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. r

^{2}= x

^{2}+ y

^{2}

Assume a unit circle r = 1. It follows from the trignometric function definitions that:

16. x=cosθr=cosθ

17. y=sinθr=sinθ

Substituting these Cartesian coordinates into the Pythagorean Theorem yields the fundamental, trigonometric identity:

18. 1

^{2}= cos

^{2}θ + sin

^{2}θ