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: 42


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

Work Step by Step

$$X=\sin\Big(\frac{3\pi}{4}-x\Big)$$ According to sine difference identity: $$\sin(A-B)=\sin A\cos B-\cos A\sin B$$ That means $$X=\sin\frac{3\pi}{4}\cos x-\cos\frac{3\pi}{4}\sin x$$ *About $\sin\frac{3\pi}{4}$ and $\cos\frac{3\pi}{4}$ $$\Big|\sin\frac{3\pi}{4}\Big|=\sin\frac{\pi}{4}\hspace{2cm}\Big|\cos\frac{3\pi}{4}\Big|=\cos\frac{\pi}{4}$$ So the values of them are not quite different. The only difference lies in the fact that while $\frac{\pi}{4}$ is in quadrant I, $\frac{3\pi}{4}$ is in quadrant II. In quadrant II, $\sin\theta\gt0$ but $\cos\theta\lt0$. Therefore, $$\sin\frac{3\pi}{4}=\sin\frac{\pi}{4}=\frac{\sqrt2}{2}\hspace{1.5cm}\cos\frac{3\pi}{4}=-\cos\frac{\pi}{4}=-\frac{\sqrt2}{2}$$ We replace the values just being found back into $X$: $$X=\frac{\sqrt2}{2}\cos x-\Big(-\frac{\sqrt2}{2}\Big)\sin x$$ $$X=\frac{\sqrt2}{2}\cos x+\frac{\sqrt2}{2}\sin x$$ $$X=\frac{\sqrt2}{2}(\cos x+\sin x)$$ Overall, $$\sin\Big(\frac{3\pi}{4}-x\Big)=\frac{\sqrt2}{2}(\cos x+\sin x)$$
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