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: 41


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

Work Step by Step

$$X=\sin\Big(\frac{\pi}{4}+x\Big)$$ According to sine sum identity: $$\sin(A+B)=\sin A\cos B+\sin B\cos A$$ That means $$X=\sin\frac{\pi}{4}\cos x+\sin x\cos\frac{\pi}{4}$$ $$X=\frac{\sqrt2}{2}\cos x+\frac{\sqrt2}{2}\sin x$$ $$X=\frac{\sqrt2}{2}(\cos x+\sin x)$$ Overall, $$\sin\Big(\frac{\pi}{4}+x\Big)=\frac{\sqrt2}{2}(\cos x+\sin x)$$
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