(ii)The amplitude of `-1 +sqrt-3` is

Options

A. Purely real complex number
B. forth quadrant
C. First quadrant
D. `2(cos(pi)/(6)+isin(pi)/(6))`

Answer

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

### Step 1: Rewrite the complex number
The complex number can be expressed as:
\[ -1 + \sqrt{-3} = -1 + \sqrt{3}i \]
Here, we have used the fact that \(\sqrt{-3} = \sqrt{3}i\).

### Step 2: Identify the real and imaginary parts
In the complex number \(x + yi\), we identify:
- \(x = -1\) (the real part)
- \(y = \sqrt{3}\) (the imaginary part)

### Step 3: Determine the quadrant
Since \(x < 0\) and \(y > 0\), the complex number lies in the second quadrant.

### Step 4: Calculate the amplitude
The amplitude (or argument) of a complex number is given by:
\[ \theta = \tan^{-1}\left(\frac{y}{x}\right) \]
Substituting the values of \(x\) and \(y\):
\[ \theta = \tan^{-1}\left(\frac{\sqrt{3}}{-1}\right) = \tan^{-1}(-\sqrt{3}) \]

### Step 5: Adjust for the quadrant
Since the complex number is in the second quadrant, we need to adjust the angle:
\[ \theta = \pi - \tan^{-1}(\sqrt{3}) \]
We know that \(\tan^{-1}(\sqrt{3}) = \frac{\pi}{3}\), so:
\[ \theta = \pi - \frac{\pi}{3} = \frac{3\pi}{3} - \frac{\pi}{3} = \frac{2\pi}{3} \]

### Final Answer
Thus, the amplitude of the complex number \(-1 + \sqrt{-3}\) is:
\[ \frac{2\pi}{3} \]

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