We're asked to find an equation
in rectangular coordinates for the spherical equation,
rho equals two cosecant phi.
To write the equation using rectangular coordinates,
we need to write the equation so it only
contains the variables x, y, and z,
and not the variables rho, phi, or theta.
Looking at the equations that relate
rectangular and spherical coordinates,
notice that none of the equations contain cosecant phi.
For the first step, we will substitute
one over sine phi for cosecant phi,
using the identity cosecant phi equals one over sine phi.
Forming this substitution gives us the equation
rho equals two divided by sine phi.
And now from here, notice that here we have
the equation z equals rho times cosine phi,
and therefore, it'll be much easier to performing
substitution for a cosine phi rather than sine phi,
and therefore, we will now square both sides
of the equation, which will give us sine squared phi
in the denominator, which we can then replace
with one minus cosine squared phi.
So for the next step, we square both sides of the equation,
on the left we have rho squared,
on the right we square the numerator and the denominator,
which gives us four divided by sine squared phi.
And now using a trig identity, sine squared phi
plus cosine squared phi equals one,
if we solve this equation for sine squared phi,
we have sine squared phi equals
one minus cosine squared phi,
which we will now use to substitute for sine squared phi.
Performing this substitution gives us rho squared equals
four divided by the quantity one minus cosine squared phi.
Again, this is helpful because now
we can use the equation, z equals rho cosine phi
and solve this for cosine phi,
which gives us cosine phi equals z divided by rho,
which means we can substitute z squared
divided by rho squared for cosine squared phi.
Performing this substitution gives us the equation,
rho squared equals four, divided by the quantity one,
minus z squared, divided by rho squared.
The next step will be to clear this fraction
from the equation by multiplying both sides of the equation
by the denominator of one minus z squared
divided by rho squared, or if we want,
we can think of cross-multiplying.
Either way, this gives us rho squared times the quantity
one minus z squared, divided by rho squared equals four.
Let's continue on the next slide.
For the next step, we distribute rho squared,
rho squared times one is equal to rho squared,
minus rho squared, times z squared,
divided by rho squared, simplifies nicely to z squared.
And finally we make the substitution for rho squared,
where rho squared is equal to x squared,
plus y squared, plus z squared.
Performing this last substitution
gives us x squared, plus y squared, plus z squared,
minus z squared equals four.
Simplifying, z squared, minus z squared is equal to zero,
giving us the rectangular equation,
x squared, plus y squared, equals four,
which graphed in three dimensions,
we should recognize as a cylinder.
Which we see graphed here, if we graph this using
three dimensional software,
here is the graph of the cylinder.
I hope you found this helpful.