Q: Which of the following statements about inverse trigonometric functions is true?

$$\cos(\cos$$^{-1}$$ x) = x$$ \text{ for } x \in [-1, 1]$

Q: If $$y = \tan$$^{-1}$$ x$$, which of the following is true?

$$\tan y = x$$ and $$y \in \left( -\frac{\pi}{2}, \frac{\pi}{2} \right)$$

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

If $$y = \sin$$^{-1}$$ x$$, which of the following statements is true?

Accepted Answer

$$\sin y = x$$ and $$y \in \left[ -\frac{\pi}{2}, \frac{\pi}{2} \right]$$

Question 3

Evaluate $$\sin$$^{-1}$$\left(\frac{\sqrt{3}}{2}\right).$$

Accepted Answer

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

Question 4

Which of the following best describes the graph of $$y = \sin$$^{-1}$$ x$$?

Accepted Answer

It is a reflection of $$y = \sin x ($$restricted) about the line $$y = x.$$

Question 5

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

Accepted Answer

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

Question 6

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

Accepted Answer

Domain: $$[-1, 1]$$, Range: $$[0, \pi]$$

Question 7

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

Accepted Answer

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

Question 8

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 9

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

Accepted Answer

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

Question 10

Which of the following expressions is NOT defined?

Accepted Answer

$$\sin$$^{-1}$$(2)$$

Question 11

If $$	heta = \cos$$^{-1}$$(-1)$$, what is the value of $$	heta$$?

Accepted Answer

$$\pi$$

Question 12

Which of the following statements about inverse trigonometric functions is true?

Accepted Answer

$$\cos(\cos$$^{-1}$$ x) = x$$ 	ext{ for } x \in [-1, 1]$

Question 13

A satellite dish is oriented so that its angle of elevation $$	heta$$ satisfies $$\sin 	heta = 0.5. $$What is the principal value of $$	heta$$?

Accepted Answer

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

Question 14

Evaluate $$\cos(\cos$$^{-1}$$ 0.7).$$

Accepted Answer

$$0.7$$

Question 15

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

Accepted Answer

$$x \leq -1 or$$ $$x \geq 1$$

Question 16

If $$y = 	an$$^{-1}$$ x$$, which of the following is true?

Accepted Answer

$$	an y = x$$ and $$y \in \left( -\frac{\pi}{2}, \frac{\pi}{2} ight)$$

Question 17

A biologist measures the angle of a plant's growth from the ground and finds $$	an 	heta = \sqrt{3}. $$What is the principal value of $$	heta$$?

Accepted Answer

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

Question 18

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

Accepted Answer

$$0$$

Question 19

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

Accepted Answer

$$[0, \pi]$$

Inverse Trigonometric Functions – Precalculus Essentials Study Notes

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