The modulus-argument form of `sqrt( 3) + i`, where `i = sqrt( -1)` is

Options

A. A) `2 (cos "" ( pi )/( 3) + i sin ""( pi )/( 3))`
B. B) `2 (cos "" ( pi )/( 6) + i sin ""( pi )/( 6))`
C. C) `4 (cos "" ( pi )/( 3) + i sin ""( pi )/( 3))`
D. D) `4 (cos "" ( pi )/( 6) + i sin ""( pi )/( 6))`

Answer

To find the modulus-argument form of the complex number \( \sqrt{3} + i \), we will follow these steps:

### Step 1: Identify the Real and Imaginary Parts
The complex number can be expressed as:
\[ z = \sqrt{3} + i \]
Here, the real part \( x = \sqrt{3} \) and the imaginary part \( y = 1 \).

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

### Step 3: Calculate the Argument
The argument \( \theta \) (or \( \alpha \)) of the complex number is given by:
\[ \tan \theta = \frac{y}{x} \]
Substituting the values of \( y \) and \( x \):
\[ \tan \theta = \frac{1}{\sqrt{3}} \]
We know that \( \tan 30^\circ = \frac{1}{\sqrt{3}} \), so:
\[ \theta = 30^\circ \quad \text{or} \quad \theta = \frac{\pi}{6} \text{ radians} \]

### Step 4: Write in Modulus-Argument Form
The modulus-argument 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 Answer
Thus, the modulus-argument form of \( \sqrt{3} + i \) is:
\[ 2 \left( \cos \frac{\pi}{6} + i \sin \frac{\pi}{6} \right) \]

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