# Gaussian approximation

Consider the following function of a variable $\theta\in[0,\frac{\pi}{2}]$ $$f(\theta)=\frac{A}{\sqrt{2\pi}\sigma\sin\theta\sin\theta_{\mu}}\exp(-\frac{(\sin\theta-\sin\theta_{\mu})^{2}}{2\sigma^{2}})$$ Numerically it seems that for small $\sigma$ this can be approximated by a Gaussian $$f(\theta)\approx\frac{A'}{\sqrt{2\pi}\sigma'}\exp(-\frac{(\theta-\theta_{\mu}')^{2}}{2\sigma'^{2}})$$ How do $\theta_{\mu}'$, $A'$ and $\sigma'$ relate to $\theta_{\mu}$, $A$ and $\sigma$? Trail and error suggests $$\theta_{\mu}'=\theta_{\mu}$$ $$\sigma'=\frac{\sigma}{\cos\theta_{\mu}}$$ $$A'=\frac{A}{\cos\theta_{\mu}\sin^{2}\theta_{\mu}}$$ Any idea on how to justify this approximation?

Edit: original question already assumed $\sin\theta\approx\sin\theta_{\mu}$ for small $\sigma$ in the factor before the exponent, thereby making $f(\theta)$ a Gaussian of $\sin\theta$

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## 1 Answer

This approximation uses the first-order Taylor series approximation of the sine around x=0 (you can see it here Wikipedia).

Since you are just using the approximation for the sine function, I think the constants would remain the same. Please take into account that, in order to approximate this function for higher values, you could use more terms in the Taylor series.

Hope this helps.

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Thanks for your suggestion: it led me to reconsider an approximation I already made (changed it in the question). –  Wox Sep 20 '12 at 9:28