Precal/Trigonometry
Midterm Review
6.1 – Graphing angles and radian measure:
1. Find an angle in the interval [−4𝜋, −2𝜋) that is conterminal with the given angle.
5𝜋
103𝜋
a.
b. −
2.
4
14
Find an angle in the interval [4𝜋, 6𝜋) that is conterminal with the given angle.
a.
5𝜋
7
b.
−
91𝜋
3
3. Scientists created a new unit of time called the farganfugan.
There are 22 farganfugans in one hour. On a clock
labeled with 22 tick marks, one for each farganfugan in an hour, find how far the tip of a 7 inch long farganfugan hand
moves in 13 farganfugans.
4.
A manufacturing accident caused a batch of clocks to be produced that had 14 hour markers and no minute markers.
As a consequence, the minute hand of the clock now makes 1 full revolution every 70 minutes. If the minute hand is 3
inches long, find the distance that the tip of the minute hand moves in 1 minute. What about in 36 minutes?
8.1 – Right triangle trig:
5.
Find the values of the six trig functions of the angle 𝜃.
6.
Label each special triangle with acceptable side lengths.
7.
𝜋/6
𝜋/4
A small plane is flying at an altitude of 10,000 ft. The radar tower of an airport spots the plane flying at an angle of
elevation of 57°. Find the distance from the plane to the base of the radar tower.
8.
A police helicopter is flying at an altitude of 800 ft. A stolen car is sighted at an angle of depression of 72°. Find the
distance from the helicopter to the car.
6.2 and 6.3 – Trig functions of any angle, the unit circle, and Fundamental identities:
9.
Find the exact value of each expression or write undefined if necessary.
a. cos 210°
b. sin
3𝜋
4
c. tan
11𝜋
6
d. sec 210°
e. cot
7𝜋
3
f. 4 tan
17𝜋
𝜋
cos
3
4
+ sin
7𝜋
𝜋
csc
6
6
10.
11.
g. cos 90°
a. sin
7𝜋
2
21𝜋
−
4
b. cos �
𝜋
4
𝜋
1+𝑔� �
4
1−𝑓� �
c. tan �−
1000𝜋�
3𝜋
2
j. sec(−630°)
13𝜋
3
k. sin �
𝜋
2
l. tan � + 31𝜋�
+ 80𝜋�
+ 801𝜋�
17𝜋
�−
2
b. 𝑓 �
13𝜋
�+
3
𝑔�
𝜋
3
35𝜋
2𝜋
5𝜋
c. 𝑓 �√3 ⋅ 𝑔 � 12 ⋅ ℎ � 4 ���
ℎ� �
Let 𝑓(𝜃) = 3 sin 𝜃, 𝑔(𝜃) = cos (3𝜃), and ℎ(𝜃) = tan3 𝜃. Find the exact value of each expression.
1−𝑔(𝜋/4)
17𝜋
�−
2
b. 𝑓 �
1+𝑓(5𝜋/6)
13𝜋
�+
3
𝑔�
𝜋
3
ℎ� �
In each case, find the exact value of each of the remaining trig functions of 𝜃.
a. csc 𝜃 = −
d. sec 𝜃 = −
14.
25𝜋
2
Let 𝑓(𝜃) = 2 sin 𝜃, 𝑔(𝜃) = cos (2𝜃), and ℎ(𝜃) = tan2 𝜃. Find the exact value of each expression.
a.
13.
i. cot
Find the exact value of each expression.
a.
12.
3𝜋
2
h. sin
7
4
5
4
and cos 𝜃 < 0
b. tan 𝜃 = −
2
3
e. tan 𝜃 = −
and sin 𝜃 ⋅ cos 𝜃 > 0
c. cos 𝜃 =
and csc 𝜃 > 0
2
3
3
5
and 5𝜋 < 5𝜃 + 2𝜋 < 8𝜋
and 0 < 5𝜃 + 4𝜋 < 4𝜋
Use identities to find the exact value of each expression. Do not use a calculator.
