based on the unit circle shown, josiah claims that sin(\\(\\frac{5\\pi}{6}\\)) = -\\(\\frac{\\sqrt{3}}{2}\\)…

Explanation:

Step1: Recall unit - circle definition

On the unit circle, for an angle $\theta$, the coordinates of the point on the terminal side of the angle are $(x,y)=(\cos\theta,\sin\theta)$.

Step2: Identify the angle

The given angle is $\theta = \frac{5\pi}{6}$. The angle $\frac{5\pi}{6}$ is in the second - quadrant.

Step3: Determine the y - coordinate

For a point $(x,y)$ on the unit circle corresponding to an angle $\theta$, $\sin\theta=y$. The angle $\frac{5\pi}{6}$ has a reference angle of $\frac{\pi}{6}$. In the second - quadrant, the $x$ - coordinate is negative and the $y$ - coordinate is positive. Since for $\theta=\frac{\pi}{6}$, $\sin\frac{\pi}{6}=\frac{1}{2}$ and $\cos\frac{\pi}{6}=\frac{\sqrt{3}}{2}$, for $\theta = \frac{5\pi}{6}$, the coordinates of the point on the unit circle are $(-\frac{\sqrt{3}}{2},\frac{1}{2})$. So, $\sin(\frac{5\pi}{6})=\frac{1}{2}$.

Answer:

The y - coordinate of the endpoint of the terminal side is positive. The coordinates of any point on the unit circle are $(\cos\theta,\sin\theta)$, so $\sin(\frac{5\pi}{6})=\frac{1}{2}$. Josiah is not correct.