Express the following angles in radian measure and centesimal measure :
`45^(@)`

Answer

To express the angle of \( 45^\circ \) in radian measure and centesimal measure, we can follow these steps:

### Step 1: Convert Degrees to Radians
We know the relationship between degrees and radians is given by:
\[ \pi \text{ radians} = 180^\circ \]
From this, we can derive that:
\[ 1^\circ = \frac{\pi}{180} \text{ radians} \]
Now, to convert \( 45^\circ \) to radians, we multiply \( 45 \) by the conversion factor:
\[ 45^\circ = 45 \times \frac{\pi}{180} \]
This simplifies to:
\[ 45^\circ = \frac{45\pi}{180} = \frac{\pi}{4} \text{ radians} \]

### Step 2: Convert Degrees to Centesimal Measure
In the centesimal system (also known as the grade system), \( 90^\circ \) is equivalent to \( 100 \) grades (or gradians). Therefore, we can find the conversion factor:
\[ 1^\circ = \frac{100}{90} \text{ grades} \]
Now, to convert \( 45^\circ \) to grades, we multiply \( 45 \) by the conversion factor:
\[ 45^\circ = 45 \times \frac{100}{90} \]
This simplifies to:
\[ 45^\circ = \frac{4500}{90} = 50 \text{ grades} \]

### Final Answers
- In radian measure: \( \frac{\pi}{4} \) radians
- In centesimal measure: \( 50 \) grades