To find two coterminal angles to [tex]\(\frac{3 \pi}{2}\)[/tex] radians, we need to either add or subtract multiples of [tex]\(2\pi\)[/tex] (a full rotation) to/from the given angle.
1. Given angle: [tex]\(\frac{3 \pi}{2}\)[/tex]
2. To find the first coterminal angle, add [tex]\(2 \pi\)[/tex]:
[tex]\[
\text{First coterminal angle} = \frac{3 \pi}{2} + 2 \pi
\][/tex]
Converting [tex]\(2 \pi\)[/tex] to a fraction over 2, we get:
[tex]\[
\frac{3 \pi}{2} + \frac{4 \pi}{2} = \frac{7 \pi}{2}
\][/tex]
3. To find the second coterminal angle, subtract [tex]\(2 \pi\)[/tex]:
[tex]\[
\text{Second coterminal angle} = \frac{3 \pi}{2} - 2 \pi
\][/tex]
Similarly, converting [tex]\(2 \pi\)[/tex] to a fraction over 2, we get:
[tex]\[
\frac{3 \pi}{2} - \frac{4 \pi}{2} = \frac{3 \pi - 4 \pi}{2} = \frac{-\pi}{2}
\][/tex]
Thus, the first pair of coterminal angles is [tex]\(\left( \frac{7 \pi}{2}, -\frac{\pi}{2} \right)\)[/tex].
To provide another set of coterminal angles:
4. We can also calculate by adding or subtracting another [tex]\(2\pi\)[/tex] (or any multiple of [tex]\(2\pi\)[/tex]) to the first obtained coterminal angles. However, let's consider equivalent negative angle outcomes directly:
5. For negative coterminal angles with [tex]\(\frac{3 \pi}{2}\)[/tex]:
Subtract [tex]\(2 \pi\)[/tex] once more from [tex]\(\frac{-\pi}{2}\)[/tex]:
[tex]\[
\text{Third coterminal angle} = \frac{-\pi}{2} - 2 \pi
\][/tex]
Converting [tex]\(2 \pi\)[/tex] into a fraction over 2, we get:
[tex]\[
\frac{-\pi}{2} - \frac{4\pi}{2} = \frac{-5\pi}{2}
\][/tex]
Another suitable combination with original [tex]\( \frac{3\pi}{2}\)[/tex]:
- Take mirror additive to closest possible range making simpler:
[tex]\[
- \left(\frac{7 \pi}{2}\right) \quad and \; resulting, \frac{\pi}{2}
\][/tex]
Thus, the other pair of coterminal angles is [tex]\(\left( -\frac{7\pi}{2}, \frac{\pi}{2} \right) \)[/tex].
Combining these findings, the two requested pairs are:
[tex]\[
(\frac{7 \pi}{2}, - \frac{\pi}{2}) \quad and \quad (-\frac{7 \pi}{2}, \frac{ \pi}{2})
\][/tex]
