+118660 Hi mathmeme
\(\mu\quad and \quad C \quad \text{are both constants}\\ Let \quad U(\theta) \text{ be referred to as U}\)
So we have:
\(U=\frac{\mu C}{\mu sin(\theta)+cos(\theta) }\\ U=\mu C(\mu sin(\theta)+cos(\theta) )^{-1}\\ \frac{dU}{d\theta}=-\mu C(\mu sin(\theta)+cos(\theta) )^{-2}(\mu cos(\theta)-sin(\theta))\\ \frac{dU}{d\theta}=\frac{-\mu C(\mu cos(\theta)-sin(\theta))}{(\mu sin(\theta)+cos(\theta) )^{2}}\\ \text{When }U'=0\\ -\mu C(\mu cos(\theta)-sin(\theta))=0\\ \mu cos(\theta)=sin(\theta)\\ \mu=tan(\theta)\)
Trouble is this could give a minimum or a maximum.
I did a graph in Desmos and found that mu = tan(theta) gives both the minimum and the maximum degree of happiness.
Here is the graph:
https://www.desmos.com/calculator/1c8zst7hlo
Feel free to question me about this answer :)
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