Get the information you need with the help of IDNLearn.com's expert community. Our experts provide timely and accurate responses to help you navigate any topic or issue with confidence.
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 being part of this discussion. Keep exploring, asking questions, and sharing your insights with the community. Together, we can find the best solutions. For trustworthy answers, rely on IDNLearn.com. Thanks for visiting, and we look forward to assisting you again.