Answered: sin 0 = -, 0 lies in quadrant III

Name: Use the given information to find the exact value of: a. sin 2θ b. c
Uploaded: 2024-03-20T06:46:55.000Z
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Description: Use the given information to find the exact value of: a. sin 2θ b. cos 2θ c. tan 2θ. sin 0 = -, 0 lies in quadrant III

Math

Calculus

sin 0 = -, 0 lies in quadrant III

Trigonometry (MindTap Course List)

10th Edition

ISBN:9781337278461

Author:Ron Larson

Publisher:Ron Larson

Chapter2: Analytic Trigonometry

Section2.2: Verifying Trigonometric Identities

Problem 3ECP

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Concept explainers

Fundamentals of Trigonometric Identities

Trigonometric Identities are the equalities that involve trigonometry functions and hold for all the values of variables given in the equation.

Topic Video

Question

Use the given information to find the exact value of:
a. sin 2θ b. cos 2θ c. tan 2θ.

Expert Solution

Step 1

Given: $\sin θ = - \frac{2}{3}, θ$ lies in quadrant III.

To determine: The exact value of

a. $\sin 2 θ$

b. $\cos 2 θ$

c. $\tan 2 θ$

Step 2

Calculation:

$\sin θ = - \frac{2}{3} \cos θ = \pm \sqrt{1 - \sin^{2} θ} \cos θ = \pm \sqrt{1 - {(- \frac{2}{3})}^{2}} \cos θ = \pm \frac{\sqrt{5}}{3}$

But $θ$ lies in quadrant III. So,

$\cos θ = - \frac{\sqrt{5}}{3} \tan θ = \frac{\sin θ}{\cos θ} = \frac{2}{\sqrt{5}}$

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