Double Angle Formulae

I posted this on my Adv. Precalculus google classroom site. I don’t know if I’ll get any responses, but I loved the problem, so I thought I’d share it here.


I mentioned in class that I had stumbled across a beautiful different proof for the double angle formulae for sine and cosine, and I would post it to the classroom. But instead of *giving* you the proof, I thought I’d share it as an (optional) challenge. Can you use this diagram to derive the formulae? You are going to have to remember a tiiiiny bit of geometry! I already included one bit (the 2*theta) using the “inscribed angle theorem.”

If you do solve it, please share it with me! If you attempt it but get stuck, feel free to show me and I can nudge you along!



Below this fold, I’m posting an image of my solutions! But I say to get maximal enjoyment, you don’t look further, take out a piece of paper, and take a stab at this!




  1. Ooooh! @eulersnephew just tweeted that they got sin(t)/cos(t)=sin(2t)/(1+cos(2t)). Which I never saw when working on this problem. But it’s JUST COOL because that is:


    That’s unexpected. I definitely wouldn’t have expected a tangent graph to be so simply related to double angle formulae! And kinda weird, because if that 1 wasn’t there, the right hand side would simplify to tan(2t). So how does that “1” change a graph of tan(2t) and make it into tan(t)?

    So I graphed the right hand side to see how the number in the denominator affected what the graph looked like:


  2. As I have never tried to prove double angle formula by pure geometry(this is great!!!), so I wonder if this technique can extend proving the general angle formula, i.e. sin(a+b)=???. Also, I really wanna know how do you come up with that pure-geometric proof, thanks!!!

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