To solve the equation [tex]\(\tan \theta - 1 = 0\)[/tex] within the interval [tex]\([0, 2\pi)\)[/tex], follow these steps:
1. Rewrite the Equation:
[tex]\[
\tan \theta - 1 = 0 \quad \Rightarrow \quad \tan \theta = 1
\][/tex]
2. Identify the General Solution:
The tangent function, [tex]\(\tan \theta\)[/tex], repeats every [tex]\(\pi\)[/tex] radians because [tex]\(\tan(\theta + \pi) = \tan \theta\)[/tex]. The specific angle for which [tex]\(\tan \theta = 1\)[/tex] is [tex]\(\frac{\pi}{4}\)[/tex]. Hence, the general solutions are:
[tex]\[
\theta = \frac{\pi}{4} + n\pi \quad \text{for integer } n
\][/tex]
3. Find Specific Solutions in the Interval [tex]\([0, 2\pi)\)[/tex]:
We need to consider values of [tex]\(n\)[/tex] such that [tex]\(\theta\)[/tex] falls within [tex]\([0, 2\pi)\)[/tex].
- For [tex]\(n = 0\)[/tex]:
[tex]\[
\theta = \frac{\pi}{4}
\][/tex]
- For [tex]\(n = 1\)[/tex]:
[tex]\[
\theta = \frac{\pi}{4} + \pi = \frac{\pi}{4} + \frac{4\pi}{4} = \frac{5\pi}{4}
\][/tex]
- For [tex]\(n = 2\)[/tex]:
[tex]\[
\theta = \frac{\pi}{4} + 2\pi = \frac{\pi}{4} + \frac{8\pi}{4} = \frac{9\pi}{4}
\][/tex]
Note that [tex]\(\frac{9\pi}{4}\)[/tex] is greater than [tex]\(2\pi\)[/tex], so it is not within the interval [tex]\([0, 2\pi)\)[/tex].
Thus, the solutions within the interval [tex]\([0, 2\pi)\)[/tex] are:
[tex]\[
\boxed{\frac{\pi}{4}, \frac{5\pi}{4}}
\][/tex]
These values in radians correspond to approximately 0.7853981633974483 and 3.9269908169872414, respectively. Therefore, the solutions to the equation [tex]\(\tan \theta - 1 = 0\)[/tex] in the interval [tex]\([0, 2\pi)\)[/tex] are:
[tex]\[
\theta = \frac{\pi}{4}, \frac{5\pi}{4}
\][/tex]
