Welcome to our video on converting
equations in polar form to rectangular form.
The first thing we should review are the equations
that relate polar and rectangular coordinates,
as you see here.
If you don't have these memorized,
you'll definitely want to have them available for reference.
Let's go and take a look at a few problems.
Determine an equivalent equation in rectangular form for r
equals tangent theta times secant theta.
There are several ways to go about converting this
to rectangular form.
We know that tangent theta is equal to y/x.
And that would be helpful.
However, there is not a secant theta
in any of these equations over here.
But if we can remember that secant theta is
a reciprocal of cosine theta, we could take this equation
and solve it for cosine theta.
Cosine theta would equal x/r Well,
if cosine theta is equal to x/r, secant theta
would have to be the reciprocal of this,
and secant theta would be r/x.
So let's go ahead and use this equation and this equation,
and see what happens.
And r equals tangent theta is y/x.
Secant theta is equal to r/x.
Let's multiply this.
We have r equals yr/x squared.
From here, we have a couple options.
If we put r/1, we have a proportion.
And then we can cross multiply.
So we have yr must equal, rx squared.
Now if we divide both sides by r, we have our equation,
y equals x squared.
And this is just a parabola with its vertex at the origin
passing through these five points.
Let's take a look at another one.
Now we have r equals 4 cosine theta.
We just talked about cosine theta in the last problem.
We know that cosine theta is equal to x/r Let's go ahead
and perform that substitution, 4 times x/r.
And that would give us 4x/r.
Let's go ahead and rewrite this.
We have r equals 4x divided by r.
Let's be consistent.
Let's go ahead and put this over 1.
And now, since we have a proportion,
we can cross multiply.
We have r squared equals 4x.
Well, now, r squared is equal to x squared plus y squared.
Let's go ahead and set this equal to 0.
You might already recognize this as a circle.
So if we complete the square on the x part,
we'll know where the center of the circle is.
And we'll also be able to figure out what the radius will be.
So we'll take 1/2 of negative 4 and then square it.
That will give us plus 4 on both sides.
So now we have a perfect square trinomial.
We have x minus 2 squared plus--
I'm going to go ahead and write this as y minus 0
squared to help us--
is equal to 4.
So we have a circle where the center is 2, 0.
And the radius will be equal to 2, since this is r square.
Let's go ahead and graph it.
Here's the center.
Since we have a radius of 2, I'll
go to the right, left, up and down 2 units.
Circle passes through these four points.
It looks something like that.
Let's go ahead and take a look at one more.
Now we have r equals 2 divided by the quantity 3 sine
theta minus cosine theta.
As you can see, I usually like to leave the r in the equation
temporarily.
So let's go ahead and do that.
We'll have r equals 2 divided by 3 times sine theta.
Well, sine theta will be equal to y/r
So we have 3y/r minus cosine theta is x/r Notice here
we have a common denominator.
Let's go ahead and subtract these two fractions.
So we have 3y/x over r.
Let's see if we can simplify this complex fraction.
Remember, this fraction bar is a division symbol.
So we have 2.
Instead of dividing by this fraction,
we multiply it by the reciprocal.
Lets go ahead and put the 2/1.
So we end up with r equals 2r all over 3y minus x.
And again, we can put this over 1.
And now we have a proportion.
So we'll cross multiply.
We have 2r must equal r times 3y minus x.
Now we can divide both sides by r.
And you can see we have 2 equals 3y minus x.
Let's go and rewrite this up here.
So we have a line this time.
I'm going to go ahead and solve it for y to put it
in slope-intercept form.
So we have 3y equals x plus 2.
We add x to both sides.
Then we'll divide the term by 3.
So our final equation will be y equals 1/3x plus 2/3.
So the y-intercept of 2/3 is roughly here.
And then we would go up 1 and right 3.
And there is our line.
Thank you for watching.