The theta comes from eulars formula
e^(ix) = cos(x) + isin(x)
Which isn't actually an angle in terms of the problem itself it's just a byproduct of solving some part of the equation linking it back to argand diagrams. So the theta comes from this and is why it seems to appear out of nowhere.
Hence where the right angled triangle comes from and hence the relationship I mentioned in my previous post.
I got this from his full solution here http://www.numberphile.com/pi/Pi_Balls_Ed.pdf
Which is in absolutely no way obviously related (or linked) in the video itself.
edit - FUN FACT
When I was doing further maths at a-level the link e(ix) = cos(x) + isin(x) made it so I didn't have to remember lots things I was meant to learn for rote in normal a-level maths. A few times I got comments form the teacher saying I have no idea what you use to solve these problems but you seem to get them right every time.
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