Q: Which of the following is the correct definition of the inverse tangent function?

$$y = \tan$$^{-1}$$ x if $$and only if $$x = \tan y$$, $$-\frac{\pi}{2} \u003c y \u003c \frac{\pi}{2}$$

Question 1

Which of the following best describes the domain and range of the inverse sine function $$y = \sin$$^{-1}$$ x$$?

Accepted Answer

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

Question 2

Find the exact value of $$\sin$$^{-1}$$\left(-1\right).$$

Accepted Answer

$$-\frac{\pi}{2}$$

Question 3

Evaluate $$\cos$$^{-1}$$\left(\frac{1}{2}\right)$$ and select the correct answer.

Accepted Answer

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

Question 4

Which of the following is the correct definition of the inverse tangent function?

Accepted Answer

$$y = 	an$$^{-1}$$ x if $$and only if $$x = 	an y$$, $$-\frac{\pi}{2} \u003c y \u003c \frac{\pi}{2}$$

Question 5

Find the exact value of $$	an$$^{-1}$$(1).$$

Accepted Answer

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

Question 6

Which of the following composite functions is NOT defined?

Accepted Answer

$$\sin(\sin$$^{-1}$$(-2.5))$$

Question 7

If $$f(x) = 2\sin x - 1$$ for $$-\frac{\pi}{2} \leq x \leq \frac{\pi}{2}$$, what is the inverse function $$f^{-1}(x$$)$$?

Accepted Answer

$$f^{-1}(x$$) = \$$sin^{-1}$$\left(\frac{x+1}{2}\right)$$

Question 8

Solve for $$x$$: $$12\sin$$^{-1}$$x = 3\pi.$$

Accepted Answer

$$x = 1$$

Question 9

Which property allows you to simplify $$\sin(\sin$$^{-1}$$x)$$ for $$x in $$the domain of $$\sin$$^{-1}$$?

Accepted Answer

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

Question 10

A medical device uses the function $$y = \cos$$^{-1}$$ x to $$determine the angle of a joint based on a sensor reading $$x. If $$the sensor reads $$x = 0$$, what is the angle in radians?

Accepted Answer

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

Question 11

Which of the following statements is TRUE about the range of $$y = 	an$$^{-1}$$ x$$?

Accepted Answer

$$-\frac{\pi}{2} \u003c y \u003c \frac{\pi}{2}$$

Question 12

Evaluate $$\cos(\cos$$^{-1}$$(-1)).$$

Accepted Answer

$$-1$$

Question 13

If $$	heta = \sin$$^{-1}$$\left(\frac{\sqrt{3}}{2}ight)$$, what is the value of $$	heta in $$radians?

Accepted Answer

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

Question 14

Which of the following is the correct domain for the function $$y = \cos$$^{-1}$$ x$$?

Accepted Answer

$$-1 \leq x \leq 1$$

Question 15

A satellite dish is pointed at an angle $$	heta$$ such that $$	an 	heta = 2. $$What is $$	heta in $$terms of inverse trigonometric functions?

Accepted Answer

$$	heta = 	an$$^{-1}$$(2)$$

Question 16

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

Accepted Answer

$$0 \leq y \leq \pi$$

Question 17

If $$\sin(	heta) = -\frac{1}{2}$$ and $$	heta is in $$the range of $$\sin$$^{-1}$$, what is $$	heta$$?

Accepted Answer

$$-\frac{\pi}{6}$$

Question 18

Which of the following is NOT a property of inverse trigonometric functions?

Accepted Answer

$$\sin$$^{-1}$$(\sin x) = x$$ for $$0 \leq x \leq \pi$$

Analytic Trigonometry: Inverse Sine, Cosine, and Tangent Functions

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