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 5.4 Sum and Difference Identities for Sine and Tangent - 5.4 Exercises - Page 227: 39



Work Step by Step

$$X=\tan(30^\circ+\theta)$$ According to tangent sum identity: $$\tan(A+B)=\frac{\tan A+\tan B}{1-\tan A\tan B}$$ That means $$X=\frac{\tan30^\circ+\tan\theta}{1-\tan30^\circ\tan\theta}$$ $$X=\frac{\frac{1}{\sqrt3}+\tan\theta}{1-\frac{1}{\sqrt3}\tan\theta}$$ $$X=\frac{\frac{1+\sqrt3\tan\theta}{\sqrt3}}{\frac{\sqrt3-\tan\theta}{\sqrt3}}$$ $$X=\frac{1+\sqrt3\tan\theta}{\sqrt3-\tan\theta}$$ Overall, $$\tan(30^\circ+\theta)=\frac{1+\sqrt3\tan\theta}{\sqrt3-\tan\theta}$$
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