Precalculus (6th Edition) Blitzer

Published by Pearson
ISBN 10: 0-13446-914-3
ISBN 13: 978-0-13446-914-0

Chapter 4 - Section 4.2 - Trigonometric Functions: The Unit Circle - Exercise Set - Page 548: 50


The value of the trigonometric function $\sin \left( -\frac{\pi }{4}-2000\pi \right)$ is $-\frac{\sqrt{2}}{2}$.

Work Step by Step

Consider the provided trigonometric function: $\sin \left( -\frac{\pi }{4}-2000\pi \right)$ Rewrite the trigonometric function as a periodic function with $2\pi $. $\sin \left( -\frac{\pi }{4}-2000\pi \right)=\sin \left( -\frac{\pi }{4}+2\pi \left( -1000 \right) \right)$ Use the property $\sin \left( t+2\pi n \right)=\sin t$. Here, the value of $t$ is $-\frac{\pi }{4}$ and the value of $n$ is $-1000$. $\sin \left( -\frac{\pi }{4}-2000\pi \right)=\sin \left( -\frac{\pi }{4} \right)$ Use the property $\sin \left( -t \right)=-\sin t$. $\sin \left( -\frac{\pi }{4}-2000\pi \right)=-\sin \frac{\pi }{4}$ The value of $\sin \frac{\pi }{4}$ is $\frac{\sqrt{2}}{2}$. So, $\sin \left( -\frac{\pi }{4}-2000\pi \right)=-\frac{\sqrt{2}}{2}$ Therefore, the value of the trigonometric function $\sin \left( -\frac{\pi }{4}-2000\pi \right)$ is$-\frac{\sqrt{2}}{2}$.
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