Q: Given that $$\sin \theta = \dfrac{3}{5}$$ and $$\theta is in $$the second quadrant, find $$\cos \theta$$ using a fundamental identity.

$$\cos \theta = -\dfrac{4}{5}$$

Q: Rewrite $$\tan x in $$terms of $$\sin x$$ and $$\cos x$$ using a fundamental identity.

$$\tan x = \dfrac{\sin x}{\cos x}$$

Q: Verify the identity: $$1 + \tan^2$ x = \sec^2$ x.$$

True, because $$\tan^2$ x = \dfrac{\sin^2$ x}{\cos^2$ x}$$ and $$\sec^2$ x = \dfrac{1}{\cos^2$ x}$$, so $$1 + \tan^2$ x = \dfrac{1}{\cos^2$ x}$$

Question 1

Given that $$\sin 	heta = \dfrac{3}{5}$$ and $$	heta is in $$the second quadrant, find $$\cos 	heta$$ using a fundamental identity.

Accepted Answer

$$\cos 	heta = -\dfrac{4}{5}$$

Question 2

Rewrite $$	an x in $$terms of $$\sin x$$ and $$\cos x$$ using a fundamental identity.

Accepted Answer

$$	an x = \dfrac{\sin x}{\cos x}$$

Question 3

Verify the identity: $$1 + 	an^2$ x = \sec^2$ x.$$

Accepted Answer

True, because $$	an^2$ x = \dfrac{\sin^2$ x}{\cos^2$ x}$$ and $$\sec^2$ x = \dfrac{1}{\cos^2$ x}$$, so $$1 + 	an^2$ x = \dfrac{1}{\cos^2$ x}$$

Question 4

Which of the following is the correct sum identity for $$\sin(A + B)$$?

Accepted Answer

$$\sin(A + B) = \sin A \cos B + \cos A \sin B$$

Question 5

Express $$\cos(\dfrac{\pi}{2} - x) in $$terms of $$\sin x$$ using a cofunction identity.

Accepted Answer

$$\cos(\dfrac{\pi}{2} - x) = \sin x$$

Question 6

A sound wave can be modeled by $$f(t) = \sin(2\pi t) + \sin(2\pi t + \dfrac{\pi}{3}). $$Use the sum-to-product identity to rewrite this as a product.

Accepted Answer

$$2 \sin\left(2\pi t + \dfrac{\pi}{6}\right) \cos\left(\dfrac{\pi}{6}\right)$$

Question 7

Given $$\sin \alpha = \dfrac{5}{13}$$ and $$\alpha is in $$quadrant II, find $$\sin(2\alpha)$$ using the double-angle identity.

Accepted Answer

$$\sin(2\alpha) = \dfrac{120}{169}$$

Question 8

Evaluate $$\cos\left(\dfrac{\pi}{8}\right)$$ using the half-angle identity.

Accepted Answer

$$\cos\left(\dfrac{\pi}{8}\right) = \sqrt{\dfrac{1 + \cos\left(\dfrac{\pi}{4}\right)}{2}}$$

Question 9

If $$	an x = 2$$ and $$x is in $$the first quadrant, find $$\sin(2x).$$

Accepted Answer

$$\sin(2x) = \dfrac{4}{5}$$

Question 10

Find the domain of $$y = \arccos x.$$

Accepted Answer

$$[-1, 1]$$

Question 11

Evaluate $$\arcsin\left(\dfrac{1}{2}\right).$$

Accepted Answer

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

Question 12

Solve for $$x in$$ $$\sin x = -\dfrac{\sqrt{2}}{2}$$, where $$0 \leq x \u003c 2\pi.$$

Accepted Answer

$$x = \dfrac{5\pi}{4}, \dfrac{7\pi}{4}$$

Question 13

Write $$\sec(\arctan x) as an $$algebraic expression in $$x.$$

Accepted Answer

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

Question 14

Solve the equation $$2\sin^2$ x - 1 = 0$$ for all solutions in $$[0, 2\pi).$$

Accepted Answer

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

Question 15

Solve for $$x in$$ $$	an(2x) = 1$$ where $$0 \leq x \u003c 2\pi.$$

Accepted Answer

$$x = \dfrac{\pi}{8}, \dfrac{5\pi}{8}, \dfrac{9\pi}{8}, \dfrac{13\pi}{8}$$

Question 16

If $$\arccos x = \dfrac{\pi}{3}$$, what is $$x$$?

Accepted Answer

$$x = \dfrac{1}{2}$$

Question 17

Solve for $$x$$: $$\sin(3x) = 0$$ for $$0 \leq x \u003c 2\pi.$$

Accepted Answer

$$x = 0, \dfrac{\pi}{3}, \dfrac{2\pi}{3}, \pi, \dfrac{4\pi}{3}, \dfrac{5\pi}{3}$$

Question 18

A Ferris wheel makes one full rotation every 60 seconds. The height $$h (in $$meters) of a passenger above the ground at time $$t$$ seconds is modeled by $$h(t) = 10 + 8\sin\left(\dfrac{\pi}{30}t\right). $$What is the maximum height reached?

Accepted Answer

$$18$$ meters

Question 19

Solve for $$x$$: $$\arcsin(\sin x) = x$$ for $$x in $$which interval?

Accepted Answer

$$\left[ -\dfrac{\pi}{2}, \dfrac{\pi}{2} \right]$$

Precalculus Trigonometric Identities and Equations Study Guide

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