Name:
Date:
Period:
Unit 8 Practice Test #2
Learning Targets: 8C and 8D
Complete the problems below, show your work, and write your answer in the blank provided.
Learning Target 8C: I can create and use graphs of transformations of inverse trigonometric
functions to solve problems.
Find the approximate value of each expression. Express your answer in radians.
1. arctan(2.37)
2. cos −1 (−0.853)
1.1715
2.5925
Find the approximate value of each expression. Express your answer in degrees rounded to
the nearest tenth.
3. arccos(0.67)
4. sin−1 (0.362)
47.9°
21.2°
Show the steps to find the exact value of the following compositions.
𝜋
5. arcsin (cos 3 )
1
𝜋
arcsin( ) =
2
6
6. sin(tan−1 (−1))
7𝜋
√2
sin ( ) = −
4
2
3
7. cot(sin 5)
4
3
Find an algebraic expression equivalent to the given expression. Justify your answer.
8. sin(cos −1 𝑥)
9. cos(tan−1 𝑥)
1
√1 − 𝑥 2
√1 + 𝑥 2
=
√1 + 𝑥 2
1 + 𝑥2
10. Samantha measures the angle of elevation, 𝜃, from where she is standing to a plane
flying overhead. The plane remains at a constant height of 650 feet. Write an
equation that relates 𝜃 to the horizontal distance, 𝑥, from Samantha’s location to the
plane.
650
tan−1 (
)=𝜃
𝑥
650 ft
θ
x
Learning Target 8D: I can create and use graphs of transformations of composite trigonometric
functions to solve problems.
Determine if the following functions will result in a sinusoidal function, and if it does, what is
the period of the function? Explain how you can tell if the function is a sinusoid.
12. 𝑓 (𝑥 ) = 2 tan 𝑥 + 3 sin 2𝑥
11. y = sin 𝜋𝑥 − 2 cos 𝜋𝑥
Yes; 𝑝𝑒𝑟𝑖𝑜𝑑: 2
Yes: 𝑝𝑒𝑟𝑖𝑜𝑑: 𝜋
Since the periods are the same it is still a
sinusoidal function
The periods are the
same
State the domain and range of the following functions.
14. 𝑓 (𝑥 ) = |cot 𝑥 |
13. 𝑦 = (cos 𝑥 )2
Domain: All Reals
Domain: {𝑥|𝑥 ≠ 𝜋 + 𝑛𝜋}
Range: [0, 1]
Range: (0, 225)
15. 𝑓 (𝑥 ) = (cos 𝑥 + 1)3
Domain: All Reals
Range: [0, 1]
Sketch the graph of the following functions for −𝟐𝝅 ≤ 𝒙 ≤ 𝟐𝝅. State whether or not the
function appears to be periodic. Explain.
3
16. 𝑦 = 2𝑥 cos 3𝑥
17. 𝑓 (𝑥 ) = 𝑥 sin 𝑥
7
6
5
4
3
2
1
-3π/2
-π
-π/2
-1
-2
-3
-4
-5
-6
-7
y
7
6
5
4
3
2
1
x
π/2
π
3π/2
-3π/2
-π
-π/2
-1
-2
-3
-4
-5
-6
-7
y
x
π/2
π
What is the dampening factor of the following functions? Explain how this factor affects the
shape of the graph.
19. 𝑓 (𝑥 ) = 4𝑥 sin 𝑥
18. 𝑦 = 𝑥 2 cos 2𝑥
Factor: 𝑥 2
Factor: 4𝑥
Bounds cos(2𝑥) by
𝑥 2 and −𝑥 2
Bounds sin 𝑥 by 4𝑥
and −4𝑥
3π/2