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The given equation is:

\[
\sin x = 1
\]

We need to find all real numbers \( x \) that satisfy this equation.

The sine function, \( \sin(x) \), reaches a value of 1 at specific points on the unit circle. The principal solution occurs when:

\[
x = \frac{\pi}{2} + 2k\pi \quad \text{for any integer } k
\]

This is because the sine function has a period of \( 2\pi \), so it repeats every \( 2\pi \) units. Therefore, the general solution is:

\[
x = \frac{\pi}{2} + 2k\pi, \quad k \in \mathbb{Z}
\]

This represents all real numbers \( x \) for which \( \sin x = 1 \).

Would you like a deeper explanation of sine functions, or have any other questions?

Here are 5 related questions for further exploration:
1. How does the sine function behave for values of \( x \) where \( \sin(x) = -1 \)?
2. What is the general solution for \( \sin x = 0 \)?
3. How does the sine function's period affect its solutions?
4. What is the graph of \( \sin x \) and how does it help in understanding periodic solutions?
5. How do we solve trigonometric equations like \( \cos x = 0.5 \)?

**Tip:** In general, trigonometric functions are periodic, meaning they repeat their values after a fixed interval. This is a key concept in solving trigonometric equations.

Find all real numbers that satisfy the following equation: sin(x) = 1.

Learn how to solve the trigonometric equation sin(x) = 1. Explore key trigonometric concepts, including the periodic nature of the sine function and unit circle properties. Suitable for students in Grades 9-12.

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