Pre-AP Pre-Calculus
1st Semester Exam Review
Show all necessary work on your own paper. These problems are NON-Calculator!
ANGLE BASICS
1.
Draw the angle 8/3. Find two coterminal angles—1 positive, 1 negative. What is
its reference angle?
2.
Convert 140 into radians.
3.
Convert 2/9 into degrees.
4.
 = 325°, find ref
5.
Which trig functions are positive in Q IV? Which trig functions are positive in Q II?
6.
sec  = 5/2 and terminates in Q IV. Find csc .
7.
cos  = 1/3 and terminates in Q I. Find cot .
8.
Find the coterminal angle of 140, where 360    720.
9.
Find sec (-7/3).
10.
Find cot (13/6).
11.
Point P is (-7, -4) and is on the terminal side of . Make of sketch of , ´, and the
reference triangle. Find the exact (fraction) ratios of the six trig functions.
TRIG GRAPHS AND SINUSOIDAL MODELS
12.
Determine whether the function is even, odd, or neither.
a) f(x) = x3
b) g(x) = x2 – 4
c) y = sin x
13.
Sketch a graph and list all transformations of y = 2 + 3 cos ( x 
14.
Sketch a graph and list all transformations: f(x) = 3 + 2 sin
15.
Write an equation for each graph shown below:
a)
y
d) y = cos x

3
).
1
(x   ) .
3
b)
y
4
200
100
2
x
0

x
0














-100

-200
16.
A mass suspended from a spring is pulled down a distance of 5 ft from its rest position.
The spring is released at time 0. The period is 1/9 cycle per hour. Sketch a graph and
write the equation that models the situation.
CALCULATOR – Round all answers to 0.001.
17.
Solve the following using algebraic methods.
a) Find f(x) for the given value.
b) Find the general solutions and the first 3 positive values of x for the given value of f(x)
f(x) = 3 + 2 sin
a) Find f(9)

4
 x  5
b) f(x) = 3.5

Pre-AP Pre-Calculus
18.
The original Ferris wheel, built by George Ferris for the 1893 World’s Fair, was much
larger and slower than its modern counterparts. It had a diameter of 250 feet and
contained 36 cars, each of which held 40 people. It made one revolution every 10
minutes and reached a maximum height of 264 feet. Grover Cleveland was given a
private ride. He got on and the wheel starting slowly turning.
a.) Sketch a graph of this sinusoid.
b.) Write an equation expressing Grover’s height above the ground in terms of time (in
minutes) since the Ferris wheel started turning.
c.) How high was Grover after 16 minutes?
d.) When was he 200 feet above the ground for the 4th time?
BACK TO NON-CALCULATOR!
VERIFYING TRIG IDENTITIES AND SOLVING TRIG EQUATIONS
19.
Simplify: cos x + sin x tan x.
20.
Simplify: (1 – cos x) (csc x + cot x).
sin 2 x  cos 2 x
21.
Simplify:
.
tan x
22.
Verify: sin2 x sec x csc x = tan x
1  cos 2 x
 tan x .
23.
Verify:
sin 2 x
24.
Find all 6 exact trig values for 75.
4
3
25.
If sin A  and is in quadrant 2, and cos B   and is in quadrant 3, find the following:
5
5
sin (A + B) = _____
cos (A + B) = _____
tan (A – B) = _____
sin 2A = _______
cos 2B = _______
2
26.
27.
28.
29.
Solve on the interval [0, 2): -2 cos x =
Solve 2 csc x – 4 = 0 for [0, 2).
Solve 4 cos2 x – 1 = 0 for [0, 2).
Find ALL solutions of 2 sin x = 3 .
30.
31.
Find ALL solutions of 3 tan 2x = 1.
Solve and give the principal value of x for 4 tan x + 4 = 0.
OTHER TRIG EQUATIONS AND TRIG INVERSES
32.
Which of the following functions are one-to-one?
sinx cosx tanx cotx secx cscx x
33.
x2
x3
|x|
x
3
x
a. Graph the parent functions for y = csc x, y = sec x, y = tan x, and y = cot x. Give the
period of each.
b. Graph the three trig inverse functions. List the domain and range of each.
Pre-AP Pre-Calculus
34.
 3
arccos 
 2 


37.
35.
tan 1
38.
36.
 3
arcsin 

 2 
 3
39.
7π 

csc-1  csc 
6 


  
sin 1  cos    
 3 


 8 
sin  arccos    
 17  

CALCULATOR – Round all answers to 0.001.
TRIANGLE TRIG – LAW OF SINES/COSINES AND VECTORS
40.
Given ∆ABC with mA = 25º , b = 10, and c = 12. Find a.
41.
Given ∆ABC with mA = 33º , b = 7, and c = 15. Find area.
42.
Given ∆ABC with mA = 40º , b = 8, and c = 10. Find perimeter.
43.
Given the following measures,  = 33;  = 79; b = 7, would you solve this triangle
using the law of sines or law of cosines? Why?
44.
In preparation for an outdoor rock concert, a stage crew must determine how far apart to
place the two large speaker columns on stage. What generally works best is to place
them at 50 angles to the center of the front row. The distance from the center of the
front row to each of the speakers is 10 ft. How far apart does the crew need to place the
speakers on stage?
45.
Two airplanes leave an airport at the same time. The first flies north at 175 km/h, and the
second plane flies southeast at 220 km/h. After 2 hours, how far apart are the planes?
46.
Two vectors, a and b , have magnitudes of 10 and 15 respectively. The angle between
them is 50º. Find a  b , and the angle this difference makes with a .
Find a  b , and the angle this difference makes with a .
47.
48.
An object moves 12 meters along a bearing of 90º and then turns and moves 18 more
meters along a bearing of 150º. Find the resultant of these two displacement vectors as a
distance.
Resolve the vector into horizontal and vertical components.
8
240º
Pre-AP Pre-Calculus
POLAR COORDINATES AND GRAPHS
49.
 
Graph the point  4,  .
 6
50.
Write the point in two additional ways—one with r > 0 and one with r < 0.
 2 
Find the rectangular coordinates for  6,
.
3 

51.
52.
53.
Convert (-1, -1) to polar coordinates.
Describe the graph: r = 3 + 5 sin θ
Describe the graph: r = 7 cos 2θ