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

Explanation

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

### Step 1: Identify the real and imaginary parts
The complex number can be expressed in the form \( z = a + bi \), where:
- \( a = 1 \) (real part)
- \( b = \sqrt{3} \) (imaginary part)

### 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 formulas:
\[ \cos \theta = \frac{a}{r} \quad \text{and} \quad \sin \theta = \frac{b}{r} \]
Substituting the values:
\[ \cos \theta = \frac{1}{2} \quad \text{and} \quad \sin \theta = \frac{\sqrt{3}}{2} \]
From trigonometric values, we know that:
\[ \theta = \frac{\pi}{3} \quad \text{(or 60 degrees)} \]

### 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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