Derivative of Sine Function/Proof 5

Proof

 * Limit-arc.png

We know that $y=\sin\left(\Theta\right)$ if and only if $\Theta=\arcsin\left(y\right)$, but what is $\Theta$? $\Theta$ is the length of the arc associated with the angle on the circle of radius $1$. One small caveat here. The length of an arc is always a positive number. But if $y$ is negative, we say that the $\arcsin$ is the negative of the length of the arc. This makes the $\arcsin$ and the $\sin$ functions odd, and puts us in line with mathematical convention. Inverse Sine is Odd Function. Without this convention, the derivative of the $\sin$ function would not be continuous.

I have chosen to make $y$ rather than $x$, the independent variable. This avoids a improper integral at $x=1$, which is where $y=0$. Now:

What is the length of this arc? According to Definition:Arc Length, this length is:

Note that we get the same answer as Derivative of Arcsine Function. This function has a positive derivative on $\left(-1,1\right)$ so it is continuous. And because of Derivative of Monotone Function it is strictly increasing on $\left(-1,1\right)$. It is therefore one to one and has a inverse. This inverse must be the $\sin$ function, because the Inverse of a Inverse function is the original function.