# Graphing Cosecant Functions

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

## Graph

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 2*π*:

same as *y* = sin *x*.

The asymptotes are *x* = *nπ*:

when sin *x* = 0.

## Example

Roughly draw one cycle of *y* = sin 2*x*

with its amplitude and period.

Amplitude: 1

Period: 2*π* / |2| = *π*

Graphing sine functions

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

Then draw *y* = csc 2*x*

by using the sine graph.

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

If *y* = sin 2*x* = 0,

then those *x* values

are the vertical asymptotes of *y* = csc 2*x*.