Find the modulus and amplitude of the following complex numbers and hence express them into polar form
`-2+2 sqrt3i`

Answer

To find the modulus and amplitude of the complex number \( z = -2 + 2\sqrt{3}i \) 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 in the form \( z = a + bi \), where:
- \( a = -2 \) (real part)
- \( b = 2\sqrt{3} \) (imaginary part)

### Step 2: Calculate the modulus
The modulus of a complex number \( z \) is given by the formula:
\[ |z| = \sqrt{a^2 + b^2} \]
Substituting the values of \( a \) and \( b \):
\[ |z| = \sqrt{(-2)^2 + (2\sqrt{3})^2} \]
Calculating the squares:
\[ |z| = \sqrt{4 + 4 \cdot 3} = \sqrt{4 + 12} = \sqrt{16} = 4 \]

### Step 3: Calculate the amplitude (argument)
The amplitude (or argument) of a complex number is given by:
\[ \theta = \tan^{-1}\left(\frac{b}{a}\right) \]
Substituting the values of \( a \) and \( b \):
\[ \theta = \tan^{-1}\left(\frac{2\sqrt{3}}{-2}\right) = \tan^{-1}(-\sqrt{3}) \]
The value \( -\sqrt{3} \) corresponds to angles in the second quadrant. The reference angle where \( \tan \theta = \sqrt{3} \) is \( \frac{\pi}{3} \). Therefore, in the second quadrant:
\[ \theta = \pi - \frac{\pi}{3} = \frac{2\pi}{3} \]

### Step 4: Express in polar form
The polar form of a complex number is given by:
\[ z = r(\cos \theta + i \sin \theta) \]
where \( r \) is the modulus and \( \theta \) is the argument. Substituting our values:
\[ z = 4\left(\cos\left(\frac{2\pi}{3}\right) + i \sin\left(\frac{2\pi}{3}\right)\right) \]

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

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