To convert the complex number $z_2 = 2\sqrt{3} - j2$ to polar form, we need to express it in terms of its magnitude $r$ and argument $\theta$.

Step-by-Step Conversion to Polar Form

1. Identify the Real and Imaginary Parts:

   • Real part $a = 2\sqrt{3}$.
   • Imaginary part $b = -2$.

2. Calculate the Magnitude $r$: The magnitude (or modulus) $r$ of a complex number $z = a + bj$ is given by: $r = \sqrt{a^2 + b^2}$

   For $z_2 = 2\sqrt{3} - j2$: $r = \sqrt{(2\sqrt{3})^2 + (-2)^2}$

   Calculating each part: $(2\sqrt{3})^2 = 4 \times 3 = 12$ $(-2)^2 = 4$

   Thus, $r = \sqrt{12 + 4} = \sqrt{16} = 4$

3. Calculate the Argument $\theta$: The argument $\theta$ is the angle the complex number makes with the positive real axis. It is given by: $\theta = \tan^{-1}\left(\frac{b}{a}\right)$

   For $z_2 = 2\sqrt{3} - j2$: $\theta = \tan^{-1}\left(\frac{-2}{2\sqrt{3}}\right)$

   Simplifying: $\theta = \tan^{-1}\left(\frac{-2}{2\sqrt{3}}\right) = \tan^{-1}\left(\frac{-1}{\sqrt{3}}\right)$

   Since $\tan^{-1}\left(\frac{-1}{\sqrt{3}}\right) = -\frac{\pi}{6}$, the argument $\theta = -\frac{\pi}{6}$.

4. Express $z_2$ in Polar Form: In polar form, the complex number $z$ is represented as: $z = r(\cos \theta + j\sin \theta)$

   Therefore, for $z_2 = 2\sqrt{3} - j2$: $z_2 = 4\left(\cos\left(-\frac{\pi}{6}\right) + j\sin\left(-\frac{\pi}{6}\right)\right)$

Answer in Polar Form

$z_2 = 4 \left(\cos\left(-\frac{\pi}{6}\right) + j\sin\left(-\frac{\pi}{6}\right)\right)$

Or, using the exponential form:

$z_2 = 4 e^{-j\frac{\pi}{6}}$

You can write either form in the box provided, depending on which format is preferred.

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Would you like to see more details on the conversion process or have any other questions?

Here are some related questions you might find helpful:

1. How do you convert a complex number from polar to rectangular form?
2. What is the significance of the argument in the polar form of a complex number?
3. How do you multiply two complex numbers in polar form?
4. How do you divide two complex numbers in polar form?
5. How is the polar form of a complex number useful in electrical engineering?

Tip: Remember that when dealing with polar forms, the magnitude represents the distance from the origin, and the argument represents the angle with the positive x-axis.