Question

given the following unit circle, rotate green dot to the appropriate angle and then find the exact value of the function. \\(\sin 150^{\circ}\\)

Explanation:

Step1: Recall the unit circle and reference angles

The angle $150^\circ$ is in the second quadrant. The reference angle for an angle $\theta$ in the second quadrant is $180^\circ - \theta$. So, the reference angle for $150^\circ$ is $180^\circ - 150^\circ = 30^\circ$.

Step2: Determine the sign of $\sin$ in the second quadrant

In the second quadrant, the sine function (which represents the $y$-coordinate on the unit circle) is positive.

Step3: Find the value of $\sin$ for the reference angle

We know that $\sin 30^\circ=\frac{1}{2}$. Since $\sin 150^\circ$ has the same absolute value as $\sin 30^\circ$ and is positive (from Step 2), we have $\sin 150^\circ = \sin(180^\circ - 30^\circ)=\sin 30^\circ=\frac{1}{2}$.

Answer:

$\frac{1}{2}$