\frac{4e\frac{\pi i}{5}}{2cos\left(\frac{3\pi }{7}\right)+isen\left(\frac{2\pi }{5}\right)}

To simplify the expression \(\frac{4e^{\frac{\pi i}{5}}}{2\cos\left(\frac{3\pi }{7}\right)+i\sin\left(\frac{2\pi }{5}\right)}\), let's break it down step by step.

### Step 1: Simplify the numerator
The numerator is \(4e^{\frac{\pi i}{5}}\). Using Euler's formula \(e^{i\theta} = \cos(\theta) + i\sin(\theta)\), we have:
\[
4e^{\frac{\pi i}{5}} = 4\left(\cos\left(\frac{\pi}{5}\right) + i\sin\left(\frac{\pi}{5}\right)\right)
\]

### Step 2: Simplify the denominator
The denominator is \(2\cos\left(\frac{3\pi }{7}\right) + i\sin\left(\frac{2\pi }{5}\right)\). This expression is already in its simplest form.

### Step 3: Combine the expressions
We need to write the whole expression as:
\[
\frac{4\left(\cos\left(\frac{\pi}{5}\right) + i\sin\left(\frac{\pi}{5}\right)\right)}{2\cos\left(\frac{3\pi }{7}\right) + i\sin\left(\frac{2\pi }{5}\right)}
\]

To simplify this fraction, we can multiply both the numerator and the denominator by the complex conjugate of the denominator. The complex conjugate of \(2\cos\left(\frac{3\pi }{7}\right) + i\sin\left(\frac{2\pi }{5}\right)\) is \(2\cos\left(\frac{3\pi }{7}\right) - i\sin\left(\frac{2\pi }{5}\right)\).

### Step 4: Multiply by the conjugate
\[
\frac{4\left(\cos\left(\frac{\pi}{5}\right) + i\sin\left(\frac{\pi}{5}\right)\right) \cdot \left(2\cos\left(\frac{3\pi }{7}\right) - i\sin\left(\frac{2\pi }{5}\right)\right)}{\left(2\cos\left(\frac{3\pi }{7}\right) + i\sin\left(\frac{2\pi }{5}\right)\right) \cdot \left(2\cos\left(\frac{3\pi }{7}\right) - i\sin\left(\frac{2\pi }{5}\right)\right)}
\]

### Step 5: Simplify the denominator
The denominator simplifies using the difference of squares:
\[
\left(2\cos\left(\frac{3\pi }{7}\right)\right)^2 - \left(\sin\left(\frac{2\pi }{5}\right)\right)^2
\]
\[
= 4\cos^2\left(\frac{3\pi }{7}\right) - \sin^2\left(\frac{2\pi }{5}\right)
\]

### Step 6: Expand and simplify the numerator
The numerator can be expanded using the distributive property:
\[
4\left(\cos\left(\frac{\pi}{5}\right) + i\sin\left(\frac{\pi}{5}\right)\right)\left(2\cos\left(\frac{3\pi }{7}\right) - i\sin\left(\frac{2\pi }{5}\right)\right)
\]
Expanding this product and combining like terms will yield a complex number with a real and imaginary part, which can be simplified further.

Would you like to proceed with the full expansion, or is there a specific part of the solution you'd like to focus on?

---

1. What is the significance of multiplying by the complex conjugate?
2. How do we apply Euler's formula in complex numbers?
3. How do we simplify trigonometric identities in complex expressions?
4. What are the steps to simplify complex fractions?
5. Why is it important to simplify both the numerator and denominator separately?

**Tip:** When simplifying complex fractions, always consider multiplying by the conjugate to eliminate imaginary parts in the denominator.

Explore how to simplify the complex fraction \(\frac{4e^{\frac{\pi i}{5}}}{2\cos\left(\frac{3\pi }{7}\right) + i\sin\left(\frac{2\pi }{5}\right)}\) using Euler's formula and trigonometric identities. Understand the significance of multiplying by the complex conjugate and simplifying both the numerator and denominator separately. Suitable for advanced high school students and above.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation