One of the student teachers I supervise is planning a lesson introducing the sine and cosine angle sum formulas. I wanted to give him some advice on how to make the lesson better – in particular, along the axes of motivation and justification – and realized that, never having taught precalculus, I barely had any! Especially re: justification. I basically understand these formulas as corollaries of the geometry of multiplication of complex numbers.[1] I have seen elementary proofs, but I remember them as feeling complicated and not that illuminating.

So: how do you teach the trig angle sum formulas? And in particular:

* How do you make them seem needed? (I offered my young acolyte the idea of asking the kids to find sin 30, sin 45, sin 60, sin 75 and sin 90 – with the intention of having them be slightly bothered by the fact that they can do all but sin 75.)

* Do you state the formulas or do you set something up to have the kids conjecture them? If the latter, how do you do it? How does it fly?

* How do you justify them? Do you do a rigorous derivation? Do you do something to make them seem intuitively reasonable? What do you do and how does it fly?

* Do you do them before or after complex numbers, and do you connect the two? If so, how do you do it and how does it fly?

Any thoughts would be much appreciated.

Addendum 3/20/11:

Thanks to John Abreu, who sent me the following in an email -

Please find attached a Word document with the proofs of the trig angle sum formulas. After opening the document you’ll see a sequence of 14 figures, the conclusions are obtained comparing the two of them in yellow. Also, I left the document in “crude” format so it’ll be easier for you to decide the format before posting.

I must say that the proofs/method is not mine, but I can’t remember where I learned them.

with an attachment containing the following figures (click to enlarge / for slideshow) -

As far as I can tell, the proof is valid for any pair of angles with an acute sum.

Notes:

[1]Let z_1,z_2 be two complex numbers on the unit circle, at angles \theta_1,\theta_2 from the positive real axis. Then z_1=\cos{\theta_1}+\imath\sin{\theta_1} and z_2=\cos{\theta_2}+\imath\sin{\theta_2}, so by sheer algebra, z_1z_2=(\cos{\theta_1}\cos{\theta_2}-\sin{\theta_1}\sin{\theta_2})+\imath(\cos{\theta_1}\sin{\theta_2}+\sin{\theta_1}\cos{\theta_2}). On the other hand, the awesome thing about multiplication of complex numbers is that the angles add – the product z_1z_2 will be at an angle of \theta_1+\theta_2 from the positive real axis; thus it is equal to \cos{(\theta_1+\theta_2)} + \imath\sin{(\theta_1+\theta_2)}. This is QED for both formulas if you believe me about the awesome thing. Of course it usually gets proven the other way – first the trig formulas, then use this to prove angles add when you multiply. But I think of the fact about multiplication of complex numbers as more essential and fundamental, and the sum formulas as byproducts.