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: 44



Work Step by Step

$$X=\tan(180^\circ+\theta)$$ According to tangent sum identity: $$\tan(A+B)=\frac{\tan A+\tan B}{1-\tan A\tan B}$$ Expand $X$: $$X=\frac{\tan180^\circ+\tan\theta}{1-\tan180^\circ\tan\theta}$$ At $180^\circ$, $\sin180^\circ=0$ and $\cos180^\circ=-1$. Therefore, $\tan180^\circ=\frac{\sin180^\circ}{\cos180^\circ}=0$ $$X=\frac{0+\tan\theta}{1-0\times\tan\theta}$$ $$X=\frac{\tan\theta}{1}$$ $$X=\tan\theta$$ Overall, $$\tan(180^\circ+\theta)=\tan\theta$$
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