To determine the reference angle for [tex]\(\theta = \frac{25\pi}{13}\)[/tex], we follow these steps:
1. Express the given angle in the range [tex]\([0, 2\pi)\)[/tex]:
The angle [tex]\(\theta = \frac{25\pi}{13}\)[/tex] is already in radians. We need to find its equivalent angle within the range [tex]\([0, 2\pi)\)[/tex].
First, we compute the equivalent angle within the standard range by using modulo [tex]\(2\pi\)[/tex]:
[tex]\[
\text{Reference Angle} = \frac{25\pi}{13} \mod 2\pi
\][/tex]
2. Simplify the angle using modulo operation:
To apply the modulo operation, we subtract [tex]\(2\pi\)[/tex] multiple times from [tex]\(\frac{25\pi}{13}\)[/tex] until the result lies within [tex]\([0, 2\pi)\)[/tex].
Given:
[tex]\[
2\pi = \frac{26\pi}{13}
\][/tex]
Thus:
[tex]\[
\frac{25\pi}{13} \mod \left( \frac{26\pi}{13} \right) = \frac{25\pi}{13}
\][/tex]
Since [tex]\(\frac{25\pi}{13}\)[/tex] is less than [tex]\(2\pi\)[/tex], we can state:
[tex]\[
\frac{25\pi}{13} \equiv \frac{25\pi}{13}
\][/tex]
3. Determine if the angle is in a quadrant that affects the reference angle adjustment:
An angle between [tex]\(0\)[/tex] and [tex]\(2\pi\)[/tex] may need to be adjusted if it lies in the second or third quadrants to find its reference angle. The reference angle is measured with respect to the x-axis.
If:
[tex]\[
0 \leq \text{angle} < \pi
\][/tex]
the reference angle is simply the angle itself.
Given that:
[tex]\[
\frac{25\pi}{13} < 2\pi
\][/tex]
To determine if it is greater than [tex]\(\pi\)[/tex]:
[tex]\[
\frac{25\pi}{13} > \pi
\][/tex]
Here:
[tex]\[
\pi = \frac{13\pi}{13}
\][/tex]
We see:
[tex]\[
\frac{25\pi}{13} > \frac{13\pi}{13}
\][/tex]
4. Adjust the reference angle if needed:
Since [tex]\(\frac{25\pi}{13}\)[/tex] is indeed greater than [tex]\(\pi\)[/tex], this means the angle is in the third or fourth quadrant. For angles greater than [tex]\(\pi\)[/tex] but less than [tex]\(2\pi\)[/tex], we need to find the reference angle as follows:
[tex]\[
\text{Reference Angle} = 2\pi - \frac{25\pi}{13}
\][/tex]
Simplifying the expression:
[tex]\[
\text{Reference Angle} = \frac{26\pi}{13} - \frac{25\pi}{13} = \frac{\pi}{13}
\][/tex]
Thus, the reference angle associated with [tex]\(\theta = \frac{25\pi}{13}\)[/tex] is:
[tex]\[
\boxed{\frac{\pi}{13}}
\][/tex]
