Express in polar form (mod-Amplitude form)
`sqrt3 +i`

Answer

To express the complex number \( z = \sqrt{3} + i \) in polar form (modulus-amplitude form), we follow these steps:

### Step 1: Calculate the Modulus
The modulus \( r \) of a complex number \( z = a + bi \) is given by the formula:
\[ r = \sqrt{a^2 + b^2} \]
For our complex number \( z = \sqrt{3} + i \), we have \( a = \sqrt{3} \) and \( b = 1 \).

Calculating the modulus:
\[ r = \sqrt{(\sqrt{3})^2 + (1)^2} = \sqrt{3 + 1} = \sqrt{4} = 2 \]

### Step 2: Calculate the Argument (Amplitude)
The argument \( \theta \) can be found using the tangent function:
\[ \tan \theta = \frac{b}{a} \]
Substituting the values of \( a \) and \( b \):
\[ \tan \theta = \frac{1}{\sqrt{3}} \]
To find \( \theta \), we recognize that \( \tan 30^\circ = \frac{1}{\sqrt{3}} \), thus:
\[ \theta = 30^\circ \]

### Step 3: Write in Polar Form
Now, we can express the complex number in polar form:
\[ z = r(\cos \theta + i \sin \theta) \]
Substituting the values of \( r \) and \( \theta \):
\[ z = 2 \left( \cos 30^\circ + i \sin 30^\circ \right) \]

### Final Result
Thus, the polar form of the complex number \( \sqrt{3} + i \) is:
\[ z = 2 \left( \cos 30^\circ + i \sin 30^\circ \right) \]

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