# Chapter 4 - Section 4.2 - Trigonometric Functions: The Unit Circle - Exercise Set - Page 548: 52

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

#### Work Step by Step

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

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