Precalculus Part 2
Final Exam Review NON-CALCULATOR SECTION
Name
1. Give the exact value of the following expressions. Show any work.

5
a. csc
b. tan
3
2
c. sec(120 )
d. cot

4
2. Solve the following equations for  , if 0    2 .
3
1
a. cos   
b. sin  
2
2
3. Find the exact values of the five remaining trigonometric functions of  .
3
Given: tan  
and 0    
4
sin   _________
cos   _________
csc   _________
sec   _________
cot   _________
4. Evaluate the following expressions.
1
a. sin 1
b. arctan 3
2
c. sec1 2
c. tan  
 3
3
Precalculus Part 2
Final Exam Review NON-CALCULATOR SECTION
5. Sketch the following graphs. Show two periods.
a. y  3sin 2 x  1
b. y   cos

2
x
c. y  tan( x   )
d. y  csc  x
page 2
Precalculus Part 2 Final Exam Review CALCULATOR SECTION
Name
All work must be shown on the test . Be sure to stay organized and neat, so that your work may be
graded properly. Remember to label your answers with the proper units, where applicable.
1. Calculate the arc length of the arc described by the following given information. Leave answers in
terms of  .
3
a.  
, r  12 cm.
b.   50 and r  9 in.
4
2. Angular Velocity. Find the angular velocity in radians per second, for a pulley turning at 1800
revolutions per minute.
3. Linear Velocity. Calculate the linear velocity of a point located 12 inches from the center of a disk
rotating at 7 radians per second.
4. A Ferris wheel 250 feet in diameter takes 45 seconds to make one complete revolution.
a. Calculate its angular velocity in radians per second.
b. If you sat on the rim of the Ferris wheel, what would your linear velocity be in feet per minute?
5. Solve for x or  . Also, find the area of each triangle.
a.
b.
10
62˚
x
50˚
x
12
Precalculus Part 2
Final Exam Review CALCULATOR SECTION
8

c.
page 2
d.
5
x
20
40˚
25
15˚
e.
f.
5
8
25

105˚
7
x
6. An observer in a lighthouse 250 feet above sea level spots a ship in the harbor. The angle of depression
of the observer is 50˚. How far is the ship from the base of the lighthouse?
7. There was a major food fight in the cafeteria on Friday afternoon. Gregg, an eyewitness to the fight,
told Mrs. Bombaci that Peter wasn’t the one who hit Kira in the head with a plate of ketchup. Gregg
claimed that Peter was 20 feet from Kira. Gregg told Mrs. Bombaci, “I was 50 feet from Peter and 75
feet from Kira when the fight began. I saw the whole thing.”
Use the law of cosines to determine whether or not Gregg was telling the truth. Be sure to justify the
reasoning behind your conclusion.
Precalculus Part 2
Final Exam Review CALCULATOR SECTION
page 3
8. Kitchen Island.
a. The kitchen island shown in the picture, has acute angles of 20˚ each. The longest side of the
triangle measure 10 feet. What are the lengths of the two legs of the triangle?
refrigerator
b. Find the total length of the sides of another triangular island that contains an angle that measures
67˚ and has adjoining sides 5 feet and 9 feet. For the greatest efficiency, the total length of the
sides of a kitchen island should be less than 26 feet. Would the island described be efficient?
Justify your answer.
9. Verify the following.
1
1

1
a.
2
sin  tan 2 
d. tan( x  y ) 
cos y  sin y cot x
cot x cos y  sin y
sec 
sin 2 
b.
1 
cos 
cos 2 
c.
cos( x  y )
 cot x cot y  1
sin x sin y
1
e. csc 2 x  sec x csc x
2
Precalculus Part 2
Final Exam Review CALCULATOR SECTION
page 4
10. Solve each of the following equations over the interval 0  x  360 . Round to the nearest hundredth.
a. 3cos x  4  2  5cos x
b. cos 2x  2cos x  0
c. 5cos 2 x  2  0
11. Polar Graphing. Graph the following points on the polar graph provided.

a. Point A (3, )
6

)
b. Point B (4,
3
2
)
c. Point C (2,
3
12. Convert from rectangular to polar.
a. ( 4, 4)
b. (5 2, 5 2)
13. Convert from polar to rectangular.
a. (3,  )
b. ( 1,
5
)
3
c. x 2  y 2  36
c.  
3
4
d. 3 x  2 y  1  0
d. r  4cos
Precalculus Part 2
Final Exam Review CALCULATOR SECTION
page 5
14. Consider the polar equation r  3cos .
a. Complete the table at the left for values of r and the given  values using
    2 3 5
7 5 4 3 5 7 11
  0, , , , , , , ,  , , , , , , ,
.
6 4 3 2 3 4 6
6 4 3 2 3 4 6
b. Plot the points ( x, y ) generated in part (a) and sketch the graph of the parametric equations. Be
sure to show the orientation of the graph.
c. Convert to a rectangular equation.
15. Bearing. A ship leaves the port of Miami with a bearing of 100˚ and a speed of 15 knots. After 1 hour,
the ship turns 90˚ toward the south.
a. What is the new bearing?
b. After 2 hours since leaving port and maintaining the same speed, what is the bearing of the ship
from port?
Precalculus Part 2
Final Exam Review CALCULATOR SECTION
page 6
16. Parametric Equations. Consider the parametric equations x  sin  and y  3cos  .
  3 5 3 7
, 2 .
c. Complete the table for x- and y-values using   0, , ,  , , ,
4 2 4
4 2 4
θ
x
y
d. Plot the points ( x, y ) generated in part (a) and sketch the graph of the parametric equations. Be
sure to show the orientation of the graph.
e. Find the rectangular equation by eliminating the parameter.
16.
Projectile Motion. Kayla hit a golf ball with an initial velocity of 100 feet per second at an
angle of 35˚ to the horizontal.
1
a. Using the formulas x  v0 (cos  )t and y   gt 2  v0 (sin  )t  h , find the parametric
2
equations that describe the position of the ball as a function of time.
b. How long is the golf ball in the air?
Precalculus Part 2
Final Exam Review CALCULATOR SECTION
page 7
c. When is the ball at its maximum height? Determine the max height of the ball.
d. Determine the distance the ball traveled.
17. Vectors.
a. Find the component form and magnitude of the vector u that has initial point P and terminal
point Q. P = (2,4) and Q = (-3, -2).
b. Find a unit vector in the same direction as u. u = 5,12 .
c.
Given: u = 2,3 and v =
4,1 . Find u v .
d. Find the component form of the vector that has the given magnitude and direction angle θ.
v  5 and   60 .