Question 1

Which of the following is the correct range for the inverse cosine function $$\cos$$^{-1}$$(x)$$?

Accepted Answer

$$[0, \pi]$$

Question 2

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

Accepted Answer

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

Question 3

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

Accepted Answer

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

Question 4

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

Accepted Answer

$$-\frac{\sqrt{2}}{10}$$

Question 5

Find all solutions to $$\sqrt{2}\sin(x) + 1 = 0$$ for $$-2\pi \leq x \leq 2\pi.$$

Accepted Answer

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

Question 6

A pilot flying at an altitude of 2.7 km sights two control towers directly in front of her. The angle of depression to the closest tower is $$58^\circ$$ and to the further tower is $$37^\circ. $$What is the distance between the towers?

Accepted Answer

$$1.896 km$$

Question 7

What is the amplitude of the function $$f(x) = 2\sin\left(\frac{5\pi}{2}x + \frac{5}{2}\right) + 1$$?

Accepted Answer

$$2$$

Question 8

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

Accepted Answer

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

Question 9

Which of the following is a correct expression for $$\cos(2x) in $$terms of $$\sin(x)$$?

Accepted Answer

$$1 - 2\sin^2$(x)$$

Question 10

Given $$	an(2	heta) = \frac{4}{3}$$ and a right triangle with legs 2 cm and 4 cm, what is the value of $$x ($$the hypotenuse)?

Accepted Answer

$$2\sqrt{3}$$

Question 11

Which of the following is the correct period of $$f(x) = 2\sin\left(\frac{5\pi}{2}x + \frac{5}{2}\right) + 1$$?

Accepted Answer

$$\frac{4}{5}$$

Question 12

Find the value of $$\sin\left[	an$$^{-1}$$\left(\frac{3}{4}ight)ight].$$

Accepted Answer

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

Question 13

Which of the following is a correct identity for $$\sec(x) - \csc(x) in $$terms of $$\sin(x)$$ and $$\cos(x)$$?

Accepted Answer

$$\frac{1}{\cos(x)} - \frac{1}{\sin(x)}$$

Question 14

If $$	an(x) = \sqrt{3}$$ and $$x is in $$Quadrant I, what is $$x in $$radians?

Accepted Answer

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

Question 15

Which of the following is the correct phase shift for $$f(x) = 2\sin\left(\frac{5\pi}{2}x + \frac{5}{2}\right) + 1$$?

Accepted Answer

$$-\frac{1}{2}$$

Question 16

Which of the following is a correct expression for $$\sin(\arccos(x))$$?

Accepted Answer

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

Question 17

Find the exact value of $$\cos\left[\arccos\left(\frac{3}{5}\right) + \arccos\left(\frac{4}{5}\right)\right].$$

Accepted Answer

$$\frac{\sqrt{10}}{10}$$

Question 18

Which of the following is a correct identity for $$\cos(x - y)$$?

Accepted Answer

$$\cos(x)\cos(y) + \sin(x)\sin(y)$$

Question 19

Which of the following is true about the cosine function?

Accepted Answer

Cosine is an even function: $$\cos(-x) = \cos(x)$$

Precalculus Study Guide: Trigonometric Functions, Identities, and Applications

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