Amplitude of `(1+sqrt3i)/(sqrt3+1)`is

Options

A. `pi/6`
B. `pi/4`
C. `pi/3`
D. `pi/2`

Answer

To find the amplitude (or argument) of the complex number \(\frac{1 + \sqrt{3}i}{\sqrt{3} + 1}\), we will follow these steps:

### Step 1: Identify the complex number
We have the complex number in the form \(z = \frac{1 + \sqrt{3}i}{\sqrt{3} + 1}\).

### Step 2: Rewrite the complex number
To find the amplitude, we can express \(z\) in the form \(a + bi\). We will multiply the numerator and the denominator by the conjugate of the denominator to simplify it.

\[ z = \frac{(1 + \sqrt{3}i)(\sqrt{3} - 1)}{(\sqrt{3} + 1)(\sqrt{3} - 1)} \]

### Step 3: Simplify the denominator
The denominator simplifies as follows:

\[ (\sqrt{3} + 1)(\sqrt{3} - 1) = 3 - 1 = 2 \]

### Step 4: Simplify the numerator
Now, we will simplify the numerator:

\[ (1 + \sqrt{3}i)(\sqrt{3} - 1) = 1 \cdot \sqrt{3} + 1 \cdot (-1) + \sqrt{3}i \cdot \sqrt{3} + \sqrt{3}i \cdot (-1) \]
\[ = \sqrt{3} - 1 + 3i - \sqrt{3}i \]
\[ = (\sqrt{3} - 1) + (3 - \sqrt{3})i \]

### Step 5: Combine the results
Now we can write \(z\) as:

\[ z = \frac{(\sqrt{3} - 1) + (3 - \sqrt{3})i}{2} \]

### Step 6: Separate into real and imaginary parts
Let \(a = \frac{\sqrt{3} - 1}{2}\) and \(b = \frac{3 - \sqrt{3}}{2}\).

### Step 7: Find the amplitude (argument)
The amplitude (or argument) \(\theta\) can be found using the formula:

\[ \tan \theta = \frac{b}{a} = \frac{\frac{3 - \sqrt{3}}{2}}{\frac{\sqrt{3} - 1}{2}} = \frac{3 - \sqrt{3}}{\sqrt{3} - 1} \]

### Step 8: Simplify the tangent expression
Now we simplify \(\tan \theta\):

\[ \tan \theta = \frac{3 - \sqrt{3}}{\sqrt{3} - 1} \]

### Step 9: Calculate the angle
Using the known values of tangent, we find that:

\[ \tan \frac{\pi}{3} = \sqrt{3} \quad \text{and} \quad \tan \frac{5\pi}{3} = -\sqrt{3} \]

Since \(\tan \theta = \sqrt{3}\), we have:

\[ \theta = \frac{\pi}{3} \]

### Final Answer
Thus, the amplitude of the complex number \(\frac{1 + \sqrt{3}i}{\sqrt{3} + 1}\) is:

\[ \theta = \frac{\pi}{3} \]