The most basic circle is r = constant i.e. the set of all points equidistant from the origin.
There are s special circles r = acos(Θ) and r = asin(Θ)
Below we see the animations but why are these circles? This will be one of the
few times we'll compare polar and rectangulat coordinates. x = rcos(Θ) and y = r sin(Θ)
x2 + y2 = r2 and tan(Θ) = y/x .
Suppose r = cos(Θ) . Multiply each side by r : r2
= rcos(Θ). We obtain:
x2 + y2
= x rearranging we obtain x2 - x + y2 = 0. ompleting the square
(x-1/2)2 + y2 = 1/4 which we recognize as
the equation of a circle of radius 1/2
centered at (1/2,0). Show r = sin(Θ) is a circle centered at (0,1/2) of radius 1/2.