Q: Which of the following identities is correct for all $$\theta in $$the domain?

$$\tan\theta \cot\theta - \sin^2$\theta = \cos^2$\theta$$

Q: Solve for $$\theta in$$ $$0 \u003c \theta \u003c 2\pi$$: $$\sin\theta = \tan\theta.$$

$$\theta = \dfrac{\pi}{4}, \dfrac{5\pi}{4}$$

Q: Solve for $$\theta in$$ $$0 \u003c \theta \u003c 2\pi$$: $$\sin\theta = -2\sin(2\theta).$$

$$\theta = 0, \dfrac{2\pi}{3}, \pi, \dfrac{4\pi}{3}$$

Question 1

Find the exact value of $$\cos(112.5^\circ). $$Which of the following is correct?

Accepted Answer

$$-\dfrac{\sqrt{2 + \sqrt{2}}}{2}$$

Question 2

Evaluate $$	an(15^\circ)$$ exactly.

Accepted Answer

Question 3

Which of the following is the exact value of $$\sin\left(\dfrac{13\pi}{12}\right)$$?

Accepted Answer

$$\dfrac{\sqrt{6} - \sqrt{2}}{4}$$

Question 4

Which of the following identities is correct for all $$	heta in $$the domain?

Accepted Answer

$$	an	heta \cot	heta - \sin^2$	heta = \cos^2$	heta$$

Question 5

Prove or disprove: $$\sec(x) - 	an(x)\sin(x) = \dfrac{1}{\sec(x)}$$

Accepted Answer

True, the identity holds for all $$x in $$the domain.

Question 6

Which of the following is equivalent to $$\csc	heta - \sin	heta$$?

Accepted Answer

$$\cos	heta \cot	heta$$

Question 7

Simplify $$\csc^2$	heta 	an^2$	heta - 1.$$

Accepted Answer

$$	an^2$	heta$$

Question 8

Find the exact value of $$\cos 80^\circ \cos 50^\circ + \sin 80^\circ \sin 50^\circ.$$

Accepted Answer

$$\cos(80^\circ - 50^\circ)$$

Question 9

Evaluate $$\sin$$^{-1}$$\left(-\dfrac{1}{2}\right) in $$radians.

Accepted Answer

$$-\dfrac{\pi}{6}$$

Question 10

Given $$\sin	heta = -\dfrac{2}{5}$$ and $$\pi \u003c 	heta \u003c \dfrac{3\pi}{2}$$, find $$\sin\dfrac{	heta}{2}.$$

Accepted Answer

$$-\dfrac{\sqrt{5 - 2}}{2\sqrt{5}}$$

Question 11

Find the exact value of $$\cos\left(\sin$$^{-1}$$\left(\dfrac{3}{4}\right) - \cos$$^{-1}$$\left(-\dfrac{1}{2}\right)\right).$$

Accepted Answer

$$\dfrac{3}{8}$$

Question 12

Solve for $$	heta in$$ $$0 \u003c 	heta \u003c 2\pi$$: $$\sin	heta = 	an	heta.$$

Accepted Answer

$$	heta = \dfrac{\pi}{4}, \dfrac{5\pi}{4}$$

Question 13

Solve for $$	heta in$$ $$0 \u003c 	heta \u003c 2\pi$$: $$\sin	heta = -2\sin(2	heta).$$

Accepted Answer

$$	heta = 0, \dfrac{2\pi}{3}, \pi, \dfrac{4\pi}{3}$$

Question 14

Which of the following best describes the intersection point of $$y = \sin(2x)$$ and $$y = \ln(x) on $$the interval $$0 \u003c x \u003c 2\pi$$?

Accepted Answer

There is exactly one intersection point between $$x = 1$$ and $$x = 2.$$

Question 15

A Ferris wheel rotates such that the height $$h (in $$meters) of a passenger above the ground at time $$t (in $$seconds) is given by $$h(t) = 10 + 8\sin\left(\dfrac{\pi}{15}t\right). $$What is the maximum height reached?

Accepted Answer

$$18$$ meters

Question 16

If $$\cos\alpha = \dfrac{3}{5}$$ and $$\alpha is in $$the first quadrant, what is $$\sin(2\alpha)$$?

Accepted Answer

$$\dfrac{24}{25}$$

Question 17

Which of the following is the period of the function $$f(x) = 	an(2x)$$?

Accepted Answer

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

Question 18

If $$\sin	heta = \dfrac{5}{13}$$ and $$	heta is in $$quadrant II, what is $$\cos	heta$$?

Accepted Answer

$$-\dfrac{12}{13}$$

Precalculus Chapter 6 Practice Test – Step-by-Step Guidance

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