It’s Professor Dave, let’s talk about polar coordinates.
Up until now, when we look at the coordinate plane and any function it contains, we have
been expressing points within those functions as having an X coordinate and a Y coordinate.
This one tells us where the point sits horizontally, and this one tells us where the point sits vertically.
We refer to these as rectangular coordinates.
But this is not the only way to plot points.
We can also use polar coordinates, which we are ready to understand now that we have learned
about trigonometric functions and the unit circle.
When we use polar coordinates, the first coordinate tells us a radius, or a distance from the
origin, or pole, and the second coordinate tells us an angle from the polar axis to a
line segment that contains the point.
This angle could be in degrees or radians.
Every point will have polar coordinates of R, theta, so if we make a line segment of
length one, and then move that line segment to an angle of quarter pi, then this endpoint
of the line segment has the coordinates one, quarter pi.
If we look at the point directly opposite the pole, we could interpret this point as
negative one, quarter pi, which would mean the same angle but extending in the negative
direction, or we could call it one, five fourths pi, which is what we would get if we kept
the same line segment but rotated it all the way to five fourths pi.
So it is the case that any point in this system will have more than one valid set of coordinates,
or infinite in fact, since we could just add multiples of two pi, or we could add pi and
switch the sign of the R value, and then add any multiple of two pi from there.
This is different from rectangular coordinates, where a point can only be represented one
specific way.
Let’s quickly practice plotting points by their polar coordinates.
Here are three points to plot, see if you can get them.
Remember, if R is negative, the point will be located R units away from the pole but
in the direction opposite the terminal side of theta.
And here are where the three points must be located.
We just go to the indicated angle, and then the specified distance away from the pole.
One thing we will want to be able to do is to convert between polar coordinates and rectangular
coordinates.
This is easy to do given our new understanding of trigonometry.
Take something like the point two, one third pi.
All we have to do is draw a vertical line from the point down to the X axis, and we
have a triangle with all angles known, as well as the length of the hypotenuse.
Then we just use the definitions of the trig functions to find the lengths of the legs
of this triangle.
The sine of one third pi is equal to the opposite side over two.
Sine of one third pi is root three over two, times two is root three, so this side has
a length of root three.
Cosine of one third pi is equal to the adjacent side over two.
Cosine of one third pi is one half, times two is one, so this side has a length of one.
Given these side lengths, this point extends one unit in the X direction, and root three
units in the Y direction, so this point must have rectangular coordinates of one, root
three.
Of course we can do the same thing in reverse, we could have some point with rectangular
coordinates, and to find its polar coordinates, we just make a right triangle with the origin
and the X axis.
Because we have the rectangular coordinates, we have the lengths of the legs of the triangle.
We can use Pythagorean theorem to get the hypotenuse, and then any trig function we
want to find this angle.
We just have to make sure that once we find the angle inside the triangle, that we use
that to find the actual angle theta from the polar axis, if those two things aren’t the
same, such as if the point is in quadrants two, three, or four.
So that’s all there is to it when we want to convert between these coordinate sytems.
The next thing we want to be able to do is convert an entire equation from a rectangular
form to a polar form.
A rectangular equation contains X and Y, while a polar equation contains R and theta, so
we have to find a way to represent X and Y in terms of R and theta.
How would we take an equation like X plus Y equals three, and turn this into one with
R’s and thetas?
It’s not too difficult if we realize that when we look at one of the triangles we constructed
earlier, we can write in X, Y, R, and theta.
What is X in terms of R and theta?
Well, cosine theta is X over R, so X is R cosine theta.
What is Y in terms of R and theta?
Sine theta is Y over R, so Y is R sine theta.
So in this equation, let’s replace X with R cosine theta and Y with R sine theta.
We can factor an R out of these terms, and then bring this term to the other side, and
we get R equals three over cosine theta plus sine theta.
Now this is the same equation as before, but with R given in terms of theta.
We could plug in any theta, get the corresponding R value, and we could plot this graph.
So to convert between equations that use rectangular coordinates and ones that use polar coordinates,
we just have to keep in mind these two equations we just used, as well as R squared equals
X squared plus Y squared, which is just the Pythagorean theorem, and tangent theta equals
Y over X.
With this information, we can see how it is much easier to graph circles in polar coordinates
rather than in rectangular coordinates.
Take the graph of R equals five.
This is a circle with a radius of five.
To convert into rectangular coordinates, we could square both sides, and then change R
squared into X squared plus Y squared.
This should look familiar, as it is the form for the equation of a circle that we already
learned, but we can see how it was much easier to interpret in the other form.
Therein lies the power of polar coordinates, which wouldn’t be very useful for lines,
unless the line goes through the origin, as these would just be equations with constant angles.
But polar coordinates are very useful for things like circles and other similar shapes.
Before we leave this topic, let’s talk a little bit about graphing polar equations.
Say we have something like R equals two cosine theta.
One way to plot this is to plug in points.
Let’s plug in multiples of a sixth pi and see what we get.
When theta is zero, cosine is one, times two is two.
So two, zero is a point.
Plugging in a sixth pi, we get two times root three over two, or root three.
We can estimate this to the tenths place and plot that point as well.
For a third pi, we get two times one half, or one, and for half pi, we get zero.
So we’ve just constructed a semicircle here.
We will find that if we continue through until pi, we get the other half of the semicircle,
and then the values just repeat from there.
So this a circle with a radius of one, with its center at one, zero.
We can verify this by converting into rectangular form.
We just multiply both sides by R, change R squared into X squared plus Y squared, and
R cosine theta into X, subtract two X, and then we can complete the square with the X
terms by adding one to both sides, giving us X minus one quantity squared plus Y squared
equals one, which should be a more familiar version of the equation for this circle.
When polar equations get more complicated, their graphs get more complex, producing all
kinds of different shapes with loops and rose curves, which can be quite beautiful.
If you have a graphing calculator, I highly recommend setting it to polar functionality
and just inputting lots of random trig functions to see what you get.
I definitely wasted a lot of time in math class making weird looking butterflies this
way, and I turned out okay.
Let’s check comprehension.