Problem

Find all solutions to $2 \sin (\theta)=-\sqrt{3}$ on the interval $0 \leq \theta< 2 \pi$
\[
\theta=
\]

Give your answers as exact values, as a list separated by commas.
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Answer

The solutions to \(2 \sin (\theta)=-\sqrt{3}\) on the interval \(0 \leq \theta<2 \pi\) are \(\theta=\boxed{\frac{4\pi}{3}, \frac{5\pi}{3}}\).

Steps

Step 1 :Given the equation is \(2 \sin (\theta)=-\sqrt{3}\). We need to find all solutions for \(\theta\) in the interval \(0 \leq \theta<2 \pi\).

Step 2 :Simplify the equation to \(\sin (\theta)=-\frac{\sqrt{3}}{2}\).

Step 3 :\(\sin (\theta)\) is negative in the third and fourth quadrants.

Step 4 :The reference angle for \(\frac{\sqrt{3}}{2}\) is \(\frac{\pi}{3}\) or \(60^\circ\).

Step 5 :The solutions in the third and fourth quadrants would be \(\pi + \frac{\pi}{3}\) and \(2\pi - \frac{\pi}{3}\) respectively.

Step 6 :The exact values of \(\theta\) in the third and fourth quadrants would be \(\frac{4\pi}{3}\) and \(\frac{5\pi}{3}\) respectively.

Step 7 :The solutions to \(2 \sin (\theta)=-\sqrt{3}\) on the interval \(0 \leq \theta<2 \pi\) are \(\theta=\boxed{\frac{4\pi}{3}, \frac{5\pi}{3}}\).

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