a. sec 8 cot 8 sin 8
d. sin 8 csc 8
b. cos2(33°) − sin2 (57°)
e. sin2(33°) + sin2 (57°)
h. csc 𝜃 sin 𝜃 − tan 𝜃 cot 𝜃
𝜋
8
i. csc sec
3𝜋
8
c. 1 − tan2 1 + sec 2 1
f. tan2 1 − sec 2 1
− tan
3𝜋
𝜋
cot
8
8
g. 1 − cot 2 20° + sec 2 70°
j. sin2 𝑥 − 2 tan2 𝑥 + 2cot 2 𝑥 + 2sec 2 𝑥 − 2csc 2 𝑥 + cos 2 𝑥
15. Let sin 𝜃 = 𝑎, cos 𝜃 = 𝑏, and tan 𝜃 = 𝑐.
a. 3 sin(−𝜃) − sin 𝜃
c. 3 sin(−𝜃) − sec(−𝜃)
Write each expression using only 𝑎, 𝑏, and 𝑐.
b. 3cos(−𝜃 − 6𝜋) + 2 sin(𝜃 + 2000𝜋) − 4cot(−𝜃 + 17𝜋)
d. 4cos(𝜃 − 6𝜋) + 2 csc(𝜃 + 2000𝜋) − 4tan(−𝜃 + 17𝜋)
6.4, 6.5, 6.6 – Graphing trig functions:
16. Use the amplitude, period, and phase shift to graph one period of each functions.
a. 𝑦 = 3 sin 4𝑥
b. 𝑦 = −2 cos 2𝑥
e. 𝑦 = −3 cos(𝑥 + 𝜋)
h. 𝑦 = sin(2𝑥) + 1
3
c. 𝑦 = 3 cos
𝜋
f. 𝑦 = cos �2𝑥 + �
2
4
𝑥
3
d. 𝑦 = − sin 𝜋𝑥
𝜋
g. 𝑦 = −3 sin � 𝑥 − 3𝜋�
3
e. sin(𝜃 + 𝜋)
17. Graph two periods of each function.
a. 𝑦 = 4 tan 𝑥
1
𝜋
4
𝜋
e. 𝑦 = − cot 𝑥
2
𝜋
b. 𝑦 = −2 tan 𝑥
2
i. 𝑦 = 3 sec(𝑥 + 𝜋)
c. 𝑦 = − tan �𝑥 − �
𝜋
4
f. 𝑦 = 2 cot �𝑥 + �
g. 𝑦 = 3 sec 2𝜋𝑥
2
5
j. 𝑦 = csc(𝑥 − 𝜋)
d. 𝑦 = 2 cot 3𝑥
h. 𝑦 = −2 csc 𝜋𝑥
2
7.1, 7.2 – Inverse trig functions:
18.
19.
Find the exact value of each expression.
a. sin−1 1
√2
2
�
4
5
e. tan �cos−1 �− ��
d. sin−1 �−
√3
�
2
3
b. sin(cos−1 0)
1
3
1
2
e. cos −1 �− �
f. tan−1 �−
√3
�
3
3
4
c. cos �tan−1 4�
d. tan �sin−1 �− ��
f. sin �tan−1 �− ��
Find the exact value of each expression or write undefined if necessary.
a. sin �sin−1
√7
8
i. cos−1 �cos �−
b. sin �sin−1
�
e. cos(cos −1 𝜋)
21.
c. tan−1 1
Find the exact value of each expression.
a. cos �sin−1
20.
b. cos−1 1
27𝜋
��
14
𝜋
7
8
√7
f. sin−1 �sin �
�
3
c. cos �cos−1 4�
g. sin−1 �sin
2𝜋
�
3
d. cos(cos−1 3.14)
𝜋
4
h. cos −1 �cos �− ��
Graph each function, and state the domain and range.
a. 𝑦 = sin−1 𝑥
b. 𝑦 = cos −1 𝑥
c. 𝑦 = tan−1 𝑥
7.3 – Trig equations:
22.
Find all solutions to each equation.
a. cos 𝑥 = −1
f. tan 2𝑥 = 0
b. sin 𝑥 = −
√3
2
𝜋
6
c. tan 𝑥 = −√3
g. 2 sin �3𝑥 − � − 5 = −6
d. 2 sin 𝑥 − 5 = 7 sin 𝑥
e. cos 3𝑥 = −
√2
2
23.
Solve on the interval [0,2𝜋).
a. 2 cos 𝑥 − 5 = 7 cos 𝑥
b. sin 3𝑥 = −
24. Find ALL vertical asymptotes for each function.
a. 𝑦 = tan 9𝑥
b. 𝑦 = csc 17𝑥
√2
2
c. cot 2𝑥 = 1
𝜋
4
c. 𝑦 = sec �10𝑥 + �
5
2
d. 6 cos 𝑥 = √27