Find the Cartesian co-ordinates of point whose polar co-ordinates are `(4, (pi)/(3))`.

Explanation

To convert the polar coordinates \((r, \theta) = (4, \frac{\pi}{3})\) into Cartesian coordinates \((x, y)\), we can use the following relationships:

1. \(x = r \cos(\theta)\)
2. \(y = r \sin(\theta)\)

### Step-by-Step Solution:

**Step 1: Identify the values of \(r\) and \(\theta\)**
From the given polar coordinates, we have:
- \(r = 4\)
- \(\theta = \frac{\pi}{3}\)

**Step 2: Calculate \(x\)**
Using the formula for \(x\):
\[ x = r \cos(\theta) = 4 \cos\left(\frac{\pi}{3}\right) \]
We know that \(\cos\left(\frac{\pi}{3}\right) = \frac{1}{2}\), so:
\[ x = 4 \times \frac{1}{2} = 2 \]

**Step 3: Calculate \(y\)**
Using the formula for \(y\):
\[ y = r \sin(\theta) = 4 \sin\left(\frac{\pi}{3}\right) \]
We know that \(\sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2}\), so:
\[ y = 4 \times \frac{\sqrt{3}}{2} = 2\sqrt{3} \]

**Step 4: Write the Cartesian coordinates**
Now that we have both \(x\) and \(y\):
- \(x = 2\)
- \(y = 2\sqrt{3}\)

Thus, the Cartesian coordinates are:
\[ (x, y) = (2, 2\sqrt{3}) \]

### Final Answer:
The Cartesian coordinates of the point whose polar coordinates are \((4, \frac{\pi}{3})\) are \((2, 2\sqrt{3})\).

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