Welcome to the first of two videos
on Graphing Polar Equations.
This video deals with the more basic polar equations.
Okay so if we want to graph the polar equation
r equals three.
There are several ways to do this.
One way would be to recognize that r squared
is equal to x squared plus y squared.
So if we squared both sides of this equation,
we would have r squared equals nine.
And then we could replace r squared
with x squared plus y squared.
So now we have a rectangular equation,
where this is a circle, where the center is
at zero zero and we have radius three.
If you want to keep this in polar form,
what we could think of is we want to graph
all of the points where r is equal to three
and theta can be any angle.
If we look at it this way, we could pick any angle
between a ray and the polar access
and pop the point where r is equal to three.
So we'd have a point here, when theta is equal to zero,
when theta is three degrees, r is three,
60 degrees, r is three,
90 degrees and so on.
And what you'll see is, we're forming a circle
with radius three centered at the pole.
It would look something like this.
Which of course matches our equation
and rectangular form as well.
Now let's take a look at theta as equal to pi over three.
Let's look at it in polar form first.
What this is telling us is the plot all of the points
where r can be any value but theta must be pi over three.
Well when theta is pi over three, we'd be looking
at this as the terminal side of the angle.
And if r is positive, we'd be plotting points
along this terminal side.
And if r is a negative number we'd be plotting points
and remember if r was negative we'd be plotting points
on the ray pointing in the opposite direction
or along here.
So theta equals pi over three is actually a line.
That would look like this.
If we really wanted to convert this to rectangular form
what we could do is take
the tangent of both sides
so tangent theta is equal to the tangent of pi over three.
Well the tangent of pi over three that's a 60 degree angle
would be the square root three over one.
Next remember that tangent theta is equal to y over x.
So let's go ahead and perform that substitution.
Now we can cross multiply,
y times one
must equal square root three times x.
So this verifies we have a line
that passes through the pole or the origin
and has slope square root three over one.
Now let's go ahead and take a look at a couple more.
Let's say we want to graph the polar equation
r equals three sine theta.
Another technique is to graph r equals three sine theta
on the coordinate plane as reference.
And this is often called the r-value analysis.
So what we're saying here, is we'll call the y axis r
and the x axis theta.
And we'll try to use this information to graph it
on the polar coordinate plane.
So if we take a look at this first blue section,
when theta increases from zero to pi over two
r increases from
zero to three.
So what that's telling us is, remember theta
would be zero here and theta would be pi over two here.
So in this region, the radius increase from zero to three.
If we plotted enough points we would see that it looks
something like this.
Then, when theta is between pi over two and pi
or in this region, the radius changes from positive three
back to zero.
So what happens is we start plotting points
and it would look something like this.
From pi over two to pi radians.
Next what we'll find between pi and three pi over two
in this region, we might be thinking we
were going to plot points in this region.
But notice that from pi to three pi over two
r is actually negative, so we'd be plotting points
in the opposite direction.
And these points would actually correspond
to the blue points that we've already plotted here.
And the same thing with this red piece
from three pi over two to two pi, r is negative
so we'll be plotting points up here in the second quadrant
rather than the fourth.
Let's also take a look at this on the graphing calculator.
I've already typed in r equals three sine theta.
And I've already set up the window,
where in degrees, the step is five degrees at a time.
Let's go ahead and graph it.
This verifies that it is a circle.
If we press trace, what this shows is the graph
and then as we move the point along,
we'll see theta changing as it plots points on this graph.
So you see that the radius would increase from zero
all the way up to three.
And then from 90 degrees to 180 it's going to go
from three to zero.
Now unfortunately this does not show the value of r
on the screen.
It just shows the rectangular coordinates and theta.
But we can see that once it reaches 180,
it starts to repeat itself, back to what we plotted
from zero to 90 degrees.
Now of course a third option would be
to use the table feature.
Or essentially complete a t-table, where you pick an angle,
find the corresponding value of r and then plot that point.
Here's another graph of what we just found
using some software.
Let's go ahead and take a look at another one.
Let's graph r equals three cosine theta.
Again we'll use this graph as a reference
where our y axis is now going to be r,
and the x axis will be our theta.
And what we'll notice, is that when theta
is equal to zero degrees, the r value is three.
But the plane would actually be right here.
So as theta approaches pi over two,
r approaches zero.
Then from pi over two to pi radians,
notice that the r value is negatives.
We're not actually plotting points here
but in the opposite direction, we're in the fourth quadrant.
So the radius starts at zero and increases to three.
And we do get another circle,
but now you can see that it's oriented to the right.
And then again from pi to three pi over two,
since r is negative, we end up plotting the points
on this blue half of the circle again.
And then from three pi over two to two pi,
we are plotting points on this red region again as well.
If we wanted to verify this on the calculator we can.
Y equals three cosine theta now.
Graph, you can see there it is.
If we trace it,
You can see from zero to 90 degrees,
we're plotting those points.
From 90 to 180 we're plotting those points.
And then again it repeats.
Or of course we can always go back to making a t-table
where we pick our theta,
and find the corresponding value of r.
Here's a nice graph of what we just found.
Hope you found this explanation helpful.
Polar equations can be a little tricky at first.
But with the use of technology and the r value analysis
hopefully you'll be successful.
I did make one more video that
goes over some additional examples.
Thank you and have a good day.