The function \( f(x) = \csc(x) \) represents the **cosecant** function, which is the reciprocal of the sine function. In mathematical terms:

\[
\csc(x) = \frac{1}{\sin(x)}
\]

### Domain and Range:
- **Domain**: The function is undefined where \( \sin(x) = 0 \), which occurs at integer multiples of \( \pi \), i.e., \( x = n\pi \), where \( n \in \mathbb{Z} \). Therefore, the domain is:

\[
x \in \mathbb{R} \setminus \{ n\pi \mid n \in \mathbb{Z} \}
\]

- **Range**: Since \( \csc(x) \) is the reciprocal of \( \sin(x) \), and \( \sin(x) \) varies between -1 and 1 (but not equal to 0), the range is:

\[
(-\infty, -1] \cup [1, \infty)
\]

### Key Properties:
- **Periodicity**: The cosecant function has a period of \( 2\pi \), which means \( \csc(x + 2\pi) = \csc(x) \).
- **Asymptotes**: Vertical asymptotes occur where \( \sin(x) = 0 \), at \( x = n\pi \).

Would you like more details on the graph of this function or other properties?

### Related Questions:
1. How does the graph of \( f(x) = \csc(x) \) compare to that of \( f(x) = \sin(x) \)?
2. Can you explain the behavior of the function near its vertical asymptotes?
3. What is the derivative of \( f(x) = \csc(x) \)?
4. What is the integral of \( f(x) = \csc(x) \)?
5. How would \( \csc(x) \) behave if it were composed with another function?

### Tip:
Always be cautious with reciprocal functions like cosecant, as they introduce vertical asymptotes where the original sine function is zero.

Explore the trigonometric function f(x) = csc(x), the reciprocal of sin(x). Learn about its domain, range, periodicity, and vertical asymptotes. This topic covers key trigonometric concepts, ideal for students in Grades 10-12.

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