Welcome to our lesson on polar coordinates.
So far, we have been graphing on the coordinate plane using
rectangular coordinates as we see here
or the x-axis is the horizontal axis,
the wax is the vertical axis and the point in the center
is called the origin.
We will now study the polar coordinate system
and the polar coordinate system is
based on a point called the pole here
and array called the polar axis, which
is usually drawn in the direction
of the positive x-axis.
So here is the polar axis.
Point P on the polar coordinate system
is located by giving the directed distance
r from the pole and the angle theta from the positive x-axis
to ray OP.
So this is our point P, the distance from point
O to point P would be the directed distance
r and the angle formed between this ray
and the polar axis would be a possible angle theta.
A point given using polar coordinates
is given in the form, r comma theta.
Now, if r is greater than 0, the point P
lies on the terminal side of angle theta.
However, if r as negative or less than zero,
the point lies on the ray pointing
in the opposite direction of the terminal side of theta.
Let's go and take a look at how this works.
If we to plot these points first, the point three, three
degrees, we would go out three units on the polar axis
and then rotate three degrees from P. So here would
be the ray, here would be our angle theta
and it's three units from the pole.
For point Q, we would go out two units
and rotate three pi over two radians,
and that would bring us to here, to point
Q. Notice the next two have negative values for r,
so we're still going to go out the absolute value
of negative four units, so we'll go out
to four on the polar axis, rotate 60 degrees
but now since r is negative, we will actually plot this point
at the end of the ray pointing in the opposite direction
or we go to the opposite side of this circle to here,
this is point R.
And for point S, again we'll go out, the absolute value
of negative one unit, so we'll go out
one unit on the polar axis, rotate negative pi
over three radians that would be to here,
but then we'll plot point S at the end of the ray pointing
in the opposite direction.
So we would be here.
So that brings us to the next point,
polar coordinates of a point P are not unique,
we can list the polar coordinates of this point P
in an infinite number of ways.
Let's see if we come up with four different ways.
First, we can see that it's three units from pole
so that means that r could equal three
or could equal negative three.
Let's stick with a positive value of r first.
So if we use a positive value for r,
the first possible angle for theta it would be 60 degrees.
Now any angle that's coterminous with 60 degrees
would also work for theta.
So we could go around the circle one more time,
60 degrees plus 360 degrees would be 420 degrees
or if we wanted to use a negative angle,
this would be negative 300 degrees.
And of course, we could keep doing this forever.
Let's see if we can use a negative three now.
If we're going to use a negative r value,
the terminal side of our angle should point
in the opposite direction.
So an angle that is terminal here,
will work if we use a negative value for r.
So we could use one 8 plus 60 or 240 degrees or any other angle
that's co-terminal two 240 degrees.
OK now let's take a look at the relationship
between rectangular and polar coordinates.
If this is our point P, what we could do
is sketch a reference triangle using this ray
as the hypotenuse.
And that would look something like this.
So this distance would be x, this
would be y using rectangular coordinates
and using polar coordinates, this
would be r on this could be theta so that away we can see
that this is a right triangle.
So x squared plus y squared equal r squared, that's
where this formula comes from.
The tangent of angle theta would be opposite over adjacent,
or y over x, that verifies this formula.
And then for these other two if we
take the cosine of angle theta, that
would be y over r, solving for y,
we would have y equals r sine theta.
And if we take the cosine of angle theta
that would be equal to x over r again, solving for x,
we would have x equals r cosine theta.
And these formulas are very helpful,
we want to convert back and forth
between rectangular and polar coordinates.
Let's go ahead and give it a try.
Give the polar coordinates of the given rectangular
coordinates two different ways, let's first go ahead
and plot the point negative one, negative one,
that would be here.
Next what we can do, is try to use these formulas to come up
with r and theta.
So using the Pythagorean theorem,
we know that R squared is equal to x squared plus y squared,
so r squared is equal to two.
So we can use the value of plus or minus the square root of two
for the value of r, we'll come back to this idea.
Let's see if we can find a possible angle for theta,
it says tan theta is equal to y over
x while negative one over negative one is equal to one.
Remember using our reference angles,
the tangent of 45 degrees is equal to one,
so theta is equal to 45 degrees.
However, remember that the tangent function is also
positive in the third quadrant and that's
actually where our point is.
So reference angle of 45 degrees in the third quadrant
would actually be an angle of 225 degrees.
So converting from rectangular to polar quadrants
does take some additional thought,
we can't just strictly rely on the Formulas.
Let's go and see if we can find two
ways to express this point using polar coordinates.
Let's see if we can use a positive value for r first.
So if we use positive square root two
for the value of r, our core terminal,
the terminal side of our angle must be in the third quadrant,
and so then we would have to use the angle of 225 degrees.
Notice if we used 45 degrees, we would actually
be over here in the first quadrant
and that won't work with a positive value of r.
But if we do want to use the negative value of r,
negative square root two, then our terminal set at the angle
would have to be in the opposite direction
and therefore if we use negative square two for r,
we would have to use theta of 45 degrees.
So again, when you're converting from rectangular to polar,
it's important that we do sketch the point
and make sure that the values we use for our end data
do match up to the exact point that we're looking for.
Let's go ahead and try the conversion
in the other direction now.
We want to convert negative one pie over three
to rectangular coordinates.
Again, let's go ahead and plot this point,
we would go out one unit on the polar axis, rotate
pi over three radians but since r is negative one,
we're going to plot this point in the opposite direction.
So this is point E. So notice here we're
in the third quadrant the x and the y-coordinates
should both be negative.
Let's go ahead and use these first two formulas now,
x is equal to r cosine theta.
So negative one times cosine of pi over three and y
is equal to negative one times sine theta.
So we need to find the sine and cosine of pi
over three radians, let's go and sketch a reference triangle
for this, that's a 60 degree angle.
So we have one, two square root of three.
So x is equal to negative one times
the cosine of pi over three, would be adjacent
over hypotenuse or one half, so we have an x-coordinate
of negative one half.
For y, we have negative one times the sine of pi
over three or 60 degrees, that would
be squared with three over two.
So we have negative square root three over two.
Notice both x and y-coordinates are negative.
So we have negative one half, negative square root
three over two using rectangular coordinates.
OK, I hope you found this very helpful.
Thank you for watching.