click the arrows to choose an answer from each menu. the y - coordinate of the endpoint of the terminal side…

Explanation:

Step1: Recall unit - circle equation

The equation of the unit circle is $x^{2}+y^{2}=1$. Given $x =-\frac{\sqrt{3}}{2}$, we substitute it into the equation: $\left(-\frac{\sqrt{3}}{2}\right)^{2}+y^{2}=1$.

Step2: Solve for $y$

First, calculate $\left(-\frac{\sqrt{3}}{2}\right)^{2}=\frac{3}{4}$. Then the equation becomes $\frac{3}{4}+y^{2}=1$. Subtract $\frac{3}{4}$ from both sides: $y^{2}=1 - \frac{3}{4}=\frac{1}{4}$. Take the square - root of both sides. Since the angle $\theta=\frac{5\pi}{6}$ is in the second quadrant where $y>0$, $y=\frac{1}{2}$.

Step3: Recall unit - circle coordinates and sine function

The coordinates of any point on the unit circle are $(\cos\theta,\sin\theta)$. For $\theta=\frac{5\pi}{6}$, $\sin\theta$ is the $y$ - coordinate of the point on the unit circle corresponding to the angle $\theta$. So $\sin\left(\frac{5\pi}{6}\right)=\frac{1}{2}$.

Answer:

The $y$-coordinate of the endpoint of the terminal side is $\frac{1}{2}$. The coordinates of any point on the unit circle are $(\cos\theta,\sin\theta)$, so $\sin\left(\frac{5\pi}{6}\right)=\frac{1}{2}$.