Question

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. use a comma to separate answers as needed. do not include the degree symbol in you
b. the solution set is ∅.
\\( \sin \frac { \theta } { 2 } = - \frac { \sqrt { 2 } } { 2 } \\)

Explanation:

Step1: Find the general solution of \(\sin\frac{\theta}{2}=-\frac{\sqrt{2}}{2}\)

We know that \(\sin x =-\frac{\sqrt{2}}{2}\) has solutions \(x = 2k\pi+\frac{5\pi}{4}\) or \(x = 2k\pi+\frac{7\pi}{4},k\in\mathbb{Z}\).
Since \(x=\frac{\theta}{2}\), then \(\frac{\theta}{2}=2k\pi+\frac{5\pi}{4}\) or \(\frac{\theta}{2}=2k\pi+\frac{7\pi}{4},k\in\mathbb{Z}\).

Step2: Solve for \(\theta\)

For \(\frac{\theta}{2}=2k\pi+\frac{5\pi}{4}\), multiply both sides by \(2\) to get \(\theta = 4k\pi+\frac{5\pi}{2}\).
For \(\frac{\theta}{2}=2k\pi+\frac{7\pi}{4}\), multiply both sides by \(2\) to get \(\theta = 4k\pi+\frac{7\pi}{2},k\in\mathbb{Z}\).

Answer:

A. The solution set is \(\{\theta|\theta = 4k\pi+\frac{5\pi}{2},\theta = 4k\pi+\frac{7\pi}{2},k\in\mathbb{Z}\}\)