To find all solutions to the equation [tex]\(2 \sin (\theta) = -\sqrt{2}\)[/tex] within the interval [tex]\(0 \leq \theta < 2\pi\)[/tex], follow these steps:
1. Isolate the sine function:
[tex]\[
2 \sin (\theta) = -\sqrt{2}
\][/tex]
Divide both sides by 2 to get:
[tex]\[
\sin (\theta) = -\frac{\sqrt{2}}{2}
\][/tex]
2. Determine the reference angle:
We know that for [tex]\(\sin (\alpha) = -\frac{\sqrt{2}}{2}\)[/tex], the reference angle [tex]\(\alpha\)[/tex] where [tex]\(\sin (\alpha) = \frac{\sqrt{2}}{2}\)[/tex] is [tex]\(\frac{\pi}{4}\)[/tex].
3. Find the general solutions:
Since the sine function is negative in the third and fourth quadrants, the angles corresponding to [tex]\(\sin (\theta) = -\frac{\sqrt{2}}{2}\)[/tex] within one period [tex]\([0, 2\pi)\)[/tex] are:
[tex]\[
\theta = \pi + \frac{\pi}{4} \quad \text{and} \quad \theta = 2\pi - \frac{\pi}{4}
\][/tex]
4. Calculate the exact values:
[tex]\[
\theta = \pi + \frac{\pi}{4} = \frac{4\pi}{4} + \frac{\pi}{4} = \frac{5\pi}{4}
\][/tex]
[tex]\[
\theta = 2\pi - \frac{\pi}{4} = \frac{8\pi}{4} - \frac{\pi}{4} = \frac{7\pi}{4}
\][/tex]
5. List the solutions within the given interval:
The solutions to [tex]\(2 \sin (\theta) = -\sqrt{2}\)[/tex] on the interval [tex]\(0 \leq \theta < 2\pi\)[/tex] are:
[tex]\[
\theta = \boxed{\frac{5\pi}{4}, \frac{7\pi}{4}}
\][/tex]
