10.6 Polar Coordinates

This section is all about polar coordinates. We are used to graphing things on the rectangular coordinate system, where (x,y) corresponds to a point x units over on the plane, and y units up.

However, there is another way to graph points: the polar coordinate system. Instead of representing points on the plane as (x,y) they are represented as (r,θ) where r is the radius of a circle with its center at the origin, and θ is the amount of rotation. Here's an example of a point graphed by polar coordinates:

One interesting thing about graphing points on a polar coordinate system is that a single point can have multiple coordinates. For instance, take the point (2, 2√3). On a rectangular grid, it can only be written as (2, 2√3) or it would be a different point. However, on a polar grid, that  exact same point can be written as (4, π/3), or (-4, 4π/3) or (4, -5π/3) or an infinite other number of ways.

Basically (r,θ) = (r,θ+2πn) or (r,θ) = (-r,θ+2πn)

To convert between polar and rectangular coordinates, just think of it like a triangle, such as the one pictured above. x = r cosθ, y = r sinθ, and r² = x² + y².

And thats really all there is to polar equations!