Name_____________________
Fall 2015
This review is an overview of the semester. This DOES NOT cover 100% of the material discussed
in class. Rather, this is a sampling of the topics you have learned. You are encouraged to go back
and look at your notes, test reviews, quizzes and tests you have taken this year in Precalculus.
1. Sketch 
4
radians in standard position. Determine two coterminal angles, one positive and one negative.
3
2. Convert 480o to radians.
3. Convert
5
radians to degrees. Round to the nearest degree.
7
4. Write an equation for each trig function described below.
a. sine function: amplitude = 3/2, period = 2 , vertical shift = -2, phase shift = 

2
.
b. cosine function: amplitude = ½, period = 2, no vertical shift, reflection in the x-axis.
c. cosecant function: period = 2, vertical shift = 2, phase shift = 1.
5. Dr. Thom Dawson standing on flat ground 62 ft from the base of a Douglas fir measures the angle of elevation
to the top of the tree as 72o. What is the height of the tree?
6. Your football has landed at the edge of the roof of your school building. When you are 25 feet from the base of
the building, the angle of elevation to your football is 21 degrees. How high off the ground is your football?
7. Write a trig function from the following graphs
a.


b.






8. Verify the following trig identities
cos 2 
 sin 
a. 1 
1  sin 
c.
b. sec a  tan a 
1  cot 2 x
 1  2 cos 2 x
1  cot 2 x
cos a
1  sin a
d. 𝑐𝑜𝑠𝑥(𝑡𝑎𝑛2 𝑥 + 1) = 𝑠𝑒𝑐𝑥
8, continued. Verify the following trig identities.
e. 𝑠𝑒𝑐 2 𝑥𝑐𝑜𝑡𝑥 − 𝑐𝑜𝑡𝑥 = 𝑡𝑎𝑛𝑥
1
f. 𝑡𝑎𝑛𝜃𝑐𝑠𝑐𝜃 = 𝑐𝑜𝑠𝜃
1
g. 𝑡𝑎𝑛𝑥𝑐𝑠𝑐𝑥𝑠𝑖𝑛𝑥 = 𝑐𝑜𝑡𝑥
9. Use the given values and trigonometric identities to evaluate (if possible) all six trigonometric functions
EXACTLY (no decimals!) (Hint: use the x, y & r ratios)
3
5
2
3
a. 𝑠𝑖𝑛𝑥 = , cos>0
2
3
b. 𝑡𝑎𝑛𝜃 = , 𝑐𝑜𝑠 < 0
c. 𝑐𝑜𝑠𝑥 = , tan<0
10. Use trigonometric identities to simplify the expressions. Then match with one of the following.
A. 𝑠𝑒𝑐𝜃𝑐𝑠𝑐𝜃
ANSWER CHOICES
D. cos2x
E. −2𝑡𝑎𝑛2 𝜃
B. cot2x C. 1+sinx
1
𝑡𝑎𝑛𝜃
F. 1
______c. 𝑡𝑎𝑛2 𝑥(𝑐𝑠𝑐 2 𝑥 − 1)
_____a. 𝑐𝑜𝑡 2 𝑥+1
______b. 1−𝑐𝑜𝑠2 𝜃
______d. 𝑐𝑜𝑡 2 𝑥(𝑠𝑖𝑛2 𝑥)
______e. 𝑐𝑜𝑠 2 𝑥 + 𝑐𝑜𝑠 2 𝑥𝑐𝑜𝑡 2 𝑥 ______f.
3 x  1

11. Given the function f ( x)  4
 x2

1
1
−
𝑐𝑠𝑐𝜃+1
𝑐𝑠𝑐𝜃−1
for
for
x  1
-1  x  1 , evaluate f(-2), f(-½ ), and f(3).
for
x 1
12. Use the graph of f to sketch the graph of g. (You can use the same graph)
G. sin2x
a) g(x) = f(x) + 2
b) g(x) = f(-x) – 1
c) g(x) = -2f(x)
d) g(x) = f(2x)+4
13. If f ( x)  x 2 and g ( x)  4 x  5 , find the following.
a) (f+g)(x)
b) (fg)(x)
c) (f o g)(x)
14. Determine the intervals over which the function is increasing, decreasing, or constant.
a)
b)
15. Determine if the function is even or odd.
a)
2 x x2  3
b) x 5  4 x  7
c) x 4  20 x 2
16. Find the inverse of the following functions.
a)
b) f ( x ) 
f ( x)  5 x  7
 x 2 -2

17. Graph f ( x)  5
8 x  5

for
for
for
x  2
-2  x  0 .
x0
x 1
c) x3  2
18. If csc   
13
13
and sec   
, find
3
2
a) sin  
b) cos  
c) tan  
d) cot  
19. Trevor is skiing down a mountain with a vertical height of 1500 feet. The vertical distance from the top of the
mountain to the base is 3000 feet. What is the angle of elevation from the base to the top of the mountain?
20. Find the period and amplitude of the following functions
a)
y  3sin 10 x 
b) y 
1
 2x 
cos  
2
 3 
21. Sketch the graph of the following functions.
a)
y  3cos  x   
 2 
x
 3 
b) y  2  sin 
22. When tuning a piano, a technician strikes a tuning fork for the A above middle C and sets up a wave motion
that can be approximated by y  0.001sin 880 x  , where x is time in seconds. What is the period of the
function?
23. Evaluate the expressions. Give an exact answer (NO DECIMALS!)
a)

arctan  3

 1
 2
b) arccos   
 3
 2 


c) sin 1 

 2 


d) sin arccos    
3

24. Use a calculator (in radians mode) to evaluate the following. Then convert your answer to degrees.
a) arccos  0.41
b) arctan  0.92 
25. A 150 foot tall cellular tower is placed on top of a mountain that is 1200 feet above sea level. What is the
angle of depression from the top of the tower to a cell phone user who is 5 horizontal miles away and 400 feet
above sea level? (HINT: 1 mile = 5280 feet)
26. An observer in a lighthouse 350 feet above sea level observes two ships directly offshore. The angles of
depression to the ships are 4o and 6.5o. How far apart are the ships?
30. If (12,16) is on the terminal side of an angle  in standard position, find the following.
(Hint: use the x, y & r ratios)
a) sin  
b) cos  
c) tan  
d) csc  
e) sec  
f) cot  
31. If sec   
6
and tan    0 , find the following.
5
a) sin  
b) cos  
c) tan  
d) csc  
e) sec  
f) cot  
32. Solve the equations in the interval [0, 2 ) .
a) sin( x)  3  sin( x)
b) 3csc2 ( x)  4
c) 2cos 2 ( x)  cos( x)  1
d) cos2 ( x)  sin( x)  1
e) 2sin(2 x )  2  0
f) cos(4 x) cos( x) 1  0
33. Use the Law of Sines and Law of Cosines to solve the triangle ∆ABC. Round sides to the nearest tenth, and
angles to the nearest whole degree.
a)
B  72, C  82, b  54
c) a  5, b  8, c  10
b) a = 7, b = 12, A = 37°
d) B  110, a  4, c  4