Convert of the complex number in the polar form: `-1-i`

Explanation

To convert the complex number \(-1 - i\) into polar form, we will follow these steps:

### Step 1: Identify the real and imaginary parts
The given complex number is:
\[ z = -1 - i \]
Here, the real part \(x = -1\) and the imaginary part \(y = -1\).

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

### Step 3: Calculate the argument of \(z\)
The argument (angle) \(\theta\) of the complex number can be calculated using:
\[ \theta = \tan^{-1}\left(\frac{y}{x}\right) \]
Substituting the values:
\[ \theta = \tan^{-1}\left(\frac{-1}{-1}\right) = \tan^{-1}(1) = \frac{\pi}{4} \]
Since both \(x\) and \(y\) are negative, the complex number lies in the third quadrant. Therefore, we adjust the angle:
\[ \theta = \frac{\pi}{4} + \pi = \frac{5\pi}{4} \]

### Step 4: Write the polar form
The polar form of a complex number is given by:
\[ z = |z|(\cos \theta + i \sin \theta) \]
Substituting the modulus and the argument:
\[ z = \sqrt{2}\left(\cos\left(\frac{5\pi}{4}\right) + i \sin\left(\frac{5\pi}{4}\right)\right) \]

### Final Result
Thus, the polar form of the complex number \(-1 - i\) is:
\[ z = \sqrt{2}\left(\cos\left(\frac{5\pi}{4}\right) + i \sin\left(\frac{5\pi}{4}\right)\right) \]

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