Study the behaviour of the function on the interval .

MATHEMATICS · Multi-part · A Level · Mathematics · Trigonometry · Trigonometry

Problem

Study the behaviour of the function

y = \csc x

on the interval

[-2\pi, 2\pi]

. (i) Draw the graphs of

y = \sin x

and

y = \csc x

on the same pair of axes for

x \in [-2\pi, 2\pi]

, clearly indicating intercepts, turning points of

y = \sin x

, and all vertical asymptotes of

y = \csc x

. (ii) Using your graph or otherwise, write the set of all

x

[-2\pi, 2\pi]

for which

\csc x \ge 2

. (iii) Explain why the function

y = \csc x

does not possess a greatest or least value on its entire domain.