Elementary Linear Algebra 7th Edition

Published by Cengage Learning
ISBN 10: 1-13311-087-8
ISBN 13: 978-1-13311-087-3

Chapter 5 - Inner Product Spaces - Review Exercises - Page 284: 18


$\theta = \frac{\pi}{6} +2n\pi$ and $n$ is an integer

Work Step by Step

Let $u=\left(\cos \frac{ \pi}{6},\sin \frac{ \pi}{6}\right),\quad v=\left(\cos \frac{5\pi}{6},\sin \frac{5\pi}{6}\right) \quad \langle u, v\rangle=u\cdot v $. The angle $\theta$ between $u$ and $v$ is given by the formula $$\cos \theta=\frac{\langle u, v\rangle}{\|u\| \cdot\|v\|}=\frac{\cos \frac{\pi}{6}\cos \frac{ \pi}{6}+\sin \frac{5\pi}{6}\sin \frac{5\pi}{6}}{\sqrt{ \cos^2 \frac{ \pi}{6}+\sin^2 \frac{ \pi}{6}}\sqrt{ \cos^2 \frac{5\pi}{6}+\sin^2 \frac{5\pi}{6}}}.$$ Using the properties of the trignometric functions, we get $$\cos \theta=\cos\left( \frac{ \pi}{6}- \frac{5\pi}{6}\right)=\cos \frac{-\pi}{6} =\cos \frac{\pi}{6}$$ that is $\theta = \frac{\pi}{6} +2n\pi$ and $n$ is an integer.
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