Half Angles from Euler's Formula
I’ve been reading the lovely Visual Complex Analysis by Tristan Needham, and the visual-style proofs he’s been throwing down have been wonderful and refreshing. I’ll write more about this book and its goals later, but I was inspired this AM to write up a proof of the half angle identities from trigonometry using some of the tools from the book. Here’s the half angle identity for cosine: This is an equation that lets you express the cosine for half of some angle in terms of the cosine of the angle itself. As you can imagine, there are double-angle, triple angle, all sorts of identities that you can sweat out next time you find yourself in a 9th grade classroom. ...