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: 37


$$\cos\Big(\frac{3\pi}{4}- x\Big)=\frac{\sqrt2(\sin x-\cos x)}{2}$$

Work Step by Step

$$X=\cos\Big(\frac{3\pi}{4}-x\Big)$$ To expand the formula, cosine difference identity would be used: $$\cos(A-B)=\cos A\cos B+\sin A\sin B$$ That means $$X=\cos\frac{3\pi}{4}\cos x+\sin\frac{3\pi}{4}\sin x$$ 1) Now we need to calculate $\cos\frac{3\pi}{4}$ and $\sin\frac{3\pi}{4}$. $\cos\frac{3\pi}{4}$ and $\sin\frac{3\pi}{4}$ are in fact like $\cos\frac{\pi}{4}$ and $\sin\frac{\pi}{4}$. The only difference is that they are in quadrant II. In quadrant II, $\sin\theta\gt0$ but $\cos\theta\lt0$. Therefore, $$\sin\frac{3\pi}{4}=\sin\frac{\pi}{4}=\frac{\sqrt2}{2}$$ while $$\cos\frac{3\pi}{4}=-\cos\frac{\pi}{4}=-\frac{\sqrt2}{2}$$ 2) Now apply back to $X$: $$X=-\frac{\sqrt2}{2}\cos x+\frac{\sqrt2}{2}\sin x$$ $$X=\frac{-\sqrt2\cos x+\sqrt2\sin x}{2}$$ $$X=\frac{\sqrt2(\sin x-\cos x)}{2}$$ Therefore, $$\cos\Big(\frac{3\pi}{4}- x\Big)=\frac{\sqrt2(\sin x-\cos x)}{2}$$
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