To solve the equation \( \sin x = e^y \), we need to express one variable in terms of the other. Let's consider the equation from two different approaches, solving for \( x \) in terms of \( y \) and solving for \( y \) in terms of \( x \).

### Solving for \( x \) in terms of \( y \):

1. The equation is \( \sin x = e^y \).
2. To solve for \( x \), we take the inverse sine (arcsine) of both sides:
   \[
   x = \arcsin(e^y)
   \]
3. However, note that \( \sin x \) only takes values between \(-1\) and \( 1 \), i.e. \( -1 \leq \sin x \leq 1 \). This means that \( e^y \) must also satisfy this condition:
   \[
   -1 \leq e^y \leq 1
   \]
4. Since the exponential function \( e^y \) is always positive and greater than or equal to 1 for all real \( y \), there is **no solution** for real \( y \) such that \( e^y \leq 1 \), except at \( y = 0 \), where \( e^y = 1 \). 

Thus, the only solution for real numbers is when \( y = 0 \), giving \( \sin x = 1 \), which implies:
   \[
   x = \frac{\pi}{2} + 2n\pi \quad \text{for any integer } n.
   \]

### Solving for \( y \) in terms of \( x \):

1. From the equation \( \sin x = e^y \), take the natural logarithm (ln) of both sides to solve for \( y \):
   \[
   y = \ln(\sin x)
   \]
2. This is valid only when \( \sin x > 0 \), because the natural logarithm is only defined for positive numbers.

### Conclusion:
- When solving for \( x \), the only real solution occurs when \( y = 0 \), which gives \( x = \frac{\pi}{2} + 2n\pi \).
- When solving for \( y \), we get \( y = \ln(\sin x) \), but this is only valid for \( \sin x > 0 \), i.e., \( 0 < x < \pi \) within one period of the sine function.

Would you like more details or have any questions?

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Here are 5 related questions to expand this topic:
1. What happens if we extend the domain of \( y \) to complex numbers?
2. Can we solve the equation \( \sin x = k \), where \( k > 1 \)?
3. How does the natural logarithm function behave for negative values?
4. What are the periodic properties of the sine function?
5. How can we visualize the relationship between exponential and trigonometric functions?

**Tip:** Always check the domain and range restrictions of the functions you are working with, especially when dealing with trigonometric and logarithmic equations.

Explore how to solve the equation sin x = e^y by analyzing both trigonometric and exponential functions. Learn about the domain and range restrictions and understand when solutions exist for real numbers. Suitable for advanced high school and college students.

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