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. −
4
2.
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 in farganfugans, 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 hour minute hand of the clock now makes 1 full revolution every 70 minutes. If the hour minute
hand is 3 inches long, find the distance that the tip of the hour 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.
𝜋/6
𝜋/4
7.
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
g. cos 90°
10.
7𝜋
2
21𝜋
−
4
b. cos (
1000𝜋)
𝜋
4
𝜋
1+𝑔( )
4
1−𝑓( )
17𝜋
)−
2
b. 𝑓 (
c. tan (−
13𝜋
)+
3
𝑔(
1−𝑔(𝜋/4)
1+𝑓(5𝜋/6)
17𝜋
)−
2
b. 𝑓 (
𝜋
k. sin ( 2 + 80𝜋)
l. tan ( 2 + 31𝜋)
13𝜋
3
+ 801𝜋)
𝜋
3
35𝜋
2𝜋
ℎ( )
5𝜋
c. 𝑓 (√3 ⋅ 𝑔 ( 12 ⋅ ℎ ( 4 )))
13𝜋
)+
3
𝑔(
𝜋
ℎ (3 )
In each case, find the exact value of each of the remaining trig functions of 𝜃.
7
a. csc 𝜃 = − 4 ⁡⁡and cos 𝜃 < 0
2
3
b. tan 𝜃 = − 3 ⁡⁡and⁡ csc 𝜃 > 0
5
c. cos 𝜃 = 5 ⁡⁡and⁡0 < 5𝜃 + 4𝜋 < 4𝜋
2
d. sec 𝜃 = − 4 ⁡⁡and sin 𝜃 ⋅ cos 𝜃 > 0
14.
3𝜋
j. sec(−630°)
Let 𝑓(𝜃) = 3 sin 𝜃, 𝑔(𝜃) = cos⁡(3𝜃), and ℎ(𝜃) = tan3 𝜃. Find the exact value of each expression.
a.
13.
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.
12.
i. cot
Find the exact value of each expression.
a. sin
11.
3𝜋
2
h. sin
e. tan 𝜃 = − 3 ⁡⁡and⁡⁡5𝜋 < 5𝜃 + 2𝜋 < 8𝜋
Use identities to find the exact value of each expression. Do not use a calculator.
b. cos2(33°) − sin2 (57°)
a. sec 8 cot 8 sin 8
d. sin 8 csc 8
e. sin2(33°) + sin2 (57°)
h. csc 𝜃 sin 𝜃 − tan 𝜃 cot 𝜃
𝜋
i. csc 8 sec
3𝜋
8
c. 1 − tan2 1 + sec 2 1
f. tan2 1 − sec 2 1
− tan
g. 1 − cot 2 20° + sec 2 70°
3𝜋
𝜋
cot 8
8
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
Write each expression using only 𝑎, 𝑏, and⁡𝑐.
b. 3cos(−𝜃 − 6𝜋) + 2 sin(𝜃 + 2000𝜋) − 4cot(−𝜃 + 17𝜋)
c. 3 sin(−𝜃) − sec(−𝜃)
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
c. 𝑦 = 3 cos
3
𝜋
2
4
f. 𝑦 = cos (2𝑥 + )
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.
𝜋
a. 𝑦 = 4 tan 𝑥
𝜋
b. 𝑦 = −2 tan 𝑥
c. 𝑦 = − tan (𝑥 − )
4
1
𝜋
2
2
𝜋
e. 𝑦 = − cot 𝑥
d. 𝑦 = 2 cot 3𝑥
4
f. 𝑦 = 2 cot (𝑥 + )
g. 𝑦 = 3 sec 2𝜋𝑥
2
h. 𝑦 = −2 csc 𝜋𝑥
5
i. 𝑦 = 3 sec(𝑥 + 𝜋)
j. 𝑦 = csc(𝑥 − 𝜋)
2
7.1, 7.2 – Inverse trig functions:
18.
Find the exact value of each expression.
a. sin−1 1⁡
19.
b. cos−1 1
d. sin−1 (−
√3
)
2
1
e. cos −1 (− 2)
f. tan−1 (−
√3
)
3
Find the exact value of each expression.
a. cos (sin−1
√2
⁡)⁡
2
3
b. sin(cos−1 0)
4
3
c. cos (tan−1 4)
d. tan [sin−1 (− 4)]⁡
1
e. tan [cos−1 (− 5)]
20.
c. tan−1 1
f. sin [tan−1 (− 3)]
Find the exact value of each expression or write undefined if necessary.
a. sin (sin−1
√7
⁡)⁡
8
e. cos(cos −1 𝜋)
i. sin−1 (sin
2𝜋
)
3
b. sin (sin−1
8
⁡)⁡
√7
1
f. tan (tan−1 2)
j. sin−1 (sin
3
c. cos (cos −1 4)
𝜋
g. tan(tan−1 2)
7𝜋
)
9
d. cos(cos−1 3.14)⁡
h. sin−1 (sin 7 )
𝜋
k. cos−1 [cos (− 4 )]
l. cos −1 [cos (−
27𝜋
)]
14
7.3 – Trig equations:
21.
For each equation, find all solutions.
1
a. cos 𝑥 = − 2
22.
b. sin 𝑥 =
√2
2
c. 2 sin 𝑥 + 1 = 0
d. √3 tan 𝑥 − 1 = 0
e. cos 2𝑥 = −1
Solve the equation on [0,2𝜋).
a. cos 2𝑥 = −1
b. 4 sin 3𝑥 − 4 = 0
e. cos2 𝑥 − 2cos 𝑥 = 3
i. sin 𝑥 = tan 𝑥
𝑥
c. tan 2 = −1
f. 2 cos 2 𝑥 − sin 𝑥 = 1
j. sin 𝑥 = −0.6031
d. tan 𝑥 = 2 cos 𝑥 tan 𝑥
g. 4 sin2 𝑥 = 1
k. 5 cos2 𝑥 − 3 = 0
h. cos 2 𝑥 − sin2 𝑥 − sin 𝑥 = 1
l. sec 2 𝑥 = 4 tan 𝑥 − 2