Arcsine as Integral/Lemma 2

Lemma
Let $\sin_G$ be the geometric sine.

$\arcsin_G$ is the inverse of this function.


 * $\ds \map {\arcsin_G} x = \int_0^x \frac {\d x} {\sqrt {1 - x^2} }$

Proof
This result will be used in proving Derivative of Sine Function in the geometric case.

So we can not use the same reasoning as 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: