To determine the reference angle, [tex]\( r \)[/tex], for an angle [tex]\(\theta = \frac{7\pi}{12}\)[/tex], we need to identify where [tex]\(\theta\)[/tex] lies on the unit circle and use the appropriate method to find the reference angle based on its position in a particular quadrant.
When [tex]\(\theta\)[/tex] is provided in radians, we must determine whether it lies in the first, second, third, or fourth quadrant. The reference angle for an angle in standard position is the acute angle formed by the terminal side of the angle and the x-axis.
[tex]\(\theta = \frac{7\pi}{12}\)[/tex]:
1. First Quadrant: [tex]\(0 < \theta < \frac{\pi}{2}\)[/tex]
[tex]\[
\text{For angles in the first quadrant, the reference angle } r \text{ is } \theta.
\][/tex]
2. Second Quadrant: [tex]\(\frac{\pi}{2} < \theta < \pi\)[/tex]
[tex]\[
\text{For angles in the second quadrant, the reference angle } r \text{ is } \pi - \theta.
\][/tex]
3. Third Quadrant: [tex]\(\pi < \theta < \frac{3\pi}{2}\)[/tex]
[tex]\[
\text{For angles in the third quadrant, the reference angle } r \text{ is } \theta - \pi.
\][/tex]
4. Fourth Quadrant: [tex]\(\frac{3\pi}{2} < \theta < 2\pi\)[/tex]
[tex]\[
\text{For angles in the fourth quadrant, the reference angle } r \text{ is } 2\pi - \theta.
\][/tex]
Now, let’s determine where [tex]\(\theta = \frac{7\pi}{12}\)[/tex] lies:
- We need to compare [tex]\(\frac{7\pi}{12}\)[/tex] with [tex]\(\pi\)[/tex]:
[tex]\[
\pi = \frac{12\pi}{12}
\][/tex]
[tex]\[
\frac{\pi}{2} = \frac{6\pi}{12}
\][/tex]
Since [tex]\(\frac{6\pi}{12} < \frac{7\pi}{12} < \frac{12\pi}{12}\)[/tex], [tex]\(\frac{7\pi}{12}\)[/tex] is between [tex]\(\frac{\pi}{2}\)[/tex] and [tex]\(\pi\)[/tex]. This places [tex]\(\theta\)[/tex] in the second quadrant.
To find the reference angle in the second quadrant, we use the following equation:
[tex]\[
r = \pi - \theta
\][/tex]
Thus, the reference angle [tex]\( r \)[/tex] is calculated as:
[tex]\[
r = \pi - \frac{7\pi}{12}
\][/tex]
So, the equation to determine the reference angle is:
[tex]\[
r = \pi - \theta
\][/tex]
