1. find the exact value of sin (3π).

Explanation:

Step1: Recall the sine - angle formula

The sine function has a period of $2\pi$, i.e., $\sin(x + 2k\pi)=\sin(x)$ for any real - number $x$ and integer $k$. We can rewrite $3\pi$ as $3\pi=\pi + 2\pi$. Since $\sin(x + 2\pi)=\sin(x)$, then $\sin(3\pi)=\sin(\pi)$.

Step2: Use the unit - circle definition

On the unit circle, for an angle $\theta=\pi$ (180 degrees), the coordinates of the corresponding point are $(- 1,0)$. The sine of an angle $\theta$ in the unit - circle definition is given by the $y$ - coordinate of the point on the unit circle corresponding to the angle $\theta$. So, $\sin(\pi)=0$.

Answer:

$0$