Q: A photographer is standing 500 ft from a hot air balloon. If the balloon is $$s$$ feet above the ground, which function gives the angle $$\theta of $$elevation from the photographer to the balloon?

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

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: $$\left[ -\frac{\pi}{2}, \frac{\pi}{2} \right]$$

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: $$\sin\left( \cos$$^{-1}$$\left( \frac{3}{5} \right) \right).$$

Accepted Answer

$$\frac{4}{5}$$

Question 5

A photographer is standing 500 ft from a hot air balloon. If the balloon is $$s$$ feet above the ground, which function gives the angle $$	heta of $$elevation from the photographer to the balloon?

Accepted Answer

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

Question 6

Which of the following best describes the end behavior of the inverse tangent function $$y = 	an$$^{-1}$$ x$$?

Accepted Answer

$$y$$ approaches $$\frac{\pi}{2} as$$ $$x$$ approaches $$\infty$$

Question 7

Evaluate $$\sin$$^{-1}$$(-0.46)$$ using a calculator and round to three decimal places.

Accepted Answer

$$-0.478$$

Question 8

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

Accepted Answer

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

Question 9

Suppose $$	heta = 	an$$^{-1}$$(x). $$Which of the following expressions is equivalent to $$\sin 	heta$$?

Accepted Answer

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

Question 10

If $$\cos 	heta = \frac{3}{5}$$ and $$	heta is in$$ $$[0, \pi]$$, what is $$\sin 	heta$$?

Accepted Answer

$$\frac{4}{5}$$

Question 11

Which of the following is NOT a property of the inverse sine function?

Accepted Answer

Its domain is $$(-\infty, \infty).$$

Question 12

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

Accepted Answer

$$\frac{\sqrt{3}}{2}$$

Question 13

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

Accepted Answer

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

Question 14

If $$	heta = 	an$$^{-1}$$\left( \frac{s}{500} ight)$$, which real-world scenario does this equation model?

Accepted Answer

The angle of elevation from a point 500 ft away to a balloon at height $$s ft.$$

Question 15

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

Accepted Answer

$$\left( -\frac{\pi}{2}, \frac{\pi}{2} \right)$$

Question 16

If $$\sin$$^{-1}$$(x) = \frac{\pi}{6}$$, what is the value of $$x$$?

Accepted Answer

$$\frac{1}{2}$$

Question 17

Which of the following is true about the rate of change of $$	heta = 	an$$^{-1}$$\left( \frac{s}{500} ight) as$$ $$s$$ increases?

Accepted Answer

The rate of change is greater for small $$s$$ and decreases as $$s$$ increases.

Inverse Trigonometric Functions: Definitions, Properties, and Applications

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