Find the modulus and amplitude of the following complex numbers and hence express them into polar form
`-1- sqrt3i`

Answer

To find the modulus and amplitude of the complex number \( z = -1 - \sqrt{3}i \) and express it in polar form, we will follow these steps:

### Step 1: Identify the real and imaginary parts
The complex number can be expressed as:
\[ z = a + bi \]
where \( a = -1 \) and \( b = -\sqrt{3} \).

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

### Step 3: Calculate the amplitude (argument)
The amplitude \( \theta \) can be found using the formula:
\[ \tan \theta = \frac{b}{a} \]
Substituting the values of \( a \) and \( b \):
\[ \tan \theta = \frac{-\sqrt{3}}{-1} = \sqrt{3} \]
The angle whose tangent is \( \sqrt{3} \) is \( \frac{\pi}{3} \). However, since both the real part and imaginary part are negative, the complex number lies in the third quadrant. Therefore, we need to add \( \pi \) to the angle:
\[ \theta = \pi + \frac{\pi}{3} = \frac{3\pi}{3} + \frac{\pi}{3} = \frac{4\pi}{3} \]

### Step 4: Express in polar form
The polar form of a complex number is given by:
\[ z = |z| \left( \cos \theta + i \sin \theta \right) \]
Substituting the values we found:
\[ z = 2 \left( \cos \frac{4\pi}{3} + i \sin \frac{4\pi}{3} \right) \]

### Final Answer
Thus, the modulus of the complex number \( -1 - \sqrt{3}i \) is \( 2 \), the amplitude is \( \frac{4\pi}{3} \), and the polar form is:
\[ z = 2 \left( \cos \frac{4\pi}{3} + i \sin \frac{4\pi}{3} \right) \]