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 5.6 Half-Angle Identities - 5.6 Exercises - Page 242: 19



Work Step by Step

$$\cos x=\frac{1}{4}\hspace{1.5cm}0\lt x\lt\frac{\pi}{2}\hspace{1.5cm}\cos\frac{x}{2}=?$$ Apply the half-angle identity for cosine $$\cos\frac{x}{2}=\pm\sqrt{\frac{1+\cos x}{2}}$$ As $0\lt x\lt\frac{\pi}{2}$, we deduce that $0\lt\frac{x}{2}\lt\frac{\pi}{4}$. That means the angle $\frac{x}{2}$ lies in quadrant I, where cosines are positive. That means $\cos\frac{x}{2}\gt0$, so we need to choose the positive square root. $$\cos\frac{x}{2}=\sqrt{\frac{1+\cos x}{2}}$$ $$\cos\frac{x}{2}=\sqrt{\frac{1+\frac{1}{4}}{2}}$$ $$\cos\frac{x}{2}=\sqrt{\frac{\frac{5}{4}}{2}}$$ $$\cos\frac{x}{2}=\sqrt{\frac{5}{8}}$$ $$\cos\frac{x}{2}=\frac{\sqrt5}{2\sqrt2}$$ $$\cos\frac{x}{2}=\frac{\sqrt{5}\times\sqrt2}{2\times2}$$ $$\cos\frac{x}{2}=\frac{\sqrt{10}}{4}$$
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