# 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 $\sin\theta$

-