How to define a trigonometric function, given standard deviation for x and y?

Question

I would like to define a trigonometric function of the form $f(x)=\alpha \cos^2(P\pi x)$

where I can define $\alpha$ as an amplitude as the function's range, and $P$ as a period related the standard deviation of $x$ values.

Is this possible? I haven't found much help when searching for a link between a cosine function's standard deviation and period. I would be grateful for any help.

Answer 1

I have found an answer from the academic paper A Cosine Approximation To The Normal Distribution:

This can be generalised as...

${\displaystyle \sigma =\sqrt{\int _b^a\left(x-\mu \right)^2f\left(x\right)dx\:}}$

With the standard deviation you would just solve for $P$ in my above equation.