Trigonometry (10th Edition)

Published by Pearson
ISBN 10: 0321671775
ISBN 13: 978-0-32167-177-6

Chapter 5 - Trigonometric Identities - Section 5.4 Sum and Difference Identities for Sine and Tangent - 5.4 Exercises - Page 221: 57

Answer

$\sin{\left(\dfrac{\pi}{2}+\theta\right)}=\cos{\theta}$

Work Step by Step

Use a graphing utility to graph the given expression. (Refer to the graph below.) Note that the graph is identical to the graph of $\cos{\theta}$. This means that $\sin{(\frac{\pi}{2}+\theta)}=\cos{\theta}$. RECALL: $\sin{(A+B)}=\sin{A}\cos{B}+\cos{A}\sin{B}$ Use the identity above with $A=\frac{\pi}{2}$ and $B=\theta$ to obtain: \begin{align*} \sin{\left(\frac{\pi}{2}+\theta\right)}&=\sin{\frac{\pi}{2}}\cos{\theta}+\cos{\frac{\pi}{2}}\sin{\theta}\\\\ &=1\cdot \cos{\theta}+0\cdot\sin{\theta}\\\\ &=\cos{\theta}+0\\\\ &=\cos{\theta} \end{align*} Therefore, $\sin{\left(\dfrac{\pi}{2}+\theta\right)}=\cos{\theta}$
Update this answer!

You can help us out by revising, improving and updating this answer.

Update this answer

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.