To express the complex number \( Z = -2 + 2i\sqrt{3} \) in polar form, we need to find its magnitude \( r \) and angle \( \theta \).

### Step 1: Calculate the magnitude \( r \)
The magnitude \( r \) of a complex number \( Z = a + bi \) is given by:
\[
r = \sqrt{a^2 + b^2}
\]
For \( Z = -2 + 2i\sqrt{3} \), where \( a = -2 \) and \( b = 2\sqrt{3} \):
\[
r = \sqrt{(-2)^2 + (2\sqrt{3})^2} = \sqrt{4 + 12} = \sqrt{16} = 4
\]

### Step 2: Calculate the angle \( \theta \)
The angle \( \theta \) (also known as the argument) is found using:
\[
\theta = \tan^{-1}\left(\frac{b}{a}\right)
\]
For \( Z = -2 + 2i\sqrt{3} \):
\[
\theta = \tan^{-1}\left(\frac{2\sqrt{3}}{-2}\right) = \tan^{-1}\left(-\sqrt{3}\right)
\]
Since the real part \( a = -2 \) is negative and the imaginary part \( b = 2\sqrt{3} \) is positive, the complex number is in the second quadrant. Therefore, we need to add \( 180^\circ \) to the principal angle found:
\[
\theta = 180^\circ + \tan^{-1}\left(-\sqrt{3}\right) = 180^\circ - 60^\circ = 120^\circ
\]

### Step 3: Write the polar form
Thus, the polar form of \( Z = -2 + 2i\sqrt{3} \) is:
\[
Z = [4, 120^\circ]
\]

### Final Answer:
\[
Z = 4\left(\cos 120^\circ + i\sin 120^\circ\right)
\]
or simply,
\[
Z = [4, 120^\circ]
\]

Would you like more details or have any questions?

Here are five related questions:
1. How would you convert a complex number from polar form back to rectangular form?
2. What is the significance of the argument in the polar form of a complex number?
3. How do you find the polar form of a complex number if it lies in the third or fourth quadrant?
4. What is the geometric interpretation of the magnitude and angle in polar form?
5. How would the polar form change if the complex number was multiplied by a scalar?

**Tip:** When determining the angle in polar form, always consider the correct quadrant of the complex number to get the proper direction of the angle.

Learn how to express the complex number -2 + 2i√3 in polar form with detailed steps and calculations. This problem covers key concepts in complex numbers, including magnitude and angle determination using trigonometric functions. Suitable for advanced high school students and those interested in complex number theory.

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