# agm variants of ordinary trigonometric functions

The ordinary sine and cosine are averages of exponentials. Here, I play around with replacing the arithmetic average with the arithmetic geometric mean

Let $\mathrm{cas}(z)=\mathrm{agm}(e^{iz},e^{-iz})$

And since the $\mathrm{agm}(x,y)$ may be represented in terms of a complete elliptic integral of the first kind $K(m)$:

$K(m)=\int_0^{\frac{\pi}{2}}\frac{d\theta}{\sqrt{1-m\sin^2(\theta)}}$

$\mathrm{agm}(x,y) = \frac{\pi}{4} (x + y) \; / \; K\!\left[\left( \frac{x - y}{x + y} \right)^2 \right]$

Therefore
$\mathrm{cas}(z) = \mathrm{agm}(e^{iz},e^{-iz}) = \frac{\pi}{4} (e^{iz} + e^{-iz}) \; / \; K\!\left[\left( \frac{e^{iz} - e^{-iz}}{e^{iz} + e^{-iz}} \right)^2 \right]$

Therefore:
$\mathrm{cas}(z) = \frac{\pi\cos(z)}{2K(-tan^{2}(z))}$

Domain colored, with mpmath, it sort of looks like this: