Graphing Cosecant Functions

Graphing Cosecant Functions

How to graph cosecant functions: graph, example, and its solution.

Graph

y = csc x = 1/sin x

Cosecant is the reciprocal of sine:
y = csc x = 1/sin x.

See the graphs of
y = sin x (gray) and
y = csc x (white, blue).

If y = sin x = ±1,
y = csc x touches y = sin x.
(y = csc x = 1/[±1] = ±1)

If y = sin x = 0,
then those x values
are the vertical asymptotes of y = csc x.
(y = sec x = 1/[±0] = ±∞)

So, to graph the cosecant function,
first draw the sine function,
then draw the cosecant function.

Graphing sine functions

The blue part is one cycle of y = csc x.

So the period of y = csc x is 2π:
same as y = sin x.

The asymptotes are x = :
when sin x = 0.

Example

Sketch the given function's graph. y = csc 2x (0 <= x <= 2pi)

Roughly draw one cycle of y = sin 2x
with its amplitude and period.

Amplitude: 1
Period: 2π / |2| = π

Graphing sine functions

Use these values to draw y = sin 2x. (gray)

Then draw y = csc 2x
by using the sine graph.

If y = sin 2x = ±1,
y = csc 2x touches y = sin 2x.

If y = sin 2x = 0,
then those x values
are the vertical asymptotes of y = csc 2x.