Answer :
To determine whether \(\frac{\pi}{6}\) is the reference angle for given angles, let's first review the concept of reference angles. The reference angle is the smallest angle between the terminal side of the given angle and the x-axis, and it must be between \(0\) and \(\frac{\pi}{2}\) (or \(0\) and \(90^\circ\)).
Given the angles:
A. \(\frac{5\pi}{6}\)
B. \(\frac{13\pi}{6}\)
C. \(\frac{8\pi}{6}\)
D. \(\frac{3\pi}{6}\)
The steps to determine if their reference angles are \(\frac{\pi}{6}\) are as follows:
1. Convert each angle into a format between \(0\) and \(2\pi\) if necessary.
2. Find the reference angle for each given angle by determining the smallest angle between the terminal side of the given angle and the x-axis.
Let's go through each option:
### A. \(\frac{5\pi}{6}\)
1. \(\frac{5\pi}{6}\) is already in the range \(0\) to \(2\pi\).
2. The reference angle is found by calculating the difference from \(\pi\): \(\pi - \frac{5\pi}{6} = \frac{\pi}{6}\).
So, \(\frac{5\pi}{6}\) has a reference angle of \(\frac{\pi}{6}\).
### B. \(\frac{13\pi}{6}\)
1. \(\frac{13\pi}{6}\) needs to be reduced to a value between \(0\) and \(2\pi\):
[tex]\[ \frac{13\pi}{6} - 2\pi = \frac{13\pi}{6} - \frac{12\pi}{6} = \frac{\pi}{6} \][/tex]
This results in \(\frac{\pi}{6}\).
2. The reference angle for \(\frac{\pi}{6}\) is itself \(\frac{\pi}{6}\).
So, \(\frac{13\pi}{6}\) has a reference angle of \(\frac{\pi}{6}\).
### C. \(\frac{8\pi}{6}\)
1. Simplify the angle: \(\frac{8\pi}{6} = \frac{4\pi}{3}\).
2. \(\frac{4\pi}{3}\) is in the third quadrant because it is greater than \(\pi\) but less than \(2\pi\). The reference angle is determined by subtracting it from \( \frac{3\pi}{2} \):
[tex]\[ \frac{4\pi}{3} - \pi = \frac{4\pi}{3} - \frac{3\pi}{3} = \frac{\pi}{3} \][/tex]
The reference angle for \(\frac{4\pi}{3}\) is \(\frac{\pi}{3}\), which is not equal to \(\frac{\pi}{6}\).
### D. \(\frac{3\pi}{6}\)
1. Simplify the angle: \(\frac{3\pi}{6} = \frac{\pi}{2}\).
2. \(\frac{\pi}{2}\) is already a right angle and does not reduce to \(\frac{\pi}{6}\).
Thus, it does not have a reference angle of \(\frac{\pi}{6}\).
After analyzing all the given angles, the correct answers are:
- A. \(\frac{5\pi}{6}\)
- B. \(\frac{13\pi}{6}\)
Therefore, the options where [tex]\(\frac{\pi}{6}\)[/tex] is the reference angle are [tex]\(A\)[/tex] and [tex]\(B\)[/tex].
Given the angles:
A. \(\frac{5\pi}{6}\)
B. \(\frac{13\pi}{6}\)
C. \(\frac{8\pi}{6}\)
D. \(\frac{3\pi}{6}\)
The steps to determine if their reference angles are \(\frac{\pi}{6}\) are as follows:
1. Convert each angle into a format between \(0\) and \(2\pi\) if necessary.
2. Find the reference angle for each given angle by determining the smallest angle between the terminal side of the given angle and the x-axis.
Let's go through each option:
### A. \(\frac{5\pi}{6}\)
1. \(\frac{5\pi}{6}\) is already in the range \(0\) to \(2\pi\).
2. The reference angle is found by calculating the difference from \(\pi\): \(\pi - \frac{5\pi}{6} = \frac{\pi}{6}\).
So, \(\frac{5\pi}{6}\) has a reference angle of \(\frac{\pi}{6}\).
### B. \(\frac{13\pi}{6}\)
1. \(\frac{13\pi}{6}\) needs to be reduced to a value between \(0\) and \(2\pi\):
[tex]\[ \frac{13\pi}{6} - 2\pi = \frac{13\pi}{6} - \frac{12\pi}{6} = \frac{\pi}{6} \][/tex]
This results in \(\frac{\pi}{6}\).
2. The reference angle for \(\frac{\pi}{6}\) is itself \(\frac{\pi}{6}\).
So, \(\frac{13\pi}{6}\) has a reference angle of \(\frac{\pi}{6}\).
### C. \(\frac{8\pi}{6}\)
1. Simplify the angle: \(\frac{8\pi}{6} = \frac{4\pi}{3}\).
2. \(\frac{4\pi}{3}\) is in the third quadrant because it is greater than \(\pi\) but less than \(2\pi\). The reference angle is determined by subtracting it from \( \frac{3\pi}{2} \):
[tex]\[ \frac{4\pi}{3} - \pi = \frac{4\pi}{3} - \frac{3\pi}{3} = \frac{\pi}{3} \][/tex]
The reference angle for \(\frac{4\pi}{3}\) is \(\frac{\pi}{3}\), which is not equal to \(\frac{\pi}{6}\).
### D. \(\frac{3\pi}{6}\)
1. Simplify the angle: \(\frac{3\pi}{6} = \frac{\pi}{2}\).
2. \(\frac{\pi}{2}\) is already a right angle and does not reduce to \(\frac{\pi}{6}\).
Thus, it does not have a reference angle of \(\frac{\pi}{6}\).
After analyzing all the given angles, the correct answers are:
- A. \(\frac{5\pi}{6}\)
- B. \(\frac{13\pi}{6}\)
Therefore, the options where [tex]\(\frac{\pi}{6}\)[/tex] is the reference angle are [tex]\(A\)[/tex] and [tex]\(B\)[/tex].
