Polar Coordinates & The Circle
Below is a circle with an angle, , and a radius, r. Move the point (r, ) around and see what shape it creates.
Think about how x and y relate to r and .
MIND CHECK:
Do you remember your trig and right triangle rules? Check them out here:
SOH CAH TOA
sin = =
cos = =
From these two things, with some moving around, we see that x = rcos and y = rsin. These equations help up convert polar coordinates, (r, ) to cartesian coordinates (x, y).
From the above activity, we see that moving around the point (r, ) gives us a circle if we go around
2 radians, a full revolution.
With our conversion above, our circle equation, and r = .
Now we have Cartesian to Polar coordinate conversion equations.
Think about the equation r = a. What is this telling us about the circle it represents? Use the graph below to help you.
If we think about r = 2acos what is this telling us? Think about what x is in polar coordinates.
Play around with the circle below to figure out what this is telling us.
If you said this means that we have a circle with radius |a| centered at (a, 0) then you are thinking correctly.
What about r = 2bsin? Explore below:
If your exploration got you to see that this equation gives you a circle with radius |b| centered at (0, b) then you are seeing things correctly.
Last but not least, let's think about r = 2acosθ + 2bsinθ
This is the general equation of a circle!