Every Triangle Is The Right Triangle

Let’s generalize! Numbers are silly—letters bring us closer to the Platonic realm. Suppose we have a right triangle with legs of length \(a\) and \(b\), a hypotenuse of length \(c\), and angles \(\alpha\) and \(\beta\) (opposite \(a\) and \(b\), respectively). Can you come up with an equation for \(c\), in terms of just \(a\) and \(b\), and maybe \(\alpha\) and \(\beta\)?

Draw a triangle with sides of length \(4\) and \(8\), and an angle in between those two sides of \(51^\circ\). Then, find the length of the third side, and of the other two angles.

(Note that this isn’t a right triangle! So you can’t use our tools that only work for right triangles, like sine, cosine, and the Pythagorean Theorem… at least, you can’t use those tools directly.)

(Don’t try to look up any fancy equations! You have all the tools you need to be able to solve this. We’re Amish woodworkers, building beautiful furniture with just very simple hand tools—no need for the i-Lab or huge CNC machines or routers or George and Zoe. Or, if you prefer: imagine you’re the solo hero of an action movie, going up against the much better-equipped enemy, and defeat them not despite your purity and simplicity, but because of it. No fanciness.)

Do the previous problem, but more generally: suppose you have a triangle with sides of length \(a\) and \(b\), and in between them an angle of \(\gamma\) (“gamma”)(when I write it by hand, I just do a downwards-swimming fishy alpha \(\alpha\)). Can you come up with all three sides and three angles of this triangle, only in terms of \(a\), \(b\), and \(\gamma\)?

(For notational consistency, maybe label the angle opposite side \(a\) as \(\alpha\), the angle opposite side \(b\) as \(\beta\), and the side opposite angle \(\gamma\) as side \(c\)?)

1. only sines
2. only cosines