Arcsine as Integral/Lemma 2

Proof
For this proof only, $\sin_G$ be the Geometric Sine from Definition:Sine/Definition from Circle.

This result will be used in proving Derivative of Sine Function in the geometric case. So we can not use the same reasoning as Derivative of Sine Function\Lemma 1 because our logic would be circular.


 * Limit-arc.png

Let $\Theta$ be the length of the arc associated with the angle on the circle of radius $1$.

By definition of arcsine:
 * $y = \sin \Theta \iff \Theta = \arcsin y$

We have that arc length is always positive.

For negative $y$, the $\arcsin$ function is defined as being the negative of the arc length.

This makes the $\arcsin$ function and the $\sin$ function odd, and puts us in line with mathematical convention:


 * Inverse Sine is Odd Function.


 * Sine Function is Odd

Without this convention, the derivative of the $\sin$ function would not be continuous.

Now:

Then: