Express in polar form (mod-Amplitude form)
`-2 + 2isqrt3`

Answer

To express the complex number \(-2 + 2i\sqrt{3}\) in polar form, we will follow these steps:

### Step 1: Identify the complex number
The given complex number is:
\[ z = -2 + 2i\sqrt{3} \]

### Step 2: Calculate the modulus \(r\)
The modulus \(r\) of a complex number \(z = a + bi\) is given by the formula:
\[ r = \sqrt{a^2 + b^2} \]
Here, \(a = -2\) and \(b = 2\sqrt{3}\).

Calculating \(r\):
\[ r = \sqrt{(-2)^2 + (2\sqrt{3})^2} = \sqrt{4 + 4 \cdot 3} = \sqrt{4 + 12} = \sqrt{16} = 4 \]

### Step 3: Calculate the angle \(\theta\)
The angle \(\theta\) can be found using the formulas:
\[ \cos \theta = \frac{a}{r} \quad \text{and} \quad \sin \theta = \frac{b}{r} \]
Substituting the values:
\[ \cos \theta = \frac{-2}{4} = -\frac{1}{2} \quad \text{and} \quad \sin \theta = \frac{2\sqrt{3}}{4} = \frac{\sqrt{3}}{2} \]

### Step 4: Determine the quadrant
Since \(\cos \theta\) is negative and \(\sin \theta\) is positive, the angle \(\theta\) is in the second quadrant.

### Step 5: Find the angle \(\theta\)
From trigonometric values, we know:
\[ \cos 120^\circ = -\frac{1}{2} \quad \text{and} \quad \sin 120^\circ = \frac{\sqrt{3}}{2} \]
Thus, \(\theta = 120^\circ\).

### Step 6: Write the polar form
The polar form of the complex number is given by:
\[ z = r(\cos \theta + i\sin \theta) \]
Substituting the values of \(r\) and \(\theta\):
\[ z = 4\left(\cos 120^\circ + i\sin 120^\circ\right) \]

### Final Answer
The polar form of the complex number \(-2 + 2i\sqrt{3}\) is:
\[ z = 4\left(\cos 120^\circ + i\sin 120^\circ\right) \]

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