Question

question
given the following unit circle, rotate green dot to the appropriate angle and then find the exact value of the function.
\sin 225^{\circ}

Explanation:

Step1: Determine the reference angle

The angle $225^\circ$ is in the third quadrant. To find the reference angle, we subtract $180^\circ$ from $225^\circ$:
$225^\circ - 180^\circ = 45^\circ$.

Step2: Recall the sine value of the reference angle

We know that $\sin 45^\circ = \frac{\sqrt{2}}{2}$.

Step3: Determine the sign of $\sin 225^\circ$

In the third quadrant, both sine and cosine values are negative (since $y$-coordinates, which correspond to sine, are negative in this quadrant).

Step4: Combine the sign and the reference angle's sine value

Thus, $\sin 225^\circ = -\sin 45^\circ = -\frac{\sqrt{2}}{2}$.

Answer:

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