If the axes are rotated through an angle of `30^@` in the anti clockwise direction, then coordinates of point `(4,-2sqrt3)` with respect to new axes are

Options

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

Explanation

To find the new coordinates of the point \((4, -2\sqrt{3})\) after rotating the axes by \(30^\circ\) in the anti-clockwise direction, we can use the following transformation formulas:

1. The new x-coordinate \(x'\) is given by:
\[ x' = x \cos \theta + y \sin \theta \]

2. The new y-coordinate \(y'\) is given by:
\[ y' = -x \sin \theta + y \cos \theta \]

Where:
- \(x\) and \(y\) are the original coordinates,
- \(\theta\) is the angle of rotation.

### Step-by-step Solution:

**Step 1: Identify the values.**
- Given point: \((x, y) = (4, -2\sqrt{3})\)
- Angle of rotation: \(\theta = 30^\circ\)

**Step 2: Calculate \(\cos 30^\circ\) and \(\sin 30^\circ\).**
- \(\cos 30^\circ = \frac{\sqrt{3}}{2}\)
- \(\sin 30^\circ = \frac{1}{2}\)

**Step 3: Substitute the values into the transformation formulas.**

**For the new x-coordinate \(x'\):**
\[ x' = 4 \cos 30^\circ + (-2\sqrt{3}) \sin 30^\circ \]
\[ x' = 4 \cdot \frac{\sqrt{3}}{2} + (-2\sqrt{3}) \cdot \frac{1}{2} \]
\[ x' = 2\sqrt{3} - \sqrt{3} \]
\[ x' = \sqrt{3} \]

**For the new y-coordinate \(y'\):**
\[ y' = -4 \sin 30^\circ + (-2\sqrt{3}) \cos 30^\circ \]
\[ y' = -4 \cdot \frac{1}{2} + (-2\sqrt{3}) \cdot \frac{\sqrt{3}}{2} \]
\[ y' = -2 - 3 \]
\[ y' = -5 \]

**Step 4: Combine the new coordinates.**
The new coordinates of the point with respect to the new axes are:
\[ (x', y') = (\sqrt{3}, -5) \]

### Final Answer:
The coordinates of the point \((4, -2\sqrt{3})\) with respect to the new axes are \((\sqrt{3}, -5)\).

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