` z= -1-isqrt(3)` find argument and modulus of given complex number

Explanation

To find the modulus and argument of the complex number \( z = -1 - i\sqrt{3} \), we will follow these steps:

### Step 1: Identify the components of the complex number
The complex number can be expressed as:
\[ z = x + iy \]
where \( x = -1 \) (the real part) and \( y = -\sqrt{3} \) (the imaginary part).

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

### Step 3: Determine the argument of \( z \)
The argument \( \theta \) of a complex number can be calculated using:
\[ \theta = \tan^{-1}\left(\frac{y}{x}\right) \]
Substituting the values of \( y \) and \( x \):
\[ \theta = \tan^{-1}\left(\frac{-\sqrt{3}}{-1}\right) = \tan^{-1}(\sqrt{3}) \]
The value of \( \tan^{-1}(\sqrt{3}) \) corresponds to \( 60^\circ \) or \( \frac{\pi}{3} \) radians.

### Step 4: Adjust the argument based on the quadrant
Since the complex number \( z \) is located in the third quadrant (both real and imaginary parts are negative), we need to adjust the angle:
\[ \text{Argument of } z = \pi + \frac{\pi}{3} = \frac{3\pi}{3} + \frac{\pi}{3} = \frac{4\pi}{3} \]
This is equivalent to \( 240^\circ \).

### Final Result
Thus, the modulus and argument of the complex number \( z = -1 - i\sqrt{3} \) are:
- **Modulus**: \( |z| = 2 \)
- **Argument**: \( \text{arg}(z) = \frac{4\pi}{3} \) or \( 240^\circ \)

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