Express the following angles in radian measure and centesimal measure :
`20^(@)35'`

Answer

To express the angle \(20^\circ 35'\) in radian measure and centesimal measure, we will follow these steps:

### Step 1: Convert the angle to decimal degrees
The angle is given as \(20^\circ 35'\). We know that:
- \(1^\circ = 60'\)

To convert \(35'\) to degrees:
\[ 35' = \frac{35}{60}^\circ = \frac{7}{12}^\circ \]

Now, we can add this to \(20^\circ\):
\[ 20^\circ + 35' = 20^\circ + \frac{7}{12}^\circ = \frac{240}{12}^\circ + \frac{7}{12}^\circ = \frac{247}{12}^\circ \]

### Step 2: Convert degrees to radians
To convert degrees to radians, we use the conversion factor:
\[ 1^\circ = \frac{\pi}{180} \text{ radians} \]

Thus, we convert \(\frac{247}{12}^\circ\) to radians:
\[ \frac{247}{12}^\circ = \frac{247}{12} \times \frac{\pi}{180} \text{ radians} \]

Calculating this gives:
\[ \frac{247 \pi}{12 \times 180} = \frac{247 \pi}{2160} \text{ radians} \]

### Step 3: Convert degrees to centesimal (grade) measure
To convert degrees to centesimal measure (grades), we know:
\[ 90^\circ = 100 \text{ grades} \]

Thus, the conversion factor is:
\[ 1^\circ = \frac{100}{90} = \frac{10}{9} \text{ grades} \]

Now, we convert \(\frac{247}{12}^\circ\) to grades:
\[ \frac{247}{12}^\circ \times \frac{10}{9} = \frac{247 \times 10}{12 \times 9} = \frac{2470}{108} \]

Simplifying this fraction:
\[ \frac{2470 \div 2}{108 \div 2} = \frac{1235}{54} \text{ grades} \]

### Final Answers
- In radians: \(\frac{247 \pi}{2160} \text{ radians}\)
- In grades: \(\frac{1235}{54} \text{ grades}\)

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