Express the following complex number in the polar form:`(2+6sqrt(3)i)/(5+sqrt(3)i)`

Options

A. `2(cos""(pi)/(4)+isin ""(pi)/(4))`
B. `sqrt2(cos""(pi)/(4)+isin ""(pi)/(4))`
C. `(cos""(pi)/(4)+isin ""(pi)/(4))`
D. None of these

Explanation

To express the complex number \(\frac{2 + 6\sqrt{3}i}{5 + \sqrt{3}i}\) in polar form, we will follow these steps:

### Step 1: Multiply by the Conjugate
To simplify the complex number, we multiply both the numerator and denominator by the conjugate of the denominator. The conjugate of \(5 + \sqrt{3}i\) is \(5 - \sqrt{3}i\).

\[ z = \frac{(2 + 6\sqrt{3}i)(5 - \sqrt{3}i)}{(5 + \sqrt{3}i)(5 - \sqrt{3}i)} \]

### Step 2: Expand the Numerator
Now, we will expand the numerator:

\[ (2 + 6\sqrt{3}i)(5 - \sqrt{3}i) = 2 \cdot 5 + 2 \cdot (-\sqrt{3}i) + 6\sqrt{3}i \cdot 5 + 6\sqrt{3}i \cdot (-\sqrt{3}i) \]

Calculating each term:
- \(2 \cdot 5 = 10\)
- \(2 \cdot (-\sqrt{3}i) = -2\sqrt{3}i\)
- \(6\sqrt{3}i \cdot 5 = 30\sqrt{3}i\)
- \(6\sqrt{3}i \cdot (-\sqrt{3}i) = -6 \cdot 3 = -18(-1) = 18\)

Combining these results:
\[ 10 + 18 + (30\sqrt{3} - 2\sqrt{3})i = 28 + 28\sqrt{3}i \]

### Step 3: Expand the Denominator
Now we will expand the denominator:

\[ (5 + \sqrt{3}i)(5 - \sqrt{3}i) = 5^2 - (\sqrt{3}i)^2 = 25 - 3(-1) = 25 + 3 = 28 \]

### Step 4: Simplify the Expression
Now we can simplify the expression:

\[ z = \frac{28 + 28\sqrt{3}i}{28} = 1 + \sqrt{3}i \]

### Step 5: Find the Modulus
Next, we find the modulus of \(z\):

\[ |z| = \sqrt{A^2 + B^2} = \sqrt{1^2 + (\sqrt{3})^2} = \sqrt{1 + 3} = \sqrt{4} = 2 \]

### Step 6: Find the Argument
Now we find the argument \(\theta\):

\[ \tan(\theta) = \frac{B}{A} = \frac{\sqrt{3}}{1} = \sqrt{3} \]

This corresponds to \(\theta = \frac{\pi}{3}\) (since both \(A\) and \(B\) are positive, it lies in the first quadrant).

### Step 7: Write in Polar Form
Finally, we can express \(z\) in polar form:

\[ z = r(\cos \theta + i \sin \theta) = 2\left(\cos\frac{\pi}{3} + i \sin\frac{\pi}{3}\right) \]

### Final Answer
Thus, the polar form of the complex number \(\frac{2 + 6\sqrt{3}i}{5 + \sqrt{3}i}\) is:

\[ z = 2\left(\cos\frac{\pi}{3} + i \sin\frac{\pi}{3}\right) \]

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