Q: Which of the following is the correct half-angle formula for $$\sin \dfrac{\theta}{2}$$?

$$\sin \dfrac{\theta}{2} = \sqrt{\dfrac{1 - \cos \theta}{2}}$$

Q: Which of the following is the correct double-angle formula for $$\cos 2\theta$$?

$$\cos 2\theta = 2\cos^2$ \theta - 1$$

Q: Which of the following expressions is equivalent to $$\tan(x + y)$$?

$$\dfrac{\tan x + \tan y}{1 + \tan x \tan y}$$

Question 1

Which of the following is the correct half-angle formula for $$\sin \dfrac{	heta}{2}$$?

Accepted Answer

$$\sin \dfrac{	heta}{2} = \sqrt{\dfrac{1 - \cos 	heta}{2}}$$

Question 2

Given $$f(x) = \sin x$$ and $$g(x) = \sin(x + \dfrac{\pi}{2})$$, which of the following statements is true about their graphs?

Accepted Answer

The graph of $$g(x) is a $$horizontal shift of $$f(x) to $$the left by $$\dfrac{\pi}{2}$$ units.

Question 3

Which trigonometric identity is used to simplify $$\sin^2$ x + \cos^2$ x$$?

Accepted Answer

$$\sin^2$ x + \cos^2$ x = 1$$

Question 4

An earthquake was felt up to 52 miles from its epicenter. You are standing 48 miles west and 20 miles south of the epicenter. What is the equation of the circle that describes the boundary of the earthquake's reach?

Accepted Answer

$$(x + 48)^2$ + (y + 20)^2$ = 52^2$

Question 5

Which of the following is the correct double-angle formula for $$\cos 2	heta$$?

Accepted Answer

$$\cos 2	heta = 2\cos^2$ 	heta - 1$$

Question 6

If $$\sin \alpha = \dfrac{\sqrt{5}}{3}$$ and $$\cos \beta = \dfrac{\sqrt{5}}{3}$$, which quadrant(s) could $$\alpha$$ and $$\beta be in$$?

Accepted Answer

Quadrant I or IV

Question 7

Which of the following expressions is equivalent to $$	an(x + y)$$?

Accepted Answer

$$\dfrac{	an x + 	an y}{1 + 	an x 	an y}$$

Question 8

Which of the following is the correct period of the function $$f(x) = 2\sin(3x - \dfrac{\pi}{2})$$?

Accepted Answer

$$\dfrac{2\pi}{3}$$

Question 9

Given $$\sin x = \dfrac{1}{2}$$, how many solutions does $$x$$ have on the interval $$[0, 2\pi)$$?

Accepted Answer

2

Question 10

Which of the following is the correct expression for $$\cos^2$ x in $$terms of $$\cos 2x$$?

Accepted Answer

$$\cos^2$ x = \dfrac{1 + \cos 2x}{2}$$

Question 11

If $$f(x) = \sin x$$ and $$g(x) = \sin(x + \dfrac{\pi}{2})$$, what is $$g(x) in $$terms of $$\sin x$$ and $$\cos x$$?

Accepted Answer

$$g(x) = \cos x$$

Question 12

Which of the following is the correct value for $$\sin 15^\circ$$ using the half-angle formula?

Accepted Answer

$$\sin 15^\circ = \dfrac{\sqrt{6} - \sqrt{2}}{4}$$

Question 13

Which of the following is the correct solution to $$2\sin x = 1$$ for $$x in$$ $$[0, 2\pi)$$?

Accepted Answer

$$x = \dfrac{\pi}{6}, \dfrac{5\pi}{6}$$

Question 14

Which of the following is the correct value for $$\arccos(0)$$?

Accepted Answer

$$\dfrac{\pi}{2}$$

Question 15

Which of the following is the correct expression for $$\sin(x - y)$$?

Accepted Answer

$$\sin(x - y) = \sin x \cos y - \cos x \sin y$$

Question 16

Which of the following is the correct range of $$f(x) = 	an x$$?

Accepted Answer

$$(-\infty, \infty)$$

Question 17

Which of the following is the correct value for $$\arcsin(1)$$?

Accepted Answer

$$\dfrac{\pi}{2}$$

Question 18

Which of the following is the correct solution to $$\cos x = 0$$ for $$x in$$ $$[0, 2\pi)$$?

Accepted Answer

$$x = \dfrac{\pi}{2}, \dfrac{3\pi}{2}$$

Question 19

Which of the following is the correct value for $$	an(45^\circ)$$?

Accepted Answer

$$1$$

Unit 3: Trigonometry – Precalculus Advanced Study Guide

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