The amplitude of `(1+ i sqrt3)/( sqrt3 + i )` is

Options

A. `(pi)/(6)`
B. `(pi)/(3)`
C. `0`
D. None of these

Answer

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

### Step 1: Identify the complex number
We start with the complex number:
\[ z = \frac{1 + i\sqrt{3}}{\sqrt{3} + i} \]

### Step 2: Multiply by the conjugate of the denominator
To simplify this expression, we multiply the numerator and the denominator by the conjugate of the denominator, which is \(\sqrt{3} - i\):
\[ z = \frac{(1 + i\sqrt{3})(\sqrt{3} - i)}{(\sqrt{3} + i)(\sqrt{3} - i)} \]

### Step 3: Simplify the denominator
The denominator simplifies as follows:
\[ (\sqrt{3} + i)(\sqrt{3} - i) = \sqrt{3}^2 - i^2 = 3 - (-1) = 3 + 1 = 4 \]

### Step 4: Simplify the numerator
Now, we simplify the numerator:
\[ (1 + i\sqrt{3})(\sqrt{3} - i) = 1 \cdot \sqrt{3} + 1 \cdot (-i) + i\sqrt{3} \cdot \sqrt{3} + i\sqrt{3} \cdot (-i) \]
Calculating each term:
\[ = \sqrt{3} - i + 3i - \sqrt{3} = \sqrt{3} - \sqrt{3} + (3i - i) = 2i \]

### Step 5: Combine the results
Now we can combine the results:
\[ z = \frac{2i}{4} = \frac{i}{2} \]

### Step 6: Write in standard form
The complex number in standard form is:
\[ z = 0 + \frac{1}{2}i \]

### Step 7: Find the amplitude
The amplitude (or argument) \(\theta\) of a complex number \(z = x + iy\) is given by:
\[ \tan(\theta) = \frac{y}{x} \]
In our case, \(x = 0\) and \(y = \frac{1}{2}\). Thus:
\[ \tan(\theta) = \frac{\frac{1}{2}}{0} \]
Since the real part \(x = 0\), the angle \(\theta\) is \(\frac{\pi}{2}\) (90 degrees).

### Final Answer
The amplitude of \(\frac{1 + i\sqrt{3}}{\sqrt{3} + i}\) is:
\[ \theta = \frac{\pi}{2} \]

---