What will be the co ordinates of the point `(4,2sqrt(3))` when the axes are rotated through an angle of `30^(@)` in clockwise sense?

Answer

To find the coordinates of the point \( (4, 2\sqrt{3}) \) when the axes are rotated through an angle of \( 30^\circ \) in the clockwise direction, we can use the formulas for rotation of coordinates.

### Step-by-Step Solution:

1. **Identify the Original Coordinates:**
The original coordinates of the point are given as:
\[ (x, y) = (4, 2\sqrt{3}) \]

2. **Determine the Angle of Rotation:**
Since the rotation is clockwise, we need to use \( -30^\circ \) (negative because clockwise is considered negative in standard mathematics).

3. **Use the Rotation Formulas:**
The formulas for the new coordinates \( (x', y') \) after rotation are:
\[ x' = x \cos(\theta) + y \sin(\theta) \]
\[ y' = -x \sin(\theta) + y \cos(\theta) \]
where \( \theta = -30^\circ \).

4. **Calculate \( \cos(-30^\circ) \) and \( \sin(-30^\circ) \):**
Using trigonometric values:
\[ \cos(-30^\circ) = \cos(30^\circ) = \frac{\sqrt{3}}{2} \]
\[ \sin(-30^\circ) = -\sin(30^\circ) = -\frac{1}{2} \]

5. **Substitute the Values into the Formulas:**
Now substitute \( x = 4 \), \( y = 2\sqrt{3} \), \( \cos(-30^\circ) \), and \( \sin(-30^\circ) \) into the rotation formulas:
\[ x' = 4 \cdot \frac{\sqrt{3}}{2} + 2\sqrt{3} \cdot \left(-\frac{1}{2}\right) \]
\[ y' = -4 \cdot \left(-\frac{1}{2}\right) + 2\sqrt{3} \cdot \frac{\sqrt{3}}{2} \]

6. **Simplify the Expressions:**
For \( x' \):
\[ x' = 2\sqrt{3} - \sqrt{3} = \sqrt{3} \]
For \( y' \):
\[ y' = 2 + 3 = 5 \]

7. **Final Coordinates:**
Therefore, the coordinates of the point after the rotation are:
\[ (x', y') = (\sqrt{3}, 5) \]

### Final Answer:
The coordinates of the point \( (4, 2\sqrt{3}) \) when the axes are rotated through an angle of \( 30^\circ \) in the clockwise sense are \( (\sqrt{3}, 5) \).