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$$ \sin 5\theta = \sin^5 \theta - 10\sin^3\theta\cos^2\theta + 5\cos^4\theta\sin\theta $$

$$ \cos 5\theta = \cos^5 \theta - 10\cos^3\theta\sin^2\theta + 5\sin^4\theta\cos\theta $$

The separating hyperplane theorem has a central application in mathematical microeconomics. If you've studied it or taken some micro courses, then you might recognize this diagram:



If not: this is an Edgeworth box, a visualization tool used in economics. It models a simple case of general equilibrium theory, a pure exchange economy with 2 agents O and A (apparently, also known as Octavio and Abby), and 2 goods X and Y. This particular example abstracts away from production of the goods, in order to focus on "pure exchange"; hence, the quantity of each good in overall economy is constant at Ω$X$ and Ω$Y$. Each agent starts with a part of this overall quantity which is his/her endowment. In vector form, O has endowment (ω$X$, ω$Y$), and A has (Ω$X$ - ω$X$, Ω$Y$ - ω$Y$). The goods can both be subdivided into any real-valued amount.

What we are interested in, is whether O and A might like to barter some of their endowments with one another, in a way that makes both of them feel more satisfied with making a deal than with not making one. Therefore, the last piece of the model is a pair of preference relations, which quantify the notion of "more satisfied": two binary relations $$(\succsim_O, \succsim_A)$$ on $$X\times Y$$ the set of possible have-able quantities of goods.

utility functions: one for each agent, U$O$ and U$A$, to quantify the notion of "more satisfied". The utility function is

Side note: you do not really need to assume that U$O$ and U$A$ exist a priori as such; you can show their existence as a consequence of some binary

makes each of them more satisfied with making a deal than s/he would have been with not making one.

Therefore we need one more element of the model: a pair of utility functions U$O$

The point prominently marked by the arrows,

https://web.stanford.edu/~jdlevin/Econ%20202/General%20Equilibrium.pdf