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.

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.

### 22 Responses to Angle Sum Formulas: Request for Ideas

1. I never liked teaching these. The complex number connection is new(ish) to me, and I wouldn’t be ready to teach with it yet. If I were teaching trig, I’d try to get solid enough on it to think about doing it that way. But I’m not sure leading trig students through complex number land would be simpler for them.

2. Aran says:

I also find these formulas to be very hard to motivate. I generally focus on the sheer coolness of the formulas themselves (this may be an exaggeration). The biggest let down is that even if you convince the kids that it’s worth it to get more exact values than 30, 45, 60, etc. they eventually realize that it only adds a couple more to the list (15, 75, etc.) Sometimes, if the kids are game for it, we use the geometry of a pentagon to find the exact values for 36 and 72. Armed with those you can find any multiple of three (75-72).

As for derivations, you can do the complex number one without using complex numbers if you use transformations and matrix multiplication. The matrix that rotates a point by angle A is [[cos A -sin A][sin A cos A]] and similarly for angle B [[cos B -sin B][sin B cos B]]. Rotating by angle (A+B) is just [[cos(A+B) -sin(A+B)][sin(A+B) cos(A+B)]]. But you can also find that by composing (multiplying) the rotation matrix for A and for B. Comparing the results gives the identities.

3. Julia says:

I’m teaching this on Monday and was planning to wait on any proofs until I do eulers formula with at least some of the kids next year. Keep them in suspense, or relief, whatever the case might be. 🙂
Looking forward to other responses!

4. jd2718 says:

Ben,

you might hate this:

I derive them, slowly, with the kids helping.

And I remark, after, that there is no need to remember them – look them up all you want. And while occasionally useful, they are nowhere near the top of the heap.

Instead I emphasize how many hard pieces there were to the derivation (distance, unit circle, angle subtraction, Law of Cosines, mult. of binomials, pythagorean ids, etc),

and how the ability to derive (even with my hints) is the equivalent to having fairly significant mathematical power.

“Anyone” I tell them “can look these up. And many people, with training, can apply them. But who could follow the derivation? Who could participate?”

Jonathan

5. Justina says:

After reading your post, I realize that I am not nearly as rigorous with this topic as I could be! I don’t derive them, and I don’t connect them with complex numbers. Although, I am teaching complex numbers next week so this gives me something to think about.
I wanted to share a story that I found a couple years ago that helps my students (and me!) to remember the formulas
http://www.pleacher.com/mp/mlessons/trig/sumdif.html

• jd2718 says:

If you are just looking to memorize them, and I can’t say that love the idea, but if that’s what you would like to do, try this: Make it a cheer.

SINE-COSine,
COSine-SINE,
COSine-COSine,
(full rest)- SINESINE (quick)

8 beats, they can clap with the first 6, pause on the rest, and double clap or double tap on the SINESINE at the end.

It works.

Jonathan

6. Mimi says:

I have not taught this topic yet, so it’s just a rough idea/suggestion, but how about giving the kids the complete derivation on a piece of paper and having them explain/annotate/justify step by step what’s happening, and then taking that sheet away and have them fill in missing parts of the same proof? (I’m thinking more along the lines of a proof that doesn’t include the unit circle.) I think there’s value in kids striving to read and understand complex proofs and derivations, even if they might not be able to come up with it by themselves.

I agree with Jonathan that they don’t need to memorize the sum/diff formulas, but just need to be able to look them up and to apply them. In general, that’s how I feel about all these trig identities except the ones that are immediately derivable from (sinx)^2 + (cosx)^2 = 1 or that are geometrically obvious such as sin(x) = cos(pi/2 – x).

7. John Abreu says:

Hi there,

I teach this topic using a combination of “Proof Without Words” and “Folding Paper” techniques, my students love it. I have prepared a the sequence figures to show how it works which I’d be delighted if the owner could post on the blog. I wait for instructions.

• John – that sounds great. If you have the figures electronically, you can email them to me and I’ll add them to the post.

8. Japheth Wood says:

A nice problem that can be solved with the angle sum formula as the right tool is:
What is the maximum value of 4cos(x)+3sin(x)? No calculus needed!
Japheth

9. Dan