Question

how many periods of the function y = csc x are there between -2π and 2π ?

Explanation:

Step1: Recall period of csc x

The function \( y = \csc x \) has a period of \( 2\pi \), but we analyze its graph segments. The cosecant function \( \csc x=\frac{1}{\sin x} \), and \( \sin x \) has period \( 2\pi \), so \( \csc x \) also has period \( 2\pi \), but its graph consists of "U - shaped" and "inverted U - shaped" parts between vertical asymptotes at \( x = n\pi \), \( n\in\mathbb{Z} \). Between two consecutive asymptotes (e.g., \( 0\) and \( \pi \), \( \pi \) and \( 2\pi \), \( -\pi \) and \( 0 \), \( - 2\pi \) and \( -\pi \)), we have one period - like segment.

Step2: Analyze intervals

• For the interval from \( - 2\pi \) to \( 0 \): The asymptotes are at \( x=-2\pi,x = -\pi,x = 0 \). So between \( - 2\pi \) and \( -\pi \), and between \( -\pi \) and \( 0 \), that's 2 periods (or period - like segments).
• For the interval from \( 0 \) to \( 2\pi \): The asymptotes are at \( x = 0,x=\pi,x = 2\pi \). So between \( 0 \) and \( \pi \), and between \( \pi \) and \( 2\pi \), that's 2 periods (or period - like segments).
• Total number of periods: \( 2 + 2=4 \). We can also count from the graph: looking at the graph, in the region from \( - 2\pi \) to \( 2\pi \), we can see 4 distinct "U" or "inverted U" shaped parts which represent the periods of \( \csc x \).

Answer:

4