Answer
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$-\frac{\sqrt{2}}{2}$.
