`- 1- i` convert in polar form.

Explanation

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

### Step 1: Identify the complex number
Let \( z = -1 - i \).

### Step 2: Calculate the modulus \( r \)
The modulus \( r \) of a complex number \( z = a + bi \) is given by:
\[ r = \sqrt{a^2 + b^2} \]
Here, \( a = -1 \) and \( b = -1 \). Thus,
\[ r = \sqrt{(-1)^2 + (-1)^2} = \sqrt{1 + 1} = \sqrt{2} \]

### Step 3: Calculate the argument \( \theta \)
The argument \( \theta \) can be found using the formulas:
\[ \cos \theta = \frac{a}{r} \quad \text{and} \quad \sin \theta = \frac{b}{r} \]
Substituting the values we have:
\[ \cos \theta = \frac{-1}{\sqrt{2}} \quad \text{and} \quad \sin \theta = \frac{-1}{\sqrt{2}} \]

### Step 4: Determine the angle \( \theta \)
From the values of \( \cos \theta \) and \( \sin \theta \), we can see that both are negative, which means the angle \( \theta \) is located in the third quadrant. The reference angle where both sine and cosine are equal (and negative) is \( \frac{\pi}{4} \). Therefore, in the third quadrant:
\[ \theta = \pi + \frac{\pi}{4} = \frac{5\pi}{4} \]

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

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

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