IDNLearn.com is designed to help you find reliable answers quickly and easily. Join our community to receive prompt and reliable responses to your questions from knowledgeable professionals.
Sagot :
To solve the equation [tex]\(\tan x = -1\)[/tex] for [tex]\(x\)[/tex] within the interval [tex]\(-2 \pi \leq x \leq 2 \pi\)[/tex], we need to determine all the [tex]\(x\)[/tex] values where the tangent function equals [tex]\(-1\)[/tex].
The tangent function, [tex]\(\tan x\)[/tex], has a period of [tex]\(\pi\)[/tex]. This means that [tex]\(\tan(x) = \tan(x + n\pi)\)[/tex] for any integer [tex]\(n\)[/tex]. Therefore, we will look for solutions within one period and then extend these solutions to cover the range [tex]\(-2\pi\)[/tex] to [tex]\(2\pi\)[/tex].
Step-by-Step Solution:
1. Identify the basic solutions in one period:
- The solutions for [tex]\(\tan x = -1\)[/tex] within one period [tex]\( [0, 2\pi) \)[/tex] are:
- [tex]\( \frac{3\pi}{4} \)[/tex] (since [tex]\(\tan \frac{3\pi}{4} = -1\)[/tex])
- [tex]\( \frac{7\pi}{4} \)[/tex] (since [tex]\(\tan \frac{7\pi}{4} = -1\)[/tex])
2. Extend basic solutions to the entire interval [tex]\(-2\pi \leq x \leq 2\pi\)[/tex]:
- From [tex]\(\frac{3\pi}{4}\)[/tex]:
- Periodic extension to the left:
- [tex]\(\frac{3\pi}{4} - 2\pi = \frac{3\pi}{4} - \frac{8\pi}{4} = \frac{3\pi - 8\pi}{4} = -\frac{5\pi}{4}\)[/tex]
- Periodic extension to the right:
- [tex]\(\frac{3\pi}{4} + 2\pi = \frac{3\pi}{4} + \frac{8\pi}{4} = \frac{3\pi + 8\pi}{4} = \frac{11\pi}{4}\)[/tex]
- However, [tex]\(\frac{11\pi}{4}\)[/tex] is not in the range [tex]\(-2\pi\)[/tex] to [tex]\(2\pi\)[/tex], so it is excluded.
- From [tex]\(\frac{7\pi}{4}\)[/tex]:
- Periodic extension to the left:
- [tex]\(\frac{7\pi}{4} - 2\pi = \frac{7\pi}{4} - \frac{8\pi}{4} = \frac{7\pi - 8\pi}{4} = -\frac{\pi}{4}\)[/tex]
- Periodic extension to the right:
- [tex]\(\frac{7\pi}{4} + 2\pi = \frac{7\pi}{4} + \frac{8\pi}{4} = \frac{15\pi}{4}\)[/tex]
- However, [tex]\(\frac{15\pi}{4}\)[/tex] is not in the range [tex]\(-2\pi\)[/tex] to [tex]\(2\pi\)[/tex], so it is excluded.
3. Collect the valid solutions within the interval [tex]\(-2\pi \leq x \leq 2\pi\)[/tex]:
- [tex]\(\frac{3\pi}{4}\)[/tex], [tex]\(-\frac{5\pi}{4}\)[/tex], [tex]\(\frac{7\pi}{4}\)[/tex], and [tex]\(-\frac{\pi}{4}\)[/tex]
4. Verify and list the final solutions:
- Ensure all values are within the desired range:
- [tex]\(\frac{3\pi}{4}\)[/tex]
- [tex]\(-\frac{5\pi}{4}\)[/tex]
- [tex]\(\frac{7\pi}{4}\)[/tex]
- [tex]\(-\frac{\pi}{4}\)[/tex]
Therefore, the solutions to the equation [tex]\(\tan x = -1\)[/tex] for [tex]\(-2\pi \leq x \leq 2\pi\)[/tex] are:
[tex]\[ x = \frac{3\pi}{4}, -\frac{5\pi}{4}, \frac{7\pi}{4}, -\frac{\pi}{4} \][/tex]
The tangent function, [tex]\(\tan x\)[/tex], has a period of [tex]\(\pi\)[/tex]. This means that [tex]\(\tan(x) = \tan(x + n\pi)\)[/tex] for any integer [tex]\(n\)[/tex]. Therefore, we will look for solutions within one period and then extend these solutions to cover the range [tex]\(-2\pi\)[/tex] to [tex]\(2\pi\)[/tex].
Step-by-Step Solution:
1. Identify the basic solutions in one period:
- The solutions for [tex]\(\tan x = -1\)[/tex] within one period [tex]\( [0, 2\pi) \)[/tex] are:
- [tex]\( \frac{3\pi}{4} \)[/tex] (since [tex]\(\tan \frac{3\pi}{4} = -1\)[/tex])
- [tex]\( \frac{7\pi}{4} \)[/tex] (since [tex]\(\tan \frac{7\pi}{4} = -1\)[/tex])
2. Extend basic solutions to the entire interval [tex]\(-2\pi \leq x \leq 2\pi\)[/tex]:
- From [tex]\(\frac{3\pi}{4}\)[/tex]:
- Periodic extension to the left:
- [tex]\(\frac{3\pi}{4} - 2\pi = \frac{3\pi}{4} - \frac{8\pi}{4} = \frac{3\pi - 8\pi}{4} = -\frac{5\pi}{4}\)[/tex]
- Periodic extension to the right:
- [tex]\(\frac{3\pi}{4} + 2\pi = \frac{3\pi}{4} + \frac{8\pi}{4} = \frac{3\pi + 8\pi}{4} = \frac{11\pi}{4}\)[/tex]
- However, [tex]\(\frac{11\pi}{4}\)[/tex] is not in the range [tex]\(-2\pi\)[/tex] to [tex]\(2\pi\)[/tex], so it is excluded.
- From [tex]\(\frac{7\pi}{4}\)[/tex]:
- Periodic extension to the left:
- [tex]\(\frac{7\pi}{4} - 2\pi = \frac{7\pi}{4} - \frac{8\pi}{4} = \frac{7\pi - 8\pi}{4} = -\frac{\pi}{4}\)[/tex]
- Periodic extension to the right:
- [tex]\(\frac{7\pi}{4} + 2\pi = \frac{7\pi}{4} + \frac{8\pi}{4} = \frac{15\pi}{4}\)[/tex]
- However, [tex]\(\frac{15\pi}{4}\)[/tex] is not in the range [tex]\(-2\pi\)[/tex] to [tex]\(2\pi\)[/tex], so it is excluded.
3. Collect the valid solutions within the interval [tex]\(-2\pi \leq x \leq 2\pi\)[/tex]:
- [tex]\(\frac{3\pi}{4}\)[/tex], [tex]\(-\frac{5\pi}{4}\)[/tex], [tex]\(\frac{7\pi}{4}\)[/tex], and [tex]\(-\frac{\pi}{4}\)[/tex]
4. Verify and list the final solutions:
- Ensure all values are within the desired range:
- [tex]\(\frac{3\pi}{4}\)[/tex]
- [tex]\(-\frac{5\pi}{4}\)[/tex]
- [tex]\(\frac{7\pi}{4}\)[/tex]
- [tex]\(-\frac{\pi}{4}\)[/tex]
Therefore, the solutions to the equation [tex]\(\tan x = -1\)[/tex] for [tex]\(-2\pi \leq x \leq 2\pi\)[/tex] are:
[tex]\[ x = \frac{3\pi}{4}, -\frac{5\pi}{4}, \frac{7\pi}{4}, -\frac{\pi}{4} \][/tex]
Thank you for using this platform to share and learn. Keep asking and answering. We appreciate every contribution you make. Thank you for visiting IDNLearn.com. We’re here to provide accurate and reliable answers, so visit us again soon.