1. θ = l⁄r.
When every point P circumscribes an arclength l such that a circle is formed, l = C and the angle measure θ is defined to be 2π. Therefore, it follows that:
2. θ = C⁄r
3. rθ = C
4. r2π = C
4'. C = 2πr.
Similarly, the degrees D of a circle are defined to be 360° when l = C. A radian R is defined to be the ratio of these 360° per archlength l.
5. R = 360°⁄l
Where l = C for a unit circle r = 1, then l = 2π. Therefore,
6. R = 360°⁄C
7. R = 360°⁄2πr
7'. 360° = 2πR
For the polar coordinate system, the point P is described by the vector components radius and polar angle:
8. P = (r,θ)
For the Cartesian coordinate system, the point P is described by the horitizonal x and vertical y vector components:
9. P = (x,y)
Notice the inequality between Cartesian and polar measures:
10. x < r
11. y < l
The Cartesian system is the foundation for triangular descriptions of a circle, or trigonometry. The trignometric functions of the polar angle θ are transformed to the Cartesian plane as:
12. sinθ = y⁄r
13. cosθ = x⁄r
And for all angle measures θ > 0,
14. tanθ = y⁄x
If r = 0, then a circle is undefined and these trignometric definitions are undefined.
The inverse trignometric functions are the multiplicative inverses of these trignometric definitions.
12'. cscθ = 1⁄sinθ
13'. secθ = 1⁄cosθ
14'. cotθ = x⁄y
According to the Pythagorean Theorem,
15. r2 = x2 + y2
Assume a unit circle r = 1. It follows from the trignometric function definitions that:
Substituting these Cartesian coordinates into the Pythagorean Theorem yields the fundamental, trigonometric identity:
18. 12 = cos2θ + sin2θ