Did Santa disappoint this year? Just how good do you have to be in order to get what you've always wanted? Is a generalization of the Law of Cosines applicable to pentagons really too much to ask for? Well, whether you wanted it or not, here it is. I doodled it today during my morning coffee.
As you'll recall, the Law of Cosines gives one side of a triangle in terms of the other two sides and the opposite angle. (See Wikipedia here, and why not give them five bucks while you're at it?) So as I sat down with my coffee, I decided there ought to be a Law of Cosines that gives the unknown side of a pentagon in terms of the other four sides and the "opposite angles" (i.e., the three angles of which the unknown side is not a part).
As an extra stocking stuffer, the simpler Law for quadrilaterals:
here and you'll have another chance to give them five bucks).
By the way, I donated $10 to Wikipedia myself today. Merry Christmas, Jimmy Wales!