Question

question given the following unit circle, rotate green dot to the appropriate angle and then find the exact value of the function. \\(sin 210^{\circ}\\)

Explanation:

Step1: Determine the reference angle

The angle $210^\circ$ is in the third quadrant. To find the reference angle, we subtract $180^\circ$ from $210^\circ$:
$210^\circ - 180^\circ = 30^\circ$.

Step2: Recall the sine value of the reference angle

We know that $\sin 30^\circ = \frac{1}{2}$.

Step3: Determine the sign of $\sin 210^\circ$

In the third quadrant, both sine and cosine are negative (since $y$-coordinates on the unit circle are negative here). Thus, $\sin 210^\circ$ will have the same magnitude as $\sin 30^\circ$ but with a negative sign.

So, $\sin 210^\circ = -\sin 30^\circ = -\frac{1}{2}$.

Answer:

$\boldsymbol{-\frac{1}{2}}$