https://file.aiquickdraw.com/mathbot/file/1725255431968ls1s3vl3.jpg

The problem provided asks to solve for the expression \(\frac{x}{y} + \frac{y}{x}\), given that:

\[
x = \cos\alpha + i\sin\alpha
\]
\[
y = \cos\beta + i\sin\beta
\]

### Solution:

First, let's express \(x\) and \(y\) in exponential form using Euler's formula:

\[
x = e^{i\alpha}
\]
\[
y = e^{i\beta}
\]

Now, compute the ratio \(\frac{x}{y}\):

\[
\frac{x}{y} = \frac{e^{i\alpha}}{e^{i\beta}} = e^{i(\alpha - \beta)}
\]

Next, compute its reciprocal \(\frac{y}{x}\):

\[
\frac{y}{x} = e^{i(\beta - \alpha)}
\]

Now, let's calculate the required expression:

\[
\frac{x}{y} + \frac{y}{x} = e^{i(\alpha - \beta)} + e^{i(\beta - \alpha)}
\]

Notice that \(e^{i(\alpha - \beta)} + e^{i(\beta - \alpha)}\) can be rewritten using the cosine identity \(e^{i\theta} + e^{-i\theta} = 2\cos\theta\):

\[
\frac{x}{y} + \frac{y}{x} = 2\cos(\alpha - \beta)
\]

### Final Answer:
\[
\frac{x}{y} + \frac{y}{x} = 2\cos(\alpha - \beta)
\]

Would you like any further details or explanations? Here are some related questions you might be interested in:

1. How does Euler's formula relate complex numbers to trigonometric functions?
2. What is the significance of the cosine identity \(e^{i\theta} + e^{-i\theta} = 2\cos\theta\)?
3. How can the result \(2\cos(\alpha - \beta)\) be interpreted geometrically?
4. What other trigonometric identities can be derived using complex numbers?
5. How can this approach be applied to more complicated expressions involving complex numbers?

**Tip:** Understanding Euler's formula and how it relates to trigonometric functions is crucial when dealing with problems involving complex numbers in exponential form.

Understand how to solve the expression x/y + y/x using complex numbers and Euler's formula. This problem explores the relationship between complex exponentials and trigonometric identities, providing a detailed step-by-step solution.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation