for which value of $\theta$ is $sin\theta = - 1$?

Explanation:

Step1: Recall sine - unit circle relationship

On the unit circle, $\sin\theta$ is the $y$ - coordinate of the point where the terminal side of the angle $\theta$ intersects the unit circle.

Step2: Locate the point with $y=-1$

We look for the angle $\theta$ on the unit circle where the $y$ - coordinate of the corresponding point is $-1$. The point on the unit circle with $y = - 1$ is $(0,-1)$.

Step3: Determine the angle

The angle $\theta$ corresponding to the point $(0,-1)$ on the unit circle is $\theta=\frac{3\pi}{2}+2k\pi$, $k\in\mathbb{Z}$. In the range $[0, 2\pi]$, the value of $\theta$ for which $\sin\theta=-1$ is $\theta = \frac{3\pi}{2}$.

Answer:

$\frac{3\pi}{2}$