To solve for the angles where [tex]\(\sin \theta = -\frac{\sqrt{2}}{2}\)[/tex], we need to identify the standard angle and then find additional possible angles within the given list.
The sine of [tex]\(\theta\)[/tex] is [tex]\(-\frac{\sqrt{2}}{2}\)[/tex] when [tex]\(\theta\)[/tex] is in one of the following quadrants:
- Quadrant III: where the reference angle is [tex]\(\frac{\pi}{4}\)[/tex]
- Quadrant IV: where the reference angle is [tex]\(\frac{\pi}{4}\)[/tex]
This means:
1. In Quadrant III, the angle can be written as [tex]\(\frac{5\pi}{4}\)[/tex].
2. In Quadrant IV, the angle can be written as [tex]\(\frac{7\pi}{4}\)[/tex].
Now consider the provided angles:
[tex]\[
\frac{3\pi}{4}, \quad \frac{5\pi}{4}, \quad \frac{7\pi}{4}, \quad \frac{9\pi}{4}, \quad \frac{13\pi}{4}
\][/tex]
Let's check which of these angles correspond to [tex]\(\theta\)[/tex] that make [tex]\(\sin \theta = -\frac{\sqrt{2}}{2}\)[/tex]:
1. [tex]\(\frac{3\pi}{4}\)[/tex]:
- [tex]\(\sin\left(\frac{3\pi}{4}\right) = \frac{\sqrt{2}}{2}\)[/tex], not [tex]\(-\frac{\sqrt{2}}{2}\)[/tex]. Therefore, this angle does not satisfy the condition.
2. [tex]\(\frac{5\pi}{4}\)[/tex]:
- [tex]\(\sin\left(\frac{5\pi}{4}\right) = -\frac{\sqrt{2}}{2}\)[/tex]. This angle satisfies the condition.
3. [tex]\(\frac{7\pi}{4}\)[/tex]:
- [tex]\(\sin\left(\frac{7\pi}{4}\right) = -\frac{\sqrt{2}}{2}\)[/tex]. This angle satisfies the condition.
4. [tex]\(\frac{9\pi}{4}\)[/tex]:
- [tex]\(\sin\left(\frac{9\pi}{4}\right) = \sin\left(2\pi + \frac{\pi}{4}\right) = \sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}\)[/tex], not [tex]\(-\frac{\sqrt{2}}{2}\)[/tex]. Therefore, this angle does not satisfy the condition.
5. [tex]\(\frac{13\pi}{4}\)[/tex]:
- [tex]\(\sin\left(\frac{13\pi}{4}\right) = \sin\left(3\pi + \frac{\pi}{4}\right) = \sin\left(\pi + \frac{\pi}{4}\right) = -\frac{\sqrt{2}}{2}\)[/tex]. This angle satisfies the condition.
Therefore, the angle measures from the given list that satisfy [tex]\(\sin \theta = -\frac{\sqrt{2}}{2}\)[/tex] are:
[tex]\[
\frac{5\pi}{4}, \quad \frac{7\pi}{4}, \quad \frac{13\pi}{4}
\][/tex]
