find the exact value of sin 7π/4. sin 7π/4 =

Explanation:

Step1: Rewrite the angle

We know that $\frac{7\pi}{4}=2\pi - \frac{\pi}{4}$. So, $\sin\frac{7\pi}{4}=\sin(2\pi - \frac{\pi}{4})$.

Step2: Use the sine - angle formula

According to the formula $\sin(A - B)=\sin A\cos B-\cos A\sin B$, when $A = 2\pi$ and $B=\frac{\pi}{4}$, since $\sin(2\pi)=0$ and $\cos(2\pi)=1$, we have $\sin(2\pi - \frac{\pi}{4})=\sin2\pi\cos\frac{\pi}{4}-\cos2\pi\sin\frac{\pi}{4}=0\times\frac{\sqrt{2}}{2}- 1\times\frac{\sqrt{2}}{2}=-\frac{\sqrt{2}}{2}$.

Answer:

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