To express $4 + 2i$ in polar form, we need to follow these steps:

1. Find the magnitude $r$:

The magnitude (or modulus) $r$ of a complex number $z = a + bi$ is given by: $r = \sqrt{a^2 + b^2}$ For $4 + 2i$: $r = \sqrt{4^2 + 2^2} = \sqrt{16 + 4} = \sqrt{20} = 2\sqrt{5}$

2. Find the argument $\theta$:

The argument $\theta$ is the angle the complex number makes with the positive real axis. It's calculated using: $\theta = \tan^{-1}\left( \frac{b}{a} \right)$ For $4 + 2i$: $\theta = \tan^{-1}\left( \frac{2}{4} \right) = \tan^{-1}(0.5)$ The value of $\tan^{-1}(0.5)$ is approximately $0.4636$ radians.

Since both the real part and the imaginary part are positive, the complex number is in the first quadrant, so we don't need to adjust the angle.

3. Polar form:

Thus, the polar form of $4 + 2i$ is: $z = 2\sqrt{5} \left( \cos(0.4636) + i\sin(0.4636) \right)$ or, using Euler's formula: $z = 2\sqrt{5} e^{i(0.4636)}$

This is the polar form of $4 + 2i$ where the argument $\theta$ is between $-\pi$ and $\pi$.

Would you like more details on any of the steps or calculations?

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Here are five related questions:

1. How do you convert a complex number in polar form back to rectangular form?
2. What is Euler's formula, and how is it used in complex numbers?
3. How do you find the magnitude of a complex number given its polar form?
4. How does the argument change for complex numbers in different quadrants?
5. How can you graphically represent the complex number $4 + 2i$ on the complex plane?

Tip: When converting a complex number to polar form, always check which quadrant the number lies in to correctly determine the argument $\theta$.