At Westonci.ca, we connect you with the best answers from a community of experienced and knowledgeable individuals. Connect with a community of professionals ready to provide precise solutions to your questions quickly and accurately. Explore comprehensive solutions to your questions from a wide range of professionals on our user-friendly platform.
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]
Thank you for your visit. We're committed to providing you with the best information available. Return anytime for more. Thanks for stopping by. We strive to provide the best answers for all your questions. See you again soon. Stay curious and keep coming back to Westonci.ca for answers to all your burning questions.
