To identify the reference angles for the given angles, we need to follow these steps:
1. Normalize the given angle [tex]\(\theta\)[/tex] to be within the range of [tex]\(0^\circ\)[/tex] to [tex]\(360^\circ\)[/tex]. This can be done by adding or subtracting multiples of [tex]\(360^\circ\)[/tex] until the angle falls within this range.
2. Determine the reference angle [tex]\(\phi\)[/tex]. The reference angle is the acute angle formed by the terminal side of the given angle and the horizontal axis, and it can be found using the following rules:
- If the angle is in the first quadrant ([tex]\(0^\circ\)[/tex] to [tex]\(90^\circ\)[/tex]), the reference angle is the same as the given angle.
- If the angle is in the second quadrant ([tex]\(90^\circ\)[/tex] to [tex]\(180^\circ\)[/tex]), the reference angle is [tex]\(180^\circ - \theta\)[/tex].
- If the angle is in the third quadrant ([tex]\(180^\circ\)[/tex] to [tex]\(270^\circ\)[/tex]), the reference angle is [tex]\(\theta - 180^\circ\)[/tex].
- If the angle is in the fourth quadrant ([tex]\(270^\circ\)[/tex] to [tex]\(360^\circ\)[/tex]), the reference angle is [tex]\(360^\circ - \theta\)[/tex].
Let's apply these steps to each given angle:
### For [tex]\(\theta = 300^\circ\)[/tex]:
1. The angle [tex]\(300^\circ\)[/tex] is already within the range [tex]\(0^\circ\)[/tex] to [tex]\(360^\circ\)[/tex].
2. Since [tex]\(300^\circ\)[/tex] is in the fourth quadrant,
[tex]\[
\phi = 360^\circ - 300^\circ = 60^\circ.
\][/tex]
So, [tex]\(\phi = 60^\circ\)[/tex].
### For [tex]\(\theta = 225^\circ\)[/tex]:
1. The angle [tex]\(225^\circ\)[/tex] is already within the range [tex]\(0^\circ\)[/tex] to [tex]\(360^\circ\)[/tex].
2. Since [tex]\(225^\circ\)[/tex] is in the third quadrant,
[tex]\[
\phi = 225^\circ - 180^\circ = 45^\circ.
\][/tex]
So, [tex]\(\phi = 45^\circ\)[/tex].
### For [tex]\(\theta = 480^\circ\)[/tex]:
1. Normalize [tex]\(480^\circ\)[/tex] by subtracting [tex]\(360^\circ\)[/tex]:
[tex]\[
480^\circ - 360^\circ = 120^\circ.
\][/tex]
2. Now, [tex]\(120^\circ\)[/tex] is in the second quadrant,
[tex]\[
\phi = 180^\circ - 120^\circ = 60^\circ.
\][/tex]
So, [tex]\(\phi = 60^\circ\)[/tex].
### For [tex]\(\theta = -210^\circ\)[/tex]:
1. Normalize [tex]\(-210^\circ\)[/tex] by adding [tex]\(360^\circ\)[/tex]:
[tex]\[
-210^\circ + 360^\circ = 150^\circ.
\][/tex]
2. Now, [tex]\(150^\circ\)[/tex] is in the second quadrant,
[tex]\[
\phi = 180^\circ - 150^\circ = 30^\circ.
\][/tex]
So, [tex]\(\phi = 30^\circ\)[/tex].
Hence, the reference angles are:
- When [tex]\(\theta = 300^\circ\)[/tex], [tex]\(\phi = 60^\circ\)[/tex].
- When [tex]\(\theta = 225^\circ\)[/tex], [tex]\(\phi = 45^\circ\)[/tex].
- When [tex]\(\theta = 480^\circ\)[/tex], [tex]\(\phi = 60^\circ\)[/tex].
- When [tex]\(\theta = -210^\circ\)[/tex], [tex]\(\phi = 30^\circ\)[/tex].
