Define more

Any point on a plane can be identified by two numbers under the Polar Coordinate system

r = distance from the origin to the point

q = angle from the 0¡ line and the line from the origin to the point

Polar Coordinates are useful to write equations for some shapes.  For example the Cartesian coordinate equation for a circle centered at the origin

x2 + y2 = a2

where a is the circle radius.

Substitute the polar conversions into this equation

(rcosq)2 + (rsinq)2 = a2

r2cos2q + r2sin2q = a2
r2(cos2q + sin2q) = a2
r2 = a2
r = a

Where cos2q + sin2q = 1, the Pythagorean Identity

This equation says take all points a distance a from the origin, for all angles -- the definition of a circle.

Pythagorean Identity

The Pythagorean Theorem

x2 + y2 = r2

Divide both sides by r2,

x2 + y2 = r2
r2        r2         r2

or 

where the terms on the left are the definitions of cosine and sine, respectively.  Thus,

cos2q + sin2q = 1