Question 1

Which of the following is the correct range for $$\sin$$^{-1}$$x ($$arcsin)?

Accepted Answer

$$[-\frac{\pi}{2}, \frac{\pi}{2}]$$

Question 2

Find the exact value of $$\arccos\left(-\frac{1}{2}\right) in $$radians.

Accepted Answer

$$\frac{2\pi}{3}$$

Question 3

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

Accepted Answer

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

Question 4

If $$\sin x = -\frac{4}{5}$$ and $$x is in $$Quadrant III, what is $$\cos x$$?

Accepted Answer

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

Question 5

Given $$f(x) = 2\sin\left(\frac{\pi}{4}x + \frac{\pi}{3}\right) + 1$$, what is the amplitude of $$f(x)$$?

Accepted Answer

$$2$$

Question 6

Find all solutions to $$	an^2$ x - \sqrt{3}	an x = 0$$ for $$-2\pi \leq x \leq 2\pi.$$

Accepted Answer

$$x = 0, x = \pi, x = \frac{\pi}{3}, x = -\frac{\pi}{3}, x = \frac{4\pi}{3}, x = -\frac{2\pi}{3}$$

Question 7

Which of the following is equivalent to $$\cos(2x)$$?

Accepted Answer

$$\cos^2$ x - \sin^2$ x$$

Question 8

If $$\sin x = -\frac{4}{5}$$ and $$x is in $$Quadrant III, and $$\cos y = \frac{2}{\sqrt{5}}$$ with $$y in $$Quadrant IV, what is $$\sin(x + y)$$?

Accepted Answer

$$-\frac{14}{25}$$

Question 9

Find the exact value of $$\sin(75^\circ)$$ using the sum formula.

Accepted Answer

$$\frac{\sqrt{6} + \sqrt{2}}{4}$$

Question 10

Which of the following is a correct identity for $$\cos\left(\frac{\pi}{2} - x\right)$$?

Accepted Answer

$$\sin x$$

Question 11

Given $$\sin$$^{-1}$$(x)$$, which of the following is an algebraic expression for $$\cos(\sin$$^{-1}$$(x))$$?

Accepted Answer

$$\sqrt{1 - x^2$}$$

Question 12

If $$\cot x = \frac{1}{2}$$, what is $$\sin x$$?

Accepted Answer

$$\frac{2}{\sqrt{5}}$$

Question 13

Which of the following is the correct period for $$f(x) = 2\sin\left(\frac{\pi}{4}x + \frac{\pi}{3}\right) + 1$$?

Accepted Answer

$$8$$

Question 14

What is the vertical shift for $$f(x) = 2\sin\left(\frac{\pi}{4}x + \frac{\pi}{3}\right) + 1$$?

Accepted Answer

$$1$$

Question 15

Which of the following is the correct phase shift for $$f(x) = 2\sin\left(\frac{\pi}{4}x + \frac{\pi}{3}\right) + 1$$?

Accepted Answer

$$-\frac{4}{\pi} \cdot \frac{\pi}{3} = -\frac{4}{3}$$

Question 16

Which of the following is equivalent to $$\frac{1}{1 - \sin x} + \frac{1}{1 + \sin x}$$?

Accepted Answer

$$2\sec^2$ x$$

Question 17

If $$\sqrt{2}\sin x + 1 = 0$$, what are the solutions for $$x in$$ $$[-2\pi, 2\pi]$$?

Accepted Answer

$$x = -\frac{\pi}{4}, x = -\frac{7\pi}{4}, x = \frac{3\pi}{4}, x = \frac{5\pi}{4}$$

Question 18

Which of the following is equivalent to $$\cos(2x)$$?

Accepted Answer

$$2\cos^2$ x - 1$$

Question 19

Use the identity for $$\cos(x - y) to $$show that cosine is an even function. What is $$\cos(-x)$$?

Accepted Answer

$$\cos x$$

Precalculus Study Guide: Trigonometric Functions, Identities, and Applications

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