solve for exact solutions over the interval 0,2π). sin 2x = -√3/2 select the correct choice below and, if…

solve for exact solutions over the interval 0,2π). sin 2x = -√3/2 select the correct choice below and, if necessary, fill in the answer box to complete your choice. a. the solution set is { }. (simplify your answer. type an exact answer, using π as needed. use integers or fractions for any numbers in the expression. use a comma to separate answers as needed.) b. the solution set is ∅.

solve for exact solutions over the interval 0,2π). sin 2x = -√3/2 select the correct choice below and, if necessary, fill in the answer box to complete your choice. a. the solution set is { }. (simplify your answer. type an exact answer, using π as needed. use integers or fractions for any numbers in the expression. use a comma to separate answers as needed.) b. the solution set is ∅.

Answer

Explanation:

Step1: Recall sine - value angles

We know that $\sin\theta=-\frac{\sqrt{3}}{2}$ has solutions $\theta=\frac{4\pi}{3}+2k\pi$ and $\theta = \frac{5\pi}{3}+2k\pi$, where $k\in\mathbb{Z}$. Since $\theta = 2x$, we set up two equations: $2x=\frac{4\pi}{3}+2k\pi$ and $2x=\frac{5\pi}{3}+2k\pi$.

Step2: Solve for $x$ in the first equation

For $2x=\frac{4\pi}{3}+2k\pi$, divide both sides by 2: $x=\frac{2\pi}{3}+k\pi$.

Step3: Solve for $x$ in the second equation

For $2x=\frac{5\pi}{3}+2k\pi$, divide both sides by 2: $x=\frac{5\pi}{6}+k\pi$.

Step4: Find solutions in the interval $[0, 2\pi)$

When $k = 0$ in $x=\frac{2\pi}{3}+k\pi$, $x=\frac{2\pi}{3}$; when $k = 1$, $x=\frac{2\pi}{3}+\pi=\frac{5\pi}{3}$. When $k = 0$ in $x=\frac{5\pi}{6}+k\pi$, $x=\frac{5\pi}{6}$; when $k = 1$, $x=\frac{5\pi}{6}+\pi=\frac{11\pi}{6}$.

Answer:

A. The solution set is $\left{\frac{5\pi}{6},\frac{2\pi}{3},\frac{11\pi}{6},\frac{5\pi}{3}\right}$