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]
