I am trying to understand, how to calculate properties of a triangle which has distributions of the side lengths and angles instead of constant values.
Example: triangle ABC with the angle $\alpha$,$\beta$ and $\gamma$ respectively opposite to the sides a, b or c. Image The distribution of the side length a and b as well as the distribution of the angle $\gamma$ (P(a), P(b), P($\gamma$)) are given: \begin{align} P(a) &\approx \exp(-{10 \times (a-1)^2}) \\ P(b) &\approx \exp(-20 \times (b-3)^2) \\ P(\gamma) &\approx \begin{cases} 1 & \text{if $\gamma\in \left[\frac{pi}{3},\frac{2\pi}{3}\right]$, and} \\ 0 & \text{otherwise.} \end{cases} \end{align}
What would be the distribution of the length of the side c (P(c))?
As far as I understand, using convolution of two probability distributions corresponds to the additions of the variables. But I am not sure, how to use e.g. the law of cosines with distributions.