Find the radian measure corresponding to the (a) following degree measure: `40^(@) 20`
following grade measure : (i) `50^(g)` (ii)150^(g)

Answer

To find the radian measure corresponding to the given degree and grade measures, we will follow these steps:

### Part (a): Convert `40° 20'` to Radians

1. **Convert Minutes to Degrees**:
- We know that 1 minute = 1/60 degrees.
- Therefore, 20 minutes = \( \frac{20}{60} = \frac{1}{3} \) degrees.

2. **Add Degrees and Minutes**:
- Now, we add the degrees and the converted minutes:
\[ 40° + \frac{1}{3}° = \frac{120}{3}° + \frac{1}{3}° = \frac{121}{3}° \]

3. **Convert Degrees to Radians**:
- We use the conversion factor \( \frac{\pi \text{ radians}}{180°} \).
- Thus,
\[ \frac{121}{3}° \times \frac{\pi}{180} = \frac{121\pi}{540} \text{ radians} \]

### Part (b): Convert Grades to Radians

#### (i) Convert `50g` to Radians
1. **Understand the Conversion**:
- We know that 100 grades correspond to \( \frac{\pi}{2} \) radians (right angle).

2. **Calculate Radians for 50 Grades**:
- Using the proportion:
\[ 50g = \frac{\pi}{2} \times \frac{50}{100} = \frac{\pi}{2} \times \frac{1}{2} = \frac{\pi}{4} \text{ radians} \]

#### (ii) Convert `150g` to Radians
1. **Calculate Radians for 150 Grades**:
- Again using the same conversion:
\[ 150g = \frac{\pi}{2} \times \frac{150}{100} = \frac{\pi}{2} \times \frac{3}{2} = \frac{3\pi}{4} \text{ radians} \]

### Final Answers:
- (a) \( 40° 20' = \frac{121\pi}{540} \) radians
- (b)(i) \( 50g = \frac{\pi}{4} \) radians
- (b)(ii) \( 150g = \frac{3\pi}{4} \) radians