Find the modulus and amplitude of the following complex numbers and hence express them into polar form
`sqrt3+i`

Answer

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

### Step 1: Identify the complex number
Let \( z = \sqrt{3} + i \).

### Step 2: Find the modulus \( r \)
The modulus of a complex number \( z = a + bi \) is given by the formula:
\[ r = |z| = \sqrt{a^2 + b^2} \]
In our case, \( a = \sqrt{3} \) and \( b = 1 \). Thus,
\[ r = \sqrt{(\sqrt{3})^2 + (1)^2} = \sqrt{3 + 1} = \sqrt{4} = 2 \]

### Step 3: Find the amplitude \( \theta \)
The amplitude (or argument) \( \theta \) of a complex number is given by:
\[ \theta = \tan^{-1} \left( \frac{b}{a} \right) \]
Substituting the values of \( a \) and \( b \):
\[ \theta = \tan^{-1} \left( \frac{1}{\sqrt{3}} \right) \]
We know that \( \tan \left( \frac{\pi}{6} \right) = \frac{1}{\sqrt{3}} \), so:
\[ \theta = \frac{\pi}{6} \]

### Step 4: Express in 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 \( \theta \):
\[ z = 2 \left( \cos \frac{\pi}{6} + i \sin \frac{\pi}{6} \right) \]

### Final Result
Thus, the modulus of \( z \) is \( 2 \), the amplitude is \( \frac{\pi}{6} \), and the polar form of the complex number is:
\[ z = 2 \left( \cos \frac{\pi}{6} + i \sin \frac{\pi}{6} \right) \]