If the axes are rotated through an angle `30^(0)`, then find the coordinates of (0, 5) in the new system.

Answer

To find the coordinates of the point (0, 5) after rotating the axes through an angle of \(30^\circ\), we will use the transformation formulas for rotating the axes.

### Step-by-Step Solution:

1. **Identify the Original Coordinates**:
The original coordinates are given as \( (x, y) = (0, 5) \).

2. **Determine the Angle of Rotation**:
The angle of rotation is \( \theta = 30^\circ \).

3. **Use the Rotation Formulas**:
The new coordinates \( (x', y') \) after rotation are given by:
\[ x' = x \cos \theta + y \sin \theta \]
\[ y' = -x \sin \theta + y \cos \theta \]

4. **Substitute the Values**:
- For \( x' \):
\[ x' = 0 \cdot \cos(30^\circ) + 5 \cdot \sin(30^\circ) \]
- For \( y' \):
\[ y' = -0 \cdot \sin(30^\circ) + 5 \cdot \cos(30^\circ) \]

5. **Calculate the Trigonometric Values**:
- We know:
\[ \cos(30^\circ) = \frac{\sqrt{3}}{2}, \quad \sin(30^\circ) = \frac{1}{2} \]

6. **Calculate \( x' \)**:
\[ x' = 0 + 5 \cdot \frac{1}{2} = \frac{5}{2} \]

7. **Calculate \( y' \)**:
\[ y' = 0 + 5 \cdot \frac{\sqrt{3}}{2} = \frac{5\sqrt{3}}{2} \]

8. **Final Result**:
The new coordinates after rotation are:
\[ (x', y') = \left( \frac{5}{2}, \frac{5\sqrt{3}}{2} \right) \]