Express each of the following angles in degrees.
(i) `((5pi)/12)^(c )`, (ii) `- ((18pi)/5)^(c ) ` , (iii) `(5/6)^(c )` , (iv) `-4^(c )`

Answer

To express the given angles in degrees, we will use the conversion factor that \( \pi \) radians is equal to \( 180 \) degrees. Thus, to convert an angle from radians to degrees, we can use the formula:

\[ \text{Degrees} = \text{Radians} \times \left(\frac{180}{\pi}\right) \]

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

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

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

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

3. **Simplify \( \frac{180}{12} \):**
\[ = 15 \quad (\text{since } 180 \div 12 = 15) \]

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

### (ii) Convert \( -\frac{18\pi}{5} \) radians to degrees

1. **Write the conversion formula:**
\[ \text{Degrees} = -\frac{18\pi}{5} \times \left(\frac{180}{\pi}\right) \]

2. **Cancel \( \pi \):**
\[ = -\frac{18 \times 180}{5} \]

3. **Calculate \( \frac{180}{5} \):**
\[ = -36 \quad (\text{since } 180 \div 5 = 36) \]

4. **Multiply:**
\[ = -18 \times 36 = -648 \text{ degrees} \]

### (iii) Convert \( \frac{5}{6} \) radians to degrees

1. **Write the conversion formula:**
\[ \text{Degrees} = \frac{5}{6} \times \left(\frac{180}{\pi}\right) \]

2. **Multiply:**
\[ = \frac{5 \times 180}{6\pi} \]

3. **Simplify \( \frac{180}{6} \):**
\[ = 30 \quad (\text{since } 180 \div 6 = 30) \]

4. **Multiply:**
\[ = \frac{5 \times 30}{\pi} = \frac{150}{\pi} \text{ degrees} \]

5. **Using \( \pi \approx \frac{22}{7} \):**
\[ = \frac{150 \times 7}{22} = \frac{1050}{22} \approx 47.727 \text{ degrees} \]

### (iv) Convert \( -4 \) radians to degrees

1. **Write the conversion formula:**
\[ \text{Degrees} = -4 \times \left(\frac{180}{\pi}\right) \]

2. **Multiply:**
\[ = -\frac{720}{\pi} \]

3. **Using \( \pi \approx \frac{22}{7} \):**
\[ = -\frac{720 \times 7}{22} = -\frac{5040}{22} \approx -229.09 \text{ degrees} \]

### Final Answers:

1. \( \frac{5\pi}{12} \) radians = \( 75 \) degrees
2. \( -\frac{18\pi}{5} \) radians = \( -648 \) degrees
3. \( \frac{5}{6} \) radians = \( 47.73 \) degrees (approximately)
4. \( -4 \) radians = \( -229.09 \) degrees (approximately)