Express each of the following angles in radians.
(i) `36^(@)` , (ii) `120^(@)` , (iii)) `225^(@)`, (iv) `330^(@)`
(v) `400^(@)`, (iv) `7^(@) 30'`, (vii) `-270^(@)` , (viii) `-(22^(@)30')`

Answer

To convert the given angles from degrees to radians, we will use the formula:

\[ \text{radians} = \frac{\pi}{180} \times \text{degrees} \]

Now, let's apply this formula step by step for each angle.

### Step 1: Convert \(36^\circ\) to radians
\[ \text{radians} = \frac{\pi}{180} \times 36 = \frac{\pi}{5} \]

### Step 2: Convert \(120^\circ\) to radians
\[ \text{radians} = \frac{\pi}{180} \times 120 = \frac{2\pi}{3} \]

### Step 3: Convert \(225^\circ\) to radians
\[ \text{radians} = \frac{\pi}{180} \times 225 = \frac{5\pi}{4} \]

### Step 4: Convert \(330^\circ\) to radians
\[ \text{radians} = \frac{\pi}{180} \times 330 = \frac{11\pi}{6} \]

### Step 5: Convert \(400^\circ\) to radians
\[ \text{radians} = \frac{\pi}{180} \times 400 = \frac{20\pi}{9} \]

### Step 6: Convert \(7^\circ 30'\) to radians
First, convert \(7^\circ 30'\) to decimal degrees:
\[ 7^\circ 30' = 7 + \frac{30}{60} = 7.5^\circ \]
Now convert to radians:
\[ \text{radians} = \frac{\pi}{180} \times 7.5 = \frac{\pi}{24} \]

### Step 7: Convert \(-270^\circ\) to radians
\[ \text{radians} = \frac{\pi}{180} \times (-270) = -\frac{3\pi}{2} \]

### Step 8: Convert \(-22^\circ 30'\) to radians
First, convert \(-22^\circ 30'\) to decimal degrees:
\[ -22^\circ 30' = -22 - \frac{30}{60} = -22.5^\circ \]
Now convert to radians:
\[ \text{radians} = \frac{\pi}{180} \times (-22.5) = -\frac{\pi}{8} \]

### Summary of Results:
1. \(36^\circ = \frac{\pi}{5}\)
2. \(120^\circ = \frac{2\pi}{3}\)
3. \(225^\circ = \frac{5\pi}{4}\)
4. \(330^\circ = \frac{11\pi}{6}\)
5. \(400^\circ = \frac{20\pi}{9}\)
6. \(7^\circ 30' = \frac{\pi}{24}\)
7. \(-270^\circ = -\frac{3\pi}{2}\)
8. \(-22^\circ 30' = -\frac{\pi}{8}\)