Welcome to four examples
of converting a cylindrical equation
to a rectangular equation.
We're asked to find an equation in rectangular coordinates
for the cylindrical equation
z equals r squared times cosine squared theta.
For a quick review, cylindrical coordinates
are in the form r comma theta comma z,
and rectangular coordinates are in the form
x comma y comma z,
which means in order to write cylindrical equation
in rectangular form,
we'll need to perform a substitution for r and theta
so that the equation only contains
the variables x, y, and z,
and we use the equations below to perform the substitutions,
which relate x, y, z, r, and theta.
So we are given the equation
z equals r squared times cosine squared theta.
Looking at our equations,
notice how x equals r cosine theta,
thus we write the right side of this equation
as the square of r cosine theta.
And now we can see we can substitute x for r cosine theta,
which gives us the equation z equals x squared.
Notice how this equation now only contains
the variables x and z,
and therefore we know it's now in rectangular form.
For our second example,
we are given the cylindrical equation
r squared plus z squared equals 16,
which we can see from below is a sphere.
To write this using rectangular coordinates,
we must perform a substitution for r squared.
Notice r squared is equal to x squared plus y squared.
So if we substitute x squared plus y squared for r squared,
we have the equation x squared plus y squared plus z squared
equals 16, which notice how it only contains
the variables x, y, and z
and therefore is in rectangular form.
We should recognize this equation
as a sphere with a radius of four.
Next, we are given the cylindrical equation
r equals two sine theta.
We must perform a substitution for r and theta
in order for this equation to be in rectangular form
and only contain the variables x, y, and z.
But notice how our equation here
is r squared equals x squared plus y squared,
so we could replace r with the square root
of the quantity r squared plus y squared.
But instead of doing this,
let's look at the right side of the equation.
Notice how we have sine theta here,
and we know y equals r sine theta.
If we were to solve this equation for sine theta,
we would divide both sides of the equation by r,
which gives us sine theta
is equal to y divided by r.
Let's perform this substitution for sine theta.
This will give us r equals two times y divided by r,
if we want r equals two y divided by r.
Now, for the next step,
let's clear the denominator of r from this fraction
by multiplying both sides of the equation by r.
Notice, by doing this, this will eliminate the r
from the right side of the equation
and give us an r squared on the left side of the equation,
which will give us a much easier substitution
using r squared equals x squared plus y squared.
So simplifying here, we have r squared equals,
here r divided by r simplifies to one.
We just have two y on the right side,
and, again, now we know r squared
equals x squared plus y squared.
Performing this substitution gives us
x squared plus y squared equals two y.
Our equation now only contains the variables x and y
and is in rectangular form.
Let's look at one more example.
Here we're given the cylindrical equation r equals 10z.
Again, we could substitute the square root
of the quantity x squared plus y squared for r here,
but it's easier to perform a substitution for r squared,
so for this equation,
let's square both sides of the equation.
On the left side, we have r squared.
On the right side, we have 100z squared,
and now we can substitute r squared plus y squared
for r squared, which gives us the equation
x squared plus y squared equals 100z squared.
And, again, our equation only contains
the variables x, y, and z,
and therefore we know it is in rectangular form.
I hope you found this helpful.