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Please fill the table with your preferred trigonometric function, denoted as Y = f(X) and its corresponding inverse function, denoted as f-1(Y). 

Let's use the sine function for \( Y = f(X) \) and its corresponding inverse function, the arcsine function, for \( f^{-1}(Y) \). The sine function is periodic and ranges from -1 to 1, with the key values falling at specific points that match the angles given in the table.

Here’s how the values would fill out:

\[
\begin{array}{|c|c|c|c|c|c|c|c|}
\hline
X & 0 & \frac{\pi}{3} & \frac{2\pi}{3} & \frac{\pi}{2} & \pi & \frac{4\pi}{3} & 2\pi \\
\hline
Y=f(X) & \sin(0) & \sin\left(\frac{\pi}{3}\right) & \sin\left(\frac{2\pi}{3}\right) & \sin\left(\frac{\pi}{2}\right) & \sin(\pi) & \sin\left(\frac{4\pi}{3}\right) & \sin(2\pi) \\
 & 0 & \frac{\sqrt{3}}{2} & \frac{\sqrt{3}}{2} & 1 & 0 & -\frac{\sqrt{3}}{2} & 0 \\
\hline
f^{-1}(Y) & \arcsin(0) & \arcsin\left(\frac{\sqrt{3}}{2}\right) & \arcsin\left(\frac{\sqrt{3}}{2}\right) & \arcsin(1) & \arcsin(0) & \arcsin\left(-\frac{\sqrt{3}}{2}\right) & \arcsin(0) \\
 & 0 & \frac{\pi}{3} & \frac{2\pi}{3} & \frac{\pi}{2} & \pi & -\frac{\pi}{3} & 2\pi \\
\hline
\end{array}
\]

So the filled-out table is:

\[
\begin{array}{|c|c|c|c|c|c|c|c|}
\hline
X & 0 & \frac{\pi}{3} & \frac{2\pi}{3} & \frac{\pi}{2} & \pi & \frac{4\pi}{3} & 2\pi \\
\hline
Y=f(X) & 0 & \frac{\sqrt{3}}{2} & \frac{\sqrt{3}}{2} & 1 & 0 & -\frac{\sqrt{3}}{2} & 0 \\
\hline
f^{-1}(Y) & 0 & \frac{\pi}{3} & \frac{2\pi}{3} & \frac{\pi}{2} & \pi & -\frac{\pi}{3} & 2\pi \\
\hline
\end{array}
\]

Would you like further details or have any questions?

Here are some related questions for deeper understanding:
1. What are the properties of the sine function?
2. How does the periodicity of sine function affect its graph?
3. What is the domain and range of the sine function?
4. How do you derive the arcsine function from the sine function?
5. What are the key points of the sine function on the unit circle?
6. How do the sine and arcsine functions relate to each other?
7. What are the practical applications of the sine function in real-world problems?
8. How do you find the inverse of other trigonometric functions like cosine and tangent?

**Tip:** When dealing with trigonometric functions, always consider their periodic nature and range to understand how they behave over different intervals.

Explore the relationship between trigonometric functions and their inverses with a detailed explanation and examples. Learn how the sine function and arcsine function are related and how they can be used to solve equations. Suitable for high school students and anyone interested in trigonometry.

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