If the axes are rotated through an angle of `45^(@)` in the clockwise direction, the coordinates of a point in the new systeme are (0,-2) then its original coordinates are

Options

A. `(sqrt(2),sqrt(2))`
B. `(-sqrt(2),sqrt(2))`
C. `(sqrt(2),-sqrt(2))`
D. `(-sqrt(2),-sqrt(2))`

Explanation

To find the original coordinates of a point after the axes have been rotated through an angle of \(45^\circ\) in the clockwise direction, we can use the transformation formulas for coordinate rotation.

### Step-by-Step Solution:

1. **Identify the Given Information**:
- The new coordinates after rotation are \((x', y') = (0, -2)\).
- The angle of rotation \(\theta = 45^\circ\).

2. **Use the Rotation Formulas**:
- The formulas to convert the new coordinates back to the original coordinates are:
\[ x = x' \cos \theta - y' \sin \theta \]
\[ y = x' \sin \theta + y' \cos \theta \]

3. **Substitute the Values**:
- Substitute \(x' = 0\), \(y' = -2\), and \(\theta = 45^\circ\) into the formulas. Recall that \(\cos 45^\circ = \frac{1}{\sqrt{2}}\) and \(\sin 45^\circ = \frac{1}{\sqrt{2}}\).

- For \(x\):
\[ x = 0 \cdot \cos 45^\circ - (-2) \cdot \sin 45^\circ \]
\[ x = 0 - (-2) \cdot \frac{1}{\sqrt{2}} = 2 \cdot \frac{1}{\sqrt{2}} = \frac{2}{\sqrt{2}} = \sqrt{2} \]

- For \(y\):
\[ y = 0 \cdot \sin 45^\circ + (-2) \cdot \cos 45^\circ \]
\[ y = 0 - 2 \cdot \frac{1}{\sqrt{2}} = -\frac{2}{\sqrt{2}} = -\sqrt{2} \]

4. **Final Result**:
- The original coordinates are \((x, y) = (\sqrt{2}, -\sqrt{2})\).

### Summary:
Thus, the original coordinates of the point before the rotation are \((\sqrt{2}, -\sqrt{2})\).

---