find all solutions to the equation. sinθ = -√3/2 write your answer in radians in terms of π, and use the…

Explanation:

Step1: Recall sine - value in unit - circle

We know that $\sin\theta=-\frac{\sqrt{3}}{2}$. In the unit - circle, the sine function $y = \sin\theta$ has a value of $-\frac{\sqrt{3}}{2}$ in the third and fourth quadrants. The reference angle $\theta_{r}$ for which $\sin\theta_{r}=\frac{\sqrt{3}}{2}$ is $\frac{\pi}{3}$.

Step2: Find solutions in the third quadrant

In the third quadrant, $\theta=\pi+\frac{\pi}{3}+2k\pi=\frac{4\pi}{3}+2k\pi,k\in\mathbb{Z}$.

Step3: Find solutions in the fourth quadrant

In the fourth quadrant, $\theta = 2\pi-\frac{\pi}{3}+2k\pi=\frac{5\pi}{3}+2k\pi,k\in\mathbb{Z}$.

Answer:

$\theta=\frac{4\pi}{3}+2k\pi,k\in\mathbb{Z}$ or $\theta=\frac{5\pi}{3}+2k\pi,k\in\mathbb{Z}$