To identify the reference angle [tex]\(\phi\)[/tex] for each given angle [tex]\(\theta\)[/tex], we need to follow these steps:
1. Normalize the given angle to lie within [tex]\(0\)[/tex] and [tex]\(360\)[/tex] degrees.
2. Based on its normalized value, determine the reference angle, which is always within [tex]\(0\)[/tex] to [tex]\(180\)[/tex] degrees.
Let's break down the process for each angle:
### 1. When [tex]\(\theta = 300^\circ\)[/tex]
1. Normalize the angle: [tex]\(300^\circ\)[/tex] is already between [tex]\(0^\circ\)[/tex] and [tex]\(360^\circ\)[/tex].
2. Since [tex]\(300^\circ\)[/tex] is greater than [tex]\(180^\circ\)[/tex], the reference angle [tex]\(\phi\)[/tex] is given by:
[tex]\[
\phi = 360^\circ - 300^\circ = 60^\circ
\][/tex]
### 2. When [tex]\(\theta = 225^\circ\)[/tex]
1. Normalize the angle: [tex]\(225^\circ\)[/tex] is already between [tex]\(0^\circ\)[/tex] and [tex]\(360^\circ\)[/tex].
2. Since [tex]\(225^\circ\)[/tex] is greater than [tex]\(180^\circ\)[/tex], the reference angle [tex]\(\phi\)[/tex] is given by:
[tex]\[
\phi = 360^\circ - 225^\circ = 135^\circ
\][/tex]
### 3. When [tex]\(\theta = 480^\circ\)[/tex]
1. Normalize the angle: [tex]\(480^\circ \mod 360^\circ = 120^\circ\)[/tex].
2. Since [tex]\(120^\circ\)[/tex] is less than [tex]\(180^\circ\)[/tex], the reference angle [tex]\(\phi\)[/tex] is:
[tex]\[
\phi = 120^\circ
\][/tex]
### 4. When [tex]\(\theta = -210^\circ\)[/tex]
1. Normalize the angle: [tex]\(-210^\circ \mod 360^\circ = 150^\circ\)[/tex].
2. Since [tex]\(150^\circ\)[/tex] is less than [tex]\(180^\circ\)[/tex], the reference angle [tex]\(\phi\)[/tex] is:
[tex]\[
\phi = 150^\circ
\][/tex]
Summarizing the reference angles:
- When [tex]\(\theta = 300^\circ, \phi = 60^\circ\)[/tex]
- When [tex]\(\theta = 225^\circ, \phi = 135^\circ\)[/tex]
- When [tex]\(\theta = 480^\circ, \phi = 120^\circ\)[/tex]
- When [tex]\(\theta = -210^\circ, \phi = 150^\circ\)[/tex]
[tex]\[
\boxed{60^\circ, 135^\circ, 120^\circ, 150^\circ}
\][/tex]
