Observe the tabular values and the graphs of Y = f(X) and f-1(Y), answer the following:

Examine the periodicity of the chosen trigonometric function, Y = f(X), as presented in the table.

Discuss the domain and range of both Y = f(X) and f-1(Y).

Determine whether the function Y=f(X) demonstrates even, odd, or neither.

Let's discuss the periodicity, domain and range, and parity of the chosen trigonometric function \( Y = f(X) = \sin(X) \) and its inverse \( f^{-1}(Y) = \arcsin(Y) \).

### Periodicity
The sine function, \( \sin(X) \), is periodic with a period of \( 2\pi \). This means that the function repeats its values every \( 2\pi \). From the table:

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

We can see that at \( X = 0 \) and \( X = 2\pi \), the function values repeat, indicating the periodic nature.

### Domain and Range
- For \( Y = f(X) = \sin(X) \):
  - **Domain**: \( (-\infty, \infty) \) because the sine function is defined for all real numbers.
  - **Range**: \([-1, 1]\) because the sine function only takes values between -1 and 1 inclusive.

- For \( f^{-1}(Y) = \arcsin(Y) \):
  - **Domain**: \([-1, 1]\) because arcsine is defined only for inputs between -1 and 1 inclusive.
  - **Range**: \([-\frac{\pi}{2}, \frac{\pi}{2}]\) because the output of the arcsine function lies within this interval.

### Parity (Even, Odd, or Neither)
To determine whether \( Y = \sin(X) \) is even, odd, or neither:
- **Even function**: \( f(-X) = f(X) \)
- **Odd function**: \( f(-X) = -f(X) \)

For the sine function:
\[
\sin(-X) = -\sin(X)
\]
This identity shows that the sine function is **odd** because \( \sin(-X) \) is the negative of \( \sin(X) \).

### Summary
1. **Periodicity**: \( \sin(X) \) is periodic with a period of \( 2\pi \).
2. **Domain and Range**:
   - \( Y = \sin(X) \): Domain \( (-\infty, \infty) \), Range \([-1, 1]\)
   - \( f^{-1}(Y) = \arcsin(Y) \): Domain \([-1, 1]\), Range \([-\frac{\pi}{2}, \frac{\pi}{2}]\)
3. **Parity**: \( \sin(X) \) is an odd function.

Would you like further details or have any questions?

Here are some related questions for deeper understanding:
1. How does the period of \( \sin(X) \) compare to other trigonometric functions like \( \cos(X) \) and \( \tan(X) \)?
2. What happens to the graph of \( \sin(X) \) if you shift it horizontally by \( \pi \)?
3. How would the domain and range change for \( \cos(X) \) and its inverse \( \arccos(X) \)?
4. Can you provide a real-world example where the sine function's periodicity is utilized?
5. What are the characteristics of an even function? Can you give an example of an even trigonometric function?
6. How can the properties of the sine function be used to solve trigonometric equations?
7. How does the concept of periodicity help in solving complex trigonometric problems?
8. How do the properties of \( \arcsin(X) \) influence its graph compared to \( \sin(X) \)?

**Tip:** Understanding the periodic nature of trigonometric functions can greatly simplify solving equations involving these functions, especially in complex scenarios.

Explore the periodicity, domain, range, and parity of trigonometric functions Y = sin(X) and its inverse function f^{-1}(Y) = arcsin(Y). This detailed explanation covers the periodic nature of sine functions, their defined domains and ranges, and their parity properties. Suitable for advanced high school students and anyone interested in trigonometry.

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