
 
sin 1  sin 
4

c.
2 

sin 1  sin

3 

Find the exact values for each of the following:
a.
31.
1

sin  sin 1 
2

 

cos 1  cos(  ) 
19 

b.
 

cos 1  cos(  ) 
19 

Find the exact value for each of the following:
a.
3 

sin 1  tan

4 

b.
   
cos 1  sin    
  6 
c.

 2 
sin  arccos    
 3 

d.
1
4

sin  arctan  arccos 
2
5

32.
If 1  x  1 , rewrite cos(sin 1 x) is terms of x without trig functions.
33.
Let sin  
34.
a. Derive the formula for tan( A  B) using the formulas for sine and cosine.
4
12
with  in the second quadrant and sin   
with  in the third
5
13
quadrant. Compute sin(   ) .
b. Find tan A if tan( A  B)  9 and tan B 
35.
1
.
3
If x is a positive number, find:
a.
cos(2sin 1 x)
b.
tan(cos1 3x)
36.
Find the exact values of:
a.
37.
Let sin A  
tan

c.
12
cot15
3
12
with A in quadrant IV and sin B 
with B in quadrant II and find:
5
13
sin( A  B )
a.
38.
b.
sin 22.5
b.
c.
cos 2B
sin
A
2
Prove the following identities:
a.
cos x
1  sin x

1  sin x
cos x
b.
sec x  sin x(tan x  cot x)
c.
sec x  cos x  sin x tan x
d.
cot x 
sin 2 x
1  cos 2 x

 

39.
If x  a sin , 
40.
Express as a single trig function and then simplify:
sin

12
cos
2
2
and a  0 , express
5

5
 cos sin
12
12
12
41.
Solve 2sin   3  0 for 0    360
42.
Solve 2 cos 2   5cos  3  0 for 0    360
43.
Solve cos 2x  3sin x  2  0 for 0  x  2
44.
Solve
45.
Solve 1  sin   3 cos for 0  x  2
3 csc  2cot   0 for 0    360
a2  x2 in terms of a trig function of  .
46.
Find all radian solutions for 2sin 2 4  2 cos 4  1
47.
In triangle ABC, B  57 , C  31 , and side a  7.3 meters. Find the missing parts of
the triangle.
48.
From a point on the ground, a person notices that a 100-ft antenna on the top of a hill
subtends an angle of 1 . If the angle of elevation to the bottom of the antenna is 37 , find
the height of the hill.
49.
How many triangles ABC satisfy the following conditions? A  140 , b  87 ft., and
a  62 meters
50.
In triangle ABC, A  27 , b  48 cm, and a  39 cm. Find angle B.
51.
In triangle ABC, if a  20 m, b  30 m, and c  40 m, find the measure of the smallest
angle.
52.
Two planes leave an airport at the same time. Their speeds are 360 mph and 420 mph
and the angle between their courses is 28 . How far apart are they after 1.5 hours?
53.
Divide
54.
Find x and y between 0 and 2 so that ( x 2  2 x)  y 2i  8  (2 y  1)i is true.
55.
Write 4  8i in trigonometric form.
56.
Write
57.
Find ( 3  i ) 7 . Express your answer in trigonometric form.
58.
Divide:
3i
. Express your answer in the form a  bi .
4  5i
2(cos 225  i sin 225 ) in standard form (use exact values).
16(cos165  i sin165 )
. Express your answer in standard form.
20(cos30  i sin 30 )
59.
Find the three cube roots of 27i .
60.
Find all the solutions for the equation x 5  1  0
61.
A vector, V, has magnitude V and forms an angle q with the positive x-axis. Find the
magnitudes of Vx and Vy if V = 850 and q  16 .
62.
A ship heads east at a constant 15 mph. The current of the water is running due north at a
constant 3 mph. Draw a vector, V, representing the true course of the ship.
63.
Find the resulting equation when the parameter t is eliminated from the equations
x  2  cos t and y  3  sin t .
64.
Graph the curve described by the parametric equations x  1  cos t and y  4  sin t .
65.
If W = 6i-8j and V = -2i-10j, then find V – W.
66.
If V = 2i - 3j and U = i + j, then find the cosine of the angle  between V and U.
67.
Express the polar equation r 2  6sin 2 in rectangular form.
68.
Write the pair 2 3, 2 in polar coordinates.
69,
The graph of r  5sin 2 is a:

a)
b)
c)
d)
70.

4-leaved rose
8-leaved rose
5-leaved rose
cardioid
Sketch the graph of r  2  2cos .