Represent the complex numbers
`z= 1 + sqrt3i` into polar form

Answer

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

### Step 1: Identify the components of the complex number
The complex number \( z \) can be expressed in the form \( z = x + yi \), where:
- \( x = 1 \)
- \( y = \sqrt{3} \)

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

### Step 3: Calculate the argument \( \theta \)
The argument \( \theta \) can be calculated using the formula:
\[ \tan \theta = \frac{y}{x} \]
Substituting the values:
\[ \tan \theta = \frac{\sqrt{3}}{1} = \sqrt{3} \]
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 \( z = 1 + \sqrt{3}i \) is:
\[ z = 2 \left( \cos \frac{\pi}{3} + i \sin \frac{\pi}{3} \right) \]

---