To solve the equation [tex]\(\tan^2(\theta) = 1\)[/tex] on the interval [tex]\([0, \pi)\)[/tex], let's follow a step-by-step approach.
1. Understand the equation:
[tex]\[
\tan^2(\theta) = 1
\][/tex]
This implies:
[tex]\[
\tan(\theta) = \pm 1
\][/tex]
2. Find the angles where [tex]\(\tan(\theta) = 1\)[/tex]:
The tangent of an angle is [tex]\(1\)[/tex] at [tex]\(\theta\)[/tex] values of:
[tex]\[
\theta = \frac{\pi}{4} + k\pi, \quad k \in \mathbb{Z}
\][/tex]
Within the interval [tex]\([0, \pi)\)[/tex], the only solution is:
[tex]\[
\theta = \frac{\pi}{4}
\][/tex]
3. Find the angles where [tex]\(\tan(\theta) = -1\)[/tex]:
The tangent of an angle is [tex]\(-1\)[/tex] at [tex]\(\theta\)[/tex] values of:
[tex]\[
\theta = \frac{3\pi}{4} + k\pi, \quad k \in \mathbb{Z}
\][/tex]
Within the interval [tex]\([0, \pi)\)[/tex], the only solution is:
[tex]\[
\theta = \frac{3\pi}{4}
\][/tex]
4. Summarize the solutions:
Within the given interval [tex]\([0, \pi)\)[/tex], the solutions to the equation [tex]\(\tan^2(\theta) = 1\)[/tex] are:
[tex]\[
\theta = \frac{\pi}{4}, \quad \theta = \frac{3\pi}{4}
\][/tex]
Thus, the answers are:
[tex]\[
A = \frac{\pi}{4} \approx 0.7854
\][/tex]
[tex]\[
B = \frac{3\pi}{4} \approx 2.3562
\][/tex]
Therefore, the solutions are:
[tex]\[
A = 0.7853981633974483
\][/tex]
[tex]\[
B = 2.356194490192345
\][/tex]
[tex]\(\boxed{0.7853981633974483}\)[/tex]
[tex]\(\boxed{2.356194490192345}\)[/tex]
