Derivative of Sine Function/Proof 5

Proof

 * 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:

Note that we get the same answer as Derivative of Arcsine Function.

By definition of real $\arcsin$ function, $\arcsin$ is bijective on its domain $\closedint 1 1$.

Thus its inverse is itself a mapping.

From Inverse of Inverse of Bijection, its inverse is the $\sin$ function.