chapter 6 quiz\n170 points possible answered: 16/17\nquestion 17\nfind all solutions to ( 2 sin ( \theta ) =…

Explanation:

Step1: Solve for (\sin(\theta))

Divide both sides of (2\sin(\theta)=-\sqrt{3}) by (2). (\sin(\theta)=-\frac{\sqrt{3}}{2})

Step2: Find the reference angle

We know that (\sin\left(\frac{\pi}{3}\right)=\frac{\sqrt{3}}{2}), so the reference angle (\theta_{r}=\frac{\pi}{3})

Step3: Determine the quadrants

Since (\sin(\theta)<0), (\theta) is in the third or fourth quadrant. In the third quadrant: (\theta=\pi+\theta_{r}=\pi +\frac{\pi}{3}=\frac{4\pi}{3}) In the fourth quadrant: (\theta = 2\pi-\theta_{r}=2\pi-\frac{\pi}{3}=\frac{5\pi}{3})

Answer:

(\frac{4\pi}{3},\frac{5\pi}{3})