Q: A photographer is 500 ft from a hot air balloon. If the balloon is $$s$$ feet above the ground, which function gives the angle $$\theta (in $$radians) between the ground and the line of sight to the balloon?

$$\theta = \tan$$^{-1}$$\left(\frac{s}{500}\right)$$

Q: A hot air balloon rises vertically from the ground. A photographer is 500 ft away from the launch site. As the balloon rises from 10 ft to 20 ft, how does the change in the viewing angle $$\theta$$ compare to the change as the balloon rises from 200 ft to 210 ft?

The change in $$\theta is $$greater from 10 ft to 20 ft.

Question 1

Which of the following is the correct domain and range for the inverse sine function $$y = \sin$$^{-1}$$(x)$$?

Accepted Answer

Domain: $$[-1, 1]$$, Range: $$[-\frac{\pi}{2}, \frac{\pi}{2}]$$

Question 2

Evaluate $$\sin$$^{-1}$$\left(\frac{\sqrt{3}}{2}\right)$$ without a calculator.

Accepted Answer

$$\frac{\pi}{3}$$

Question 3

Which of the following statements about the inverse cosine function $$y = \cos$$^{-1}$$(x) is $$TRUE?

Accepted Answer

Its range is $$[0, \pi]$$

Question 4

Find the exact value: $$\cos$$^{-1}$$\left(\frac{1}{2}\right).$$

Accepted Answer

$$\frac{\pi}{3}$$

Question 5

Which of the following best describes the end behavior of the inverse tangent function $$y = 	an$$^{-1}$$(x) as$$ $$x 	o \infty$$?

Accepted Answer

$$y 	o \frac{\pi}{2}$$

Question 6

A photographer is 500 ft from a hot air balloon. If the balloon is $$s$$ feet above the ground, which function gives the angle $$	heta (in $$radians) between the ground and the line of sight to the balloon?

Accepted Answer

$$	heta = 	an$$^{-1}$$\left(\frac{s}{500}ight)$$

Question 7

Evaluate $$\sin(\sin$$^{-1}$$(0.6)).$$

Accepted Answer

$$0.6$$

Question 8

Which of the following is NOT a property of the inverse tangent function $$y = 	an$$^{-1}$$(x)$$?

Accepted Answer

Its range is $$[0, \pi]$$

Question 9

If $$	heta = \cos$$^{-1}$$(x)$$, which of the following expressions is equivalent to $$\sin(	heta)$$?

Accepted Answer

$$\sqrt{1-x^2$}$$

Question 10

Suppose $$\sin(	heta) = -0.46$$ and $$	heta is in $$the range of the inverse sine function. What is $$	heta ($$rounded to two decimal places)?

Accepted Answer

$$-0.48$$ radians

Question 11

Which of the following compositions is always true for all $$x in $$the domain of $$\sin$$^{-1}$$?

Accepted Answer

$$\sin(\sin$$^{-1}$$(x)) = x$$

Question 12

Which of the following best explains why the basic trigonometric functions do not have inverses on their entire domains?

Accepted Answer

They are not one-to-one due to periodicity.

Question 13

If $$	heta = 	an$$^{-1}$$(x)$$, what is $$\sin(	heta) in $$terms of $$x$$?

Accepted Answer

$$\frac{x}{\sqrt{1+x^2$}}$$

Question 14

A hot air balloon rises vertically from the ground. A photographer is 500 ft away from the launch site. As the balloon rises from 10 ft to 20 ft, how does the change in the viewing angle $$	heta$$ compare to the change as the balloon rises from 200 ft to 210 ft?

Accepted Answer

The change in $$	heta is $$greater from 10 ft to 20 ft.

Question 15

Which of the following is the correct range for the inverse cosine function $$y = \cos$$^{-1}$$(x)$$?

Accepted Answer

$$[0, \pi]$$

Question 16

If $$	heta = \sin$$^{-1}$$(x)$$, what is $$\cos(	heta) in $$terms of $$x$$?

Accepted Answer

$$\sqrt{1-x^2$}$$

Question 17

Which of the following is the correct domain for the inverse tangent function $$y = 	an$$^{-1}$$(x)$$?

Accepted Answer

$$(-\infty, \infty)$$

Question 18

Evaluate $$	an$$^{-1}$$(1).$$

Accepted Answer

$$\frac{\pi}{4}$$

Question 19

Which of the following is a correct interpretation of $$\sin$$^{-1}$$(x)$$?

Accepted Answer

The unique angle $$y in$$ $$[-\frac{\pi}{2}, \frac{\pi}{2}]$$ such that $$\sin(y) = x$$

Inverse Trigonometric Functions: Definitions, Properties, and Applications

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