Trigonometry (11th Edition) Clone

Published by Pearson
ISBN 10: 978-0-13-421743-7
ISBN 13: 978-0-13421-743-7

Chapter 5 - Trigonometric Identities - Section 5.1 Fundamental Identities - 5.1 Exercises - Page 200: 37

Answer

$$\sin\theta=-\frac{\sqrt7}{4}$$ $$\cos\theta=\frac{3}{4}$$ $$\csc\theta=-\frac{4\sqrt7}{7}$$ $$\tan\theta=-\frac{\sqrt7}{3}$$ $$\cot\theta=-\frac{3\sqrt7}{7}$$

Work Step by Step

$$\sec\theta=\frac{4}{3}\hspace{1.5cm}\sin\theta\lt0$$ 1) Find $\cos\theta$ - Reciprocal Identities: $$\sec\theta=\frac{1}{\cos\theta}$$ $$\cos\theta=\frac{1}{\sec\theta}=\frac{1}{\frac{4}{3}}=\frac{3}{4}$$ 2) Find $\sin\theta$ - Pythagorean Identities: $$\sin^2\theta=1-\cos^2\theta=1-(\frac{3}{4})^2=1-\frac{9}{16}=\frac{7}{16}$$ $$\sin\theta=\pm\frac{\sqrt7}{4}$$ But since $\sin\theta\lt0$, $$\sin\theta=-\frac{\sqrt7}{4}$$ 3) Find $\csc\theta$ - Reciprocal Identities: $$\csc\theta=\frac{1}{\sin\theta}=\frac{1}{-\frac{\sqrt7}{4}}=-\frac{4}{\sqrt7}=-\frac{4\sqrt7}{7}$$ 4) Find $\tan\theta$ and $\cot\theta$ - Quotient Identities: $$\tan\theta=\frac{\sin\theta}{\cos\theta}=\frac{-\frac{\sqrt7}{4}}{\frac{3}{4}}=-\frac{\sqrt7}{3}$$ - Reciprocal Identities: $$\cot\theta=\frac{1}{\tan\theta}=\frac{1}{-\frac{\sqrt7}{3}}=-\frac{3}{\sqrt7}=-\frac{3\sqrt7}{7}$$
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