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!!!

  3. oh man, i love this approach–it’s the way I do it with my precal kids (…stole it from an IB book). i basically give them the same blank diagram and (after a little steering) they can totally come up with the double angle rules for themselves.

    p.s. this often happens pretty early in precal when my kids think that “proofs” are those boring two-column things they got dragged through in geometry. i feel this is one of the earliest instances where they get to justify why something is true but don’t feel restricted by what they perceive as official “rules” for how a proof should go.

    1. David!!! I miss you! Hope all is well! Let me know if you have other things like this. So far, no takers from the kids in my class… Sigh. Next year, though, I’m definitely doing this.

      I recently had (with a lot of steering, so it only took 15-20 minutes or so) my kids prove the law of sines using a triangle-in-a-circle argument… but after they had first proved it by using the area of the triangle argument. I think next year I might just skip to the triangle in the circle argument. If you don’t do it already, you may love it! I’ll try to blog about it soon, both for you and so I can remember for next year.

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