Express the following angle in degrees.
(i) `((5pi)/(12))^(c)` `" "` (ii) `-((7pi)/(12))^(c)`
(iii) `(1^(c))/(3)` `" "` (iv) `-(2pi^(c))/(9)`

Answer

To convert the given angles from radians to degrees, we will use the conversion factor that \( \pi \) radians is equal to 180 degrees. The formula to convert radians to degrees is:

\[ \text{Degrees} = \text{Radians} \times \frac{180}{\pi} \]

Now, let's solve each part step by step.

### (i) Convert \( \frac{5\pi}{12} \) radians to degrees

1. **Write the conversion formula**:
\[ \text{Degrees} = \frac{5\pi}{12} \times \frac{180}{\pi} \]

2. **Cancel \( \pi \)**:
\[ \text{Degrees} = \frac{5 \times 180}{12} \]

3. **Calculate \( \frac{180}{12} \)**:
\[ \frac{180}{12} = 15 \]

4. **Multiply by 5**:
\[ \text{Degrees} = 5 \times 15 = 75 \]

**Final Answer**: \( 75^\circ \)

---

### (ii) Convert \( -\frac{7\pi}{12} \) radians to degrees

1. **Write the conversion formula**:
\[ \text{Degrees} = -\frac{7\pi}{12} \times \frac{180}{\pi} \]

2. **Cancel \( \pi \)**:
\[ \text{Degrees} = -\frac{7 \times 180}{12} \]

3. **Calculate \( \frac{180}{12} \)**:
\[ \frac{180}{12} = 15 \]

4. **Multiply by -7**:
\[ \text{Degrees} = -7 \times 15 = -105 \]

**Final Answer**: \( -105^\circ \)

---

### (iii) Convert \( \frac{1}{3} \) radians to degrees

1. **Write the conversion formula**:
\[ \text{Degrees} = \frac{1}{3} \times \frac{180}{\pi} \]

2. **Calculate \( \frac{180}{\pi} \)**:
\[ \text{Degrees} = \frac{180}{3\pi} \]

3. **Calculate \( \frac{180}{3} \)**:
\[ \frac{180}{3} = 60 \]
Thus,
\[ \text{Degrees} = \frac{60}{\pi} \]

4. **Approximate \( \frac{60}{\pi} \)**:
\[ \text{Degrees} \approx 19.1 \]

5. **Convert to degrees, minutes, and seconds**:
- \( 19 \) degrees
- \( 0.1 \) degrees \( \approx 6 \) minutes
- \( 0.1 \) degrees \( \approx 36 \) seconds

**Final Answer**: \( 19^\circ 6' 36'' \)

---

### (iv) Convert \( -\frac{2\pi}{9} \) radians to degrees

1. **Write the conversion formula**:
\[ \text{Degrees} = -\frac{2\pi}{9} \times \frac{180}{\pi} \]

2. **Cancel \( \pi \)**:
\[ \text{Degrees} = -\frac{2 \times 180}{9} \]

3. **Calculate \( \frac{180}{9} \)**:
\[ \frac{180}{9} = 20 \]

4. **Multiply by -2**:
\[ \text{Degrees} = -2 \times 20 = -40 \]

**Final Answer**: \( -40^\circ \)

---

### Summary of Answers:
(i) \( 75^\circ \)
(ii) \( -105^\circ \)
(iii) \( 19^\circ 6' 36'' \)
(iv) \( -40^\circ \)