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Sagot :
To solve for the angle measures where [tex]\(\sin \theta = -\frac{\sqrt{2}}{2}\)[/tex], we need to understand where the sine function has this specific value on the unit circle. The sine of an angle is -[tex]\(\frac{\sqrt{2}}{2}\)[/tex] at specific points in the unit circle, particularly in the 3rd and 4th quadrants.
Here’s a detailed step-by-step solution:
1. Understanding the unit circle:
- The sine function is negative in the 3rd and 4th quadrants.
- The reference angle for [tex]\(\frac{\sqrt{2}}{2}\)[/tex] is [tex]\(\frac{\pi}{4}\)[/tex].
2. Finding the specific angles in the 3rd and 4th quadrants:
- In the 3rd quadrant, the angle is: [tex]\(\theta = \pi + \frac{\pi}{4} = \frac{5\pi}{4}\)[/tex].
- In the 4th quadrant, the angle is: [tex]\(\theta = 2\pi - \frac{\pi}{4} = \frac{7\pi}{4}\)[/tex].
Therefore, the angles within one full cycle [tex]\([0, 2\pi]\)[/tex] where [tex]\(\sin \theta = -\frac{\sqrt{2}}{2}\)[/tex] are [tex]\(\frac{5\pi}{4}\)[/tex] and [tex]\(\frac{7\pi}{4}\)[/tex].
3. Considering angles greater than [tex]\(2\pi\)[/tex]:
- Angles can be extended to cycles beyond [tex]\([0, 2\pi]\)[/tex] by adding multiples of [tex]\(2\pi\)[/tex].
- For [tex]\(\frac{5\pi}{4}\)[/tex]:
- Adding [tex]\(2\pi\)[/tex] to [tex]\(\frac{5\pi}{4}\)[/tex]:
[tex]\[\frac{5\pi}{4} + 2\pi = \frac{5\pi}{4} + \frac{8\pi}{4} = \frac{13\pi}{4}\][/tex]
- For [tex]\(\frac{7\pi}{4}\)[/tex]:
- Adding [tex]\(2\pi\)[/tex] to [tex]\(\frac{7\pi}{4}\)[/tex]:
[tex]\[\frac{7\pi}{4} + 2\pi = \frac{7\pi}{4} + \frac{8\pi}{4} = \frac{15\pi}{4}\][/tex]
- [tex]\(\frac{15\pi}{4}\)[/tex] is out of the options given.
4. Listing relevant angles from options:
- [tex]\(\frac{3\pi}{4}\)[/tex]: This angle is not in the 3rd or 4th quadrant. It is in the 2nd quadrant.
- [tex]\(\frac{5\pi}{4}\)[/tex]: This angle is valid as calculated.
- [tex]\(\frac{7\pi}{4}\)[/tex]: This angle is valid as calculated.
- [tex]\(\frac{9\pi}{4}\)[/tex]: [tex]\(\frac{9\pi}{4} = 2\pi + \frac{\pi}{4}\)[/tex]. This angle is in the 1st quadrant, so it is not calculated.
- [tex]\(\frac{13\pi}{4}\)[/tex]: This angle is valid as calculated.
Therefore, the valid angle measures for which [tex]\(\sin \theta = -\frac{\sqrt{2}}{2}\)[/tex] from the provided options are:
[tex]\[ \boxed{\frac{5\pi}{4}, \frac{7\pi}{4}, \frac{13\pi}{4}} \][/tex]
Here’s a detailed step-by-step solution:
1. Understanding the unit circle:
- The sine function is negative in the 3rd and 4th quadrants.
- The reference angle for [tex]\(\frac{\sqrt{2}}{2}\)[/tex] is [tex]\(\frac{\pi}{4}\)[/tex].
2. Finding the specific angles in the 3rd and 4th quadrants:
- In the 3rd quadrant, the angle is: [tex]\(\theta = \pi + \frac{\pi}{4} = \frac{5\pi}{4}\)[/tex].
- In the 4th quadrant, the angle is: [tex]\(\theta = 2\pi - \frac{\pi}{4} = \frac{7\pi}{4}\)[/tex].
Therefore, the angles within one full cycle [tex]\([0, 2\pi]\)[/tex] where [tex]\(\sin \theta = -\frac{\sqrt{2}}{2}\)[/tex] are [tex]\(\frac{5\pi}{4}\)[/tex] and [tex]\(\frac{7\pi}{4}\)[/tex].
3. Considering angles greater than [tex]\(2\pi\)[/tex]:
- Angles can be extended to cycles beyond [tex]\([0, 2\pi]\)[/tex] by adding multiples of [tex]\(2\pi\)[/tex].
- For [tex]\(\frac{5\pi}{4}\)[/tex]:
- Adding [tex]\(2\pi\)[/tex] to [tex]\(\frac{5\pi}{4}\)[/tex]:
[tex]\[\frac{5\pi}{4} + 2\pi = \frac{5\pi}{4} + \frac{8\pi}{4} = \frac{13\pi}{4}\][/tex]
- For [tex]\(\frac{7\pi}{4}\)[/tex]:
- Adding [tex]\(2\pi\)[/tex] to [tex]\(\frac{7\pi}{4}\)[/tex]:
[tex]\[\frac{7\pi}{4} + 2\pi = \frac{7\pi}{4} + \frac{8\pi}{4} = \frac{15\pi}{4}\][/tex]
- [tex]\(\frac{15\pi}{4}\)[/tex] is out of the options given.
4. Listing relevant angles from options:
- [tex]\(\frac{3\pi}{4}\)[/tex]: This angle is not in the 3rd or 4th quadrant. It is in the 2nd quadrant.
- [tex]\(\frac{5\pi}{4}\)[/tex]: This angle is valid as calculated.
- [tex]\(\frac{7\pi}{4}\)[/tex]: This angle is valid as calculated.
- [tex]\(\frac{9\pi}{4}\)[/tex]: [tex]\(\frac{9\pi}{4} = 2\pi + \frac{\pi}{4}\)[/tex]. This angle is in the 1st quadrant, so it is not calculated.
- [tex]\(\frac{13\pi}{4}\)[/tex]: This angle is valid as calculated.
Therefore, the valid angle measures for which [tex]\(\sin \theta = -\frac{\sqrt{2}}{2}\)[/tex] from the provided options are:
[tex]\[ \boxed{\frac{5\pi}{4}, \frac{7\pi}{4}, \frac{13\pi}{4}} \][/tex]
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