Q: Which of the following is the correct formula for the cotangent function in terms of $$x$$ and $$y on $$the unit circle?

$$\cot t = \frac{x}{y}$$, $$y \neq 0$$

Question 1

Which of the following best describes the domain and range of the function $$f(x) = \sin x$$?

Accepted Answer

Domain: $$(-\infty, \infty)$$, Range: $$[-1,1]$$

Question 2

Given the function $$y = 2\sin(3x - \frac{\pi}{2})$$, what is the amplitude, period, and phase shift?

Accepted Answer

Amplitude: $$2$$, Period: $$\frac{2\pi}{3}$$, Phase shift: $$\frac{\pi}{6}$$ right

Question 3

Which of the following statements is true about the function $$y = 	an x$$?

Accepted Answer

It is an odd function with period $$\pi.$$

Question 4

If $$\sin$$^{-1}$$(x) is $$defined, which of the following must be true?

Accepted Answer

$$x \in [-1,1]$$

Question 5

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

Accepted Answer

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

Question 6

A ladder leans against a wall, forming a $$65^\circ$$ angle with the ground. If the foot of the ladder is $$4$$ meters from the wall, how long is the ladder? (Round to two decimal places.)

Accepted Answer

$$4 / \cos 65^\circ \approx 9.48 m$$

Question 7

Which transformation will shift the graph of $$y = \sin x to $$the left by $$\frac{\pi}{4}$$ units?

Accepted Answer

$$y = \sin(x + \frac{\pi}{4})$$

Question 8

If $$	an$$^{-1}$$(\sqrt{3}) = 	heta$$, what is $$	heta in$$ $$[ -\frac{\pi}{2}, \frac{\pi}{2} ]$$?

Accepted Answer

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

Question 9

Given $$\sin 	heta = -\frac{1}{2}$$ and $$	heta is in$$ $$[ -\frac{\pi}{2}, \frac{\pi}{2} ]$$, what is $$	heta$$?

Accepted Answer

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

Question 10

Which of the following is the correct equation for the graph shown in the third image (cosine curve)?

Accepted Answer

$$y = \cos x$$

Question 11

If $$	heta = \sin$$^{-1}$$\left(\frac{\sqrt{2}}{2}ight)$$, what is $$\cos 	heta$$?

Accepted Answer

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

Question 12

Which of the following is NOT a property of the function $$y = \cos x$$?

Accepted Answer

Its range is $$[0,1].$$

Question 13

A right triangle has legs $$a = 5$$ and $$b = 12. $$What is the length of the hypotenuse $$c$$?

Accepted Answer

$$13$$

Question 14

Which of the following is the correct period for the function $$y = 	an x$$?

Accepted Answer

$$\pi$$

Question 15

If $$\sin(\cos$$^{-1}$$(\frac{1}{2})) = x$$, what is $$x$$?

Accepted Answer

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

Question 16

Which of the following is undefined?

Accepted Answer

$$	an(\frac{\pi}{2})$$

Question 17

If $$y = 3\cos(2x)$$, what is the amplitude and period?

Accepted Answer

Amplitude: $$3$$, Period: $$\pi$$

Question 18

Which of the following is the correct formula for the cotangent function in terms of $$x$$ and $$y on $$the unit circle?

Accepted Answer

$$\cot t = \frac{x}{y}$$, $$y 
eq 0$$

Question 19

If $$	heta is an $$angle in a right triangle such that $$	an 	heta = \frac{3}{4}$$, what is $$\sin 	heta$$?

Accepted Answer

$$\frac{3}{5}$$

Study Guide: Trigonometric Functions and Their Applications (Sections 5.4–5.8)

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