To express the complex number \( -1 - i \) in polar form, we need to find the modulus \( r \) and the argument \( \theta \) of the complex number.

### Step 1: Find the modulus \( r \)

The modulus \( r \) of a complex number \( a + bi \) is given by:

\[
r = \sqrt{a^2 + b^2}
\]

For \( -1 - i \), we have \( a = -1 \) and \( b = -1 \). So:

\[
r = \sqrt{(-1)^2 + (-1)^2} = \sqrt{1 + 1} = \sqrt{2}
\]

### Step 2: Find the argument \( \theta \)

The argument \( \theta \) is the angle the complex number makes with the positive real axis, and it can be found using the formula:

\[
\theta = \tan^{-1}\left(\frac{b}{a}\right)
\]

However, in this case, \( a = -1 \) and \( b = -1 \), so the complex number lies in the third quadrant. Thus, we need to add \( \pi \) to the principal value of the argument (since the arctangent function alone will give us the wrong quadrant).

First, calculate:

\[
\theta = \tan^{-1}\left(\frac{-1}{-1}\right) = \tan^{-1}(1) = \frac{\pi}{4}
\]

Since the number is in the third quadrant, we add \( \pi \) to get:

\[
\theta = \frac{\pi}{4} + \pi = \frac{5\pi}{4}
\]

### Step 3: Write in polar form

The polar form of a complex number is given by:

\[
r (\cos \theta + i \sin \theta)
\]

Substitute \( r = \sqrt{2} \) and \( \theta = \frac{5\pi}{4} \):

\[
-1 - i = \sqrt{2} \left( \cos\left( \frac{5\pi}{4} \right) + i \sin\left( \frac{5\pi}{4} \right) \right)
\]

This is the polar form of the complex number \( -1 - i \).

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Would you like further details or have any questions about this process? 

Here are some related questions:
1. How do you convert a complex number from polar to rectangular form?
2. What is the geometric interpretation of a complex number in polar form?
3. How do you find the argument of a complex number in the second quadrant?
4. How does the modulus relate to the distance of a point from the origin in the complex plane?
5. Can polar form be used for operations like multiplication or division of complex numbers?

**Tip:** Always check which quadrant the complex number is in before determining the argument, since it will affect the angle!

Learn how to express the complex number -1 - i in polar form. This solution covers the process of finding the modulus and argument of the complex number, and how to write it in polar form using key formulas. Suitable for high school students.

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