To express \( 2e^{\frac{2\pi i}{4}} \) in polar form, follow these steps:

### Step 1: Break down the exponential
You are given \( 2e^{\frac{2\pi i}{4}} \), which is already in polar form since it's expressed as \( re^{i\theta} \). In this case:
- \( r = 2 \) (the modulus)
- \( \theta = \frac{2\pi}{4} = \frac{\pi}{2} \) (the argument)

### Step 2: Verify the form
The general form for polar representation of a complex number is:
\[
z = r (\cos \theta + i \sin \theta)
\]
So, \( 2e^{\frac{2\pi i}{4}} \) can be written as:
\[
2 \left( \cos \frac{\pi}{2} + i \sin \frac{\pi}{2} \right)
\]

### Step 3: Compute trigonometric values
- \( \cos \frac{\pi}{2} = 0 \)
- \( \sin \frac{\pi}{2} = 1 \)

Thus, the expression becomes:
\[
2 (0 + i \cdot 1) = 2i
\]

### Final Polar Form:
So, \( 2e^{\frac{2\pi i}{4}} \) is \( 2i \), which is the polar form of the number.

Would you like more details or have any questions?

Here are 5 related questions:
1. What is the geometric interpretation of the polar form?
2. How do you convert from polar to rectangular coordinates?
3. What is Euler's formula and how does it relate to polar form?
4. How do you find the magnitude and angle of a complex number?
5. How do you add and multiply complex numbers in polar form?

**Tip:** In polar form, multiplying complex numbers is easier since you just multiply the moduli and add the angles!

Learn how to convert the complex number 2e^(2πi/4) to polar form using Euler's formula and trigonometric identities. This step-by-step solution covers key concepts in complex numbers, polar representation, and Euler’s theorem. Ideal for students in Grades 10-12.

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