Convert the complex number `(1+isqrt(3))` into polar form.

Options

A. `2(cos""(pi)/(3)+isin ""(pi)/(3))`
B. `(cos""(pi)/(3)+isin ""(pi)/(3))`
C. `2(cos""(pi)/(6)+isin ""(pi)/(6))`
D. None of these

Explanation

To convert the complex number \( z = 1 + i\sqrt{3} \) into polar form, we will follow these steps:

### Step 1: Identify the real and imaginary parts
The complex number can be expressed as:
- Real part \( a = 1 \)
- Imaginary part \( b = \sqrt{3} \)

### Step 2: Calculate the modulus \( r \)
The modulus \( r \) of a complex number is given by the formula:
\[ r = \sqrt{a^2 + b^2} \]
Substituting the values of \( a \) and \( b \):
\[ r = \sqrt{1^2 + (\sqrt{3})^2} = \sqrt{1 + 3} = \sqrt{4} = 2 \]

### Step 3: Calculate the argument \( \theta \)
The argument \( \theta \) can be found using the tangent function:
\[ \tan \theta = \frac{b}{a} = \frac{\sqrt{3}}{1} = \sqrt{3} \]
The angle \( \theta \) for which \( \tan \theta = \sqrt{3} \) is:
\[ \theta = \frac{\pi}{3} \quad \text{(or 60 degrees)} \]

### Step 4: Write the polar form
The polar form of a complex number is given by:
\[ z = r(\cos \theta + i \sin \theta) \]
Substituting the values of \( r \) and \( \theta \):
\[ z = 2 \left( \cos \frac{\pi}{3} + i \sin \frac{\pi}{3} \right) \]

### Final Result
Thus, the polar form of the complex number \( 1 + i\sqrt{3} \) is:
\[ z = 2 \left( \cos \frac{\pi}{3} + i \sin \frac{\pi}{3} \right) \]

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