If the axes are rotated through an angle of `30^(@)` in the clockwise direction, the point `(4,-2 sqrt3)` in the new system was formerly

Options

A. `(2, sqrt3)`
B. `(sqrt3, -5)`
C. `(sqrt3,2)`
D. `(2,3)`

Answer

To solve the problem of finding the original coordinates of the point (4, -2√3) after the axes have been rotated through an angle of 30 degrees in the clockwise direction, we will use the transformation formulas for coordinates under rotation.

### Step-by-Step Solution:

1. **Understand the Rotation Transformation**:
When the axes are rotated clockwise by an angle θ, the new coordinates (X, Y) can be expressed in terms of the old coordinates (x, y) as follows:
\[ X = x \cos \theta + y \sin \theta \]
\[ Y = -x \sin \theta + y \cos \theta \]

2. **Identify the Given Values**:
Here, we have:
- New coordinates: \( (X, Y) = (4, -2\sqrt{3}) \)
- Angle of rotation: \( \theta = 30^\circ \)

3. **Calculate Cosine and Sine Values**:
For \( \theta = 30^\circ \):
\[ \cos 30^\circ = \frac{\sqrt{3}}{2}, \quad \sin 30^\circ = \frac{1}{2} \]

4. **Set Up the Equations**:
Substitute the known values into the transformation equations:
\[ 4 = x \cdot \frac{\sqrt{3}}{2} + y \cdot \frac{1}{2} \quad \text{(1)} \]
\[ -2\sqrt{3} = -x \cdot \frac{1}{2} + y \cdot \frac{\sqrt{3}}{2} \quad \text{(2)} \]

5. **Multiply Equation (1) by 2**:
To eliminate the fractions, multiply the first equation by 2:
\[ 8 = x \sqrt{3} + y \quad \text{(3)} \]

6. **Multiply Equation (2) by 2**:
Similarly, multiply the second equation by 2:
\[ -4\sqrt{3} = -x + y \sqrt{3} \quad \text{(4)} \]

7. **Rearranging Equation (4)**:
Rearranging equation (4) gives:
\[ x + y \sqrt{3} = 4\sqrt{3} \quad \text{(5)} \]

8. **Solve the System of Equations**:
Now we have two equations:
- From (3): \( y = 8 - x \sqrt{3} \)
- Substitute \( y \) from (3) into (5):
\[ x + (8 - x \sqrt{3})\sqrt{3} = 4\sqrt{3} \]
\[ x + 8\sqrt{3} - x \cdot 3 = 4\sqrt{3} \]
\[ -2x + 8\sqrt{3} = 4\sqrt{3} \]
\[ -2x = 4\sqrt{3} - 8\sqrt{3} \]
\[ -2x = -4\sqrt{3} \]
\[ x = 2\sqrt{3} \]

9. **Substitute Back to Find y**:
Substitute \( x = 2\sqrt{3} \) back into equation (3):
\[ 8 = 2\sqrt{3} \cdot \sqrt{3} + y \]
\[ 8 = 6 + y \]
\[ y = 2 \]

10. **Final Result**:
The original coordinates before the rotation are:
\[ (x, y) = (2\sqrt{3}, 2) \]