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August 24th, 2006

A circle c is defined by any point P equidistant from an origin O forming a circumference C, where the equidistant displacement of P from O is the radius r. An angle measure θ is defined as a fraction of this circumference C, called the arclength l, circumscribed over radius r.

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:

16. x=cosθr=cosθ
17. y=sinθr=sinθ

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

18. 12 = cos2θ + sin2θ

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