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

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

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

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

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

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

Q: Which of the following statements is TRUE about the range of $y = \tan^{-1} x$?

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

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

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

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 $x = 	an y$, $-\frac{\pi}{2} < y < \frac{\pi}{2}$

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

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

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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