How do you find the asymptote for $y = csc (4x + pi)$ ?

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Answer 1 · 2018-02-12T19:23:02

See below.

Identity:

$color(red)bb(cscx=1/sinx)$

Vertical asymptotes occur where $1/sinx$ is undefined.

i.e. where $sinx=0$

So:

$1/(sin(4x+pi)$ will be undefined when. $sin(4x+pi)=0$

$:.$

$4x+pi=arcsin(sin(4x+pi)=arcsin(0)=0,pi,2pi$ etc.

Using $2pi$

$4x+pi=2pi=>x=pi/4$

We can write all solutions as:

$npi/4$

Where n is an integer.

This is where all vertical asymptotes occur:

This is confirmed by the graph of $csc(4x+pi)$

enter image source here

How do you find the asymptote for y = csc (4x + pi)y = csc (4x + pi)?