# 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. 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:

 $\text {(1)}: \quad$ $\ds x^2 + y^2$ $=$ $\ds 1$ Equation of Circle $\ds \dfrac {\d x} {\d y}$ $=$ $\ds -\dfrac y x$ Implicit Differentiation $\ds$ $=$ $\ds -\dfrac y {\sqrt {1 - y^2} }$ substituting for $x$

Then:

 $\ds \arcsin_G y$ $=$ $\ds \int_0^y \sqrt {1 + \paren {\dfrac {\d x} {\d y} }^2} \rd y$ Definition of Arc Length $\ds$ $=$ $\ds \int_0^y \sqrt {1 + \paren {-\dfrac y x}^2}$ substituting for $\dfrac {\d x} {\d y}$ $\ds$ $=$ $\ds \int_0^y \sqrt {1 + \dfrac {y^2} {x^2} } \rd y$ $\ds$ $=$ $\ds \int_0^y \sqrt {\dfrac {x^2} {x^2} + \dfrac {y^2} {x^2} } \rd y$ rewriting $1$ to create common denominator $\ds$ $=$ $\ds \int_0^y \sqrt {\dfrac {x^2 + y^2} {x^2} } \rd y$ combining terms with common denominator $\ds$ $=$ $\ds \int_0^y \sqrt {\dfrac 1 {x^2} } \rd y$ Equation of Circle $(1)$ $\ds$ $=$ $\ds \int_0^y \dfrac 1 x \rd y$ in Quadrant $\text I$ and Quadrant $\text {IV}$ $\ds$ $=$ $\ds \int_0^y \dfrac 1 {\sqrt {1 - y^2} } \rd y$ substituting for $x$ in Quadrant $\text I$ and Quadrant $\text {IV}$

$\Box$