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: 31



Work Step by Step

$$X=\sin\frac{\pi}{5}\cos\frac{3\pi}{10}+\cos\frac{\pi}{5}\sin\frac{3\pi}{10}$$ Recall the sine sum identity that $$\sin A\cos B+\cos A\sin B=\sin(A+B)$$ Looking back at $X$, we find $X$ is indeed the above identity with $A=\frac{\pi}{5}$ and $B=\frac{3\pi}{10}$. Therefore, we can rewrite $X$ as $$X=\sin\Big(\frac{\pi}{5}+\frac{3\pi}{10}\Big)$$ $$X=\sin\frac{5\pi}{10}=\sin\frac{\pi}{2}$$ $$X=1$$ In conclusion, $$\sin\frac{\pi}{5}\cos\frac{3\pi}{10}+\cos\frac{\pi}{5}\sin\frac{3\pi}{10}=1$$
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