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 5.5 Double-Angle Identities - 5.5 Exercises - Page 236: 12


$$\sin2\theta=\frac{2\sqrt{66}}{25}$$ $$\cos2\theta=-\frac{19}{25}$$

Work Step by Step

$$\cos\theta=\frac{\sqrt3}{5} \hspace{2cm}\sin\theta\gt0$$ $$\sin 2\theta=?\hspace{2cm}\cos2\theta=?$$ 1) First, we need to figure out the unknown $\sin\theta$ as they are essential to the calculations of $\sin 2\theta$ and $\cos 2\theta$. According to Pythagorean Identities: $$\sin^2\theta=1-\cos^2\theta=1-\Big(\frac{\sqrt3}{5}\Big)^2=1-\frac{3}{25}=\frac{22}{25}$$ $$\sin\theta=\frac{\sqrt{22}}{5}\hspace{1cm}(\sin\theta\gt0)$$ 2) Now we can calculate $\sin2\theta$ and $\cos2\theta$ using Double-Angle Identities, which states $$\sin2\theta=2\sin\theta\cos\theta$$ $$\cos2\theta=2\cos^2\theta-1$$ Thus, $$\sin2\theta=2\times\frac{\sqrt{22}}{5}\times\frac{\sqrt{3}}{5}=\frac{2\sqrt{66}}{25}$$ $$\cos 2\theta=2\times\Big(\frac{\sqrt{3}}{5}\Big)^2-1=2\times\frac{3}{25}-1=\frac{6}{25}-1=-\frac{19}{25}$$
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