solve the equation. give a general formula for all the solutions. list six solutions.\n\nfind all the…

solve the equation. give a general formula for all the solutions. list six solutions.\n\nfind all the solutions to ( sin (2 \theta)=-\frac{sqrt{2}}{2} ) in ( 0, pi) ).\n\n( \theta=square ) (simplify your answer. type an exact answer, using ( pi ) as needed. use integers

solve the equation. give a general formula for all the solutions. list six solutions.\n\nfind all the solutions to ( sin (2 \theta)=-\frac{sqrt{2}}{2} ) in ( 0, pi) ).\n\n( \theta=square ) (simplify your answer. type an exact answer, using ( pi ) as needed. use integers

Answer

Explanation:

Step1: Find the general solutions

We know that if (\sin x =-\frac{\sqrt{2}}{2}), then (x = 2k\pi+\frac{5\pi}{4}) or (x = 2k\pi+\frac{7\pi}{4},k\in\mathbb{Z}). Since (x = 2\theta), we have (2\theta=2k\pi+\frac{5\pi}{4}) or (2\theta=2k\pi+\frac{7\pi}{4}). Solving for (\theta), we get (\theta=k\pi+\frac{5\pi}{8}) or (\theta=k\pi+\frac{7\pi}{8},k\in\mathbb{Z}).

Step2: List six solutions

When (k = 0): (\theta=\frac{5\pi}{8}) or (\theta=\frac{7\pi}{8}) When (k = 1): (\theta=\pi+\frac{5\pi}{8}=\frac{13\pi}{8}) or (\theta=\pi+\frac{7\pi}{8}=\frac{15\pi}{8}) When (k=- 1): (\theta=-\pi+\frac{5\pi}{8}=-\frac{3\pi}{8}) or (\theta=-\pi+\frac{7\pi}{8}=-\frac{\pi}{8})

Answer:

The general solution is (\theta = k\pi+\frac{5\pi}{8}) or (\theta=k\pi+\frac{7\pi}{8},k\in\mathbb{Z}). Six solutions are (-\frac{3\pi}{8},-\frac{\pi}{8},\frac{5\pi}{8},\frac{7\pi}{8},\frac{13\pi}{8},\frac{15\pi}{8})