express 4+2i in it's polar form between negative pi and pi

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 \( \theta \).

Express 4+2i in its polar form between negative pi and pi.

Learn how to convert the complex number 4+2i into polar form with a step-by-step solution. This involves finding the magnitude, calculating the argument, and using Euler's formula. Ideal for high school students in Grades 10-12 studying complex numbers and polar coordinates.

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