Q: A photographer is 500 ft from a hot air balloon rising vertically. 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}$$\left( \frac{s}{500} \right)$$

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

Accepted Answer

$$\frac{2\pi}{3}$$

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 rising vertically. 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 $$\sin\left( \sin$$^{-1}$$\left( -\frac{1}{2} \right) \right).$$

Accepted Answer

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

Question 8

Which of the following is the correct composition for $$\cos\left( \sin$$^{-1}$$ x \right)$$ for $$x in$$ $$[-1, 1]$$?

Accepted Answer

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

Question 9

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

Accepted Answer

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

Question 10

Use a calculator to approximate $$\sin$$^{-1}$$(-0.46) in $$radians. Which value is closest?

Accepted Answer

$$-0.48$$

Question 11

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

Accepted Answer

Its domain is $$(-1, 1)$$

Question 12

Suppose $$	heta = \cos$$^{-1}$$(x)$$ for $$x in$$ $$[-1, 1]. $$What is $$\sin(	heta) in $$terms of $$x$$?

Accepted Answer

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

Question 13

Which of the following best explains why the 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 $$\sin(	heta) = x$$ and $$	heta is in$$ $$\left[ -\frac{\pi}{2}, \frac{\pi}{2} ight]$$, what is $$	heta$$?

Accepted Answer

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

Question 15

A hot air balloon rises from ground level. The angle of elevation from a point 500 ft away increases from $$\$$theta_1$$ to $$\$$theta_2 as $$the balloon rises from 10 ft to 20 ft. Why is the change in angle greater at lower heights?

Accepted Answer

Because the inverse tangent function increases more rapidly for small values of $$s.$$

Question 16

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 17

If $$	heta = 	an$$^{-1}$$\left( \frac{s}{500} ight)$$, what does $$	an(	heta)$$ equal?

Accepted Answer

$$\frac{s}{500}$$

Question 18

Which of the following is the correct composition for $$	an\left( \cos$$^{-1}$$ x ight)$$ for $$x in$$ $$[-1, 1]$$?

Accepted Answer

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

Inverse Trigonometric Functions: Definitions, Properties, and Applications

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