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

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.

• I don't understand what you are asking. I am also wary of the "standard deviation" terminology. What do you mean by that? Jun 12, 2020 at 14:56
• Well, I guess it could also be "variance". But I am trying to create an empirical model where standard deviations can define the period. I have currently got a trig function by plugging in values and seeing if it fits the data: $cos^2\left(\frac{\pi \sigma _y}{\sigma _x}x\right)$. However, as you might have guessed, it doesn't work for all data sets, and I feel it's 'unmathematical'. So, I would like a way of being able to define the period using standard deviation 'mathematically'. I hope this helps and thanks for your reply btw. Jun 13, 2020 at 0:26
• Oh, so, you really did mean the "standard deviation" in the statistical sense. That sounds quite tough. Jun 13, 2020 at 1:15
• That's exactly it! And yeah, I haven't had much luck so far. Jun 13, 2020 at 1:48

I have found an answer from the academic paper A Cosine Approximation To The Normal Distribution: $$\sigma =\sqrt{\int _b^a\left(x-\mu \right)^2f\left(x\right)dx\:}}$$
With the standard deviation you would just solve for $$P$$ in my above equation.