Angle Sum Formulae: Proof using Ptolemy's Theorem

We prove the angle sum and difference identities for sin(θ ± ϕ), using Ptolemy's theorem.

00:00 Setup
00:51 sin(θ + ϕ)
02:24 Finding |BD|
05:56 Lemma
06:25 sin(θ - ϕ)

[FildasFilipi]

Nice.

My favorite method for remembering those equalites is by using e^i(x+y)=e^ix * e^iy (Its not a proof unless you know that initial equation holds without using the geometric identities, but there are definitions of cos and sin demanding it)

[MichaelRothwell]

Very nice indeed & very unexpected application of Ptolemy's Theorem!

I think we should also consider the case where O lies on BD. In this case θ+φ=90°, so sin(θ+φ)=1 and BD=1 (diameter) so we're good to go in this case also.

BTW, your result that sin α=opposite side for circumscribed circle of diameter 1 is a particular case of the well known result that for a ΔABC, a/sin A=b/sin B=c/sin C=2R where R is the radius of the circumscribed circle. Moreover, your proof of this particular case becomes the proof of the general case if you take the radius of the circumscribed circle to be R instead of ½.

[txikitofandango]

Strange to think that old Ptolemy could've come up with our modern trig functions with just a little more work. Maybe there was a good reason to avoid them though