Find positive and negative coterminal angles for each indicated angle on the coordinate plane.

To solve the problem of finding positive and negative coterminal angles for each given angle, we need to understand the concept of coterminal angles. Coterminal angles are angles that share the same initial and terminal sides but differ by full rotations (360° or \(2\pi\) radians).

Steps to Find Coterminal Angles:

1. Positive Coterminal Angle: Add \(360^\circ\) (or \(2\pi\) radians) to the given angle.
2. Negative Coterminal Angle: Subtract \(360^\circ\) (or \(2\pi\) radians) from the given angle.

Let's solve each part step by step.

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1) Given: \( \frac{5\pi}{4} \) radians

- Positive Coterminal Angle:
\[
\frac{5\pi}{4} + 2\pi = \frac{5\pi}{4} + \frac{8\pi}{4} = \frac{13\pi}{4}
\]

- Negative Coterminal Angle:
\[
\frac{5\pi}{4} - 2\pi = \frac{5\pi}{4} - \frac{8\pi}{4} = -\frac{3\pi}{4}
\]

Answer:
\[
\text{Positive coterminal angle: } \frac{13\pi}{4}, \quad \text{Negative coterminal angle: } -\frac{3\pi}{4}
\]

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2) Given: \( \frac{\pi}{6} \) radians

- Positive Coterminal Angle:
\[
\frac{\pi}{6} + 2\pi = \frac{\pi}{6} + \frac{12\pi}{6} = \frac{13\pi}{6}
\]

- Negative Coterminal Angle:
\[
\frac{\pi}{6} - 2\pi = \frac{\pi}{6} - \frac{12\pi}{6} = -\frac{11\pi}{6}
\]

Answer:
\[
\text{Positive coterminal angle: } \frac{13\pi}{6}, \quad \text{Negative coterminal angle: } -\frac{11\pi}{6}
\]

---

3) Given: \( \frac{7\pi}{3} \) radians

- Positive Coterminal Angle:
\[
\frac{7\pi}{3} + 2\pi = \frac{7\pi}{3} + \frac{6\pi}{3} = \frac{13\pi}{3}
\]

- Negative Coterminal Angle:
\[
\frac{7\pi}{3} - 2\pi = \frac{7\pi}{3} - \frac{6\pi}{3} = \frac{\pi}{3}
\]
Since \(\frac{\pi}{3}\) is already positive, we subtract another \(2\pi\) to get a negative coterminal angle:
\[
\frac{\pi}{3} - 2\pi = \frac{\pi}{3} - \frac{6\pi}{3} = -\frac{5\pi}{3}
\]

Answer:
\[
\text{Positive coterminal angle: } \frac{13\pi}{3}, \quad \text{Negative coterminal angle: } -\frac{5\pi}{3}
\]

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4) Given: \( \frac{11\pi}{6} \) radians

- Positive Coterminal Angle:
\[
\frac{11\pi}{6} + 2\pi = \frac{11\pi}{6} + \frac{12\pi}{6} = \frac{23\pi}{6}
\]

- Negative Coterminal Angle:
\[
\frac{11\pi}{6} - 2\pi = \frac{11\pi}{6} - \frac{12\pi}{6} = -\frac{\pi}{6}
\]

Answer:
\[
\text{Positive coterminal angle: } \frac{23\pi}{6}, \quad \text{Negative coterminal angle: } -\frac{\pi}{6}
\]

---

5) Given: \( 80^\circ \)

- Positive Coterminal Angle:
\[
80^\circ + 360^\circ = 440^\circ
\]

- Negative Coterminal Angle:
\[
80^\circ - 360^\circ = -280^\circ
\]

Answer:
\[
\text{Positive coterminal angle: } 440^\circ, \quad \text{Negative coterminal angle: } -280^\circ
\]

---

6) Given: \( \frac{3\pi}{2} \) radians

- Positive Coterminal Angle:
\[
\frac{3\pi}{2} + 2\pi = \frac{3\pi}{2} + \frac{4\pi}{2} = \frac{7\pi}{2}
\]

- Negative Coterminal Angle:
\[
\frac{3\pi}{2} - 2\pi = \frac{3\pi}{2} - \frac{4\pi}{2} = -\frac{\pi}{2}
\]

Answer:
\[
\text{Positive coterminal angle: } \frac{7\pi}{2}, \quad \text{Negative coterminal angle: } -\frac{\pi}{2}
\]

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Final Answers:

\[
\boxed{
\begin{array}{ll}
1) & \text{Positive: } \frac{13\pi}{4}, \text{ Negative: } -\frac{3\pi}{4} \\
2) & \text{Positive: } \frac{13\pi}{6}, \text{ Negative: } -\frac{11\pi}{6} \\
3) & \text{Positive: } \frac{13\pi}{3}, \text{ Negative: } -\frac{5\pi}{3} \\
4) & \text{Positive: } \frac{23\pi}{6}, \text{ Negative: } -\frac{\pi}{6} \\
5) & \text{Positive: } 440^\circ, \text{ Negative: } -280^\circ \\
6) & \text{Positive: } \frac{7\pi}{2}, \text{ Negative: } -\frac{\pi}{2}
\end{array}
}
\]