If the axes are rotated through an angle`30^(0)` then find the coordinates of (-2,4) in the new system.

Answer

To find the coordinates of the point (-2, 4) after rotating the axes through an angle of 30 degrees, we can use the following transformation formulas:

1. **Transformation Formulas**:
- \( x' = x \cos \theta + y \sin \theta \)
- \( y' = -x \sin \theta + y \cos \theta \)

Where:
- \( (x, y) \) are the original coordinates,
- \( (x', y') \) are the new coordinates after rotation,
- \( \theta \) is the angle of rotation (30 degrees in this case).

2. **Substituting Values**:
- Given \( (x, y) = (-2, 4) \) and \( \theta = 30^\circ \),
- We need to calculate \( \cos 30^\circ \) and \( \sin 30^\circ \):
- \( \cos 30^\circ = \frac{\sqrt{3}}{2} \)
- \( \sin 30^\circ = \frac{1}{2} \)

3. **Calculating \( x' \)**:
\[ x' = (-2) \cos 30^\circ + 4 \sin 30^\circ \]
\[ x' = (-2) \left(\frac{\sqrt{3}}{2}\right) + 4 \left(\frac{1}{2}\right) \]
\[ x' = -\sqrt{3} + 2 \]
\[ x' = 2 - \sqrt{3} \]

4. **Calculating \( y' \)**:
\[ y' = -(-2) \sin 30^\circ + 4 \cos 30^\circ \]
\[ y' = 2 \left(\frac{1}{2}\right) + 4 \left(\frac{\sqrt{3}}{2}\right) \]
\[ y' = 1 + 2\sqrt{3} \]

5. **Final Coordinates**:
Thus, the new coordinates after rotation are:
\[ (x', y') = (2 - \sqrt{3}, 1 + 2\sqrt{3}) \]

### Summary of the Solution:
The coordinates of the point (-2, 4) in the new system after rotating the axes through an angle of 30 degrees are \( (2 - \sqrt{3}, 1 + 2\sqrt{3}) \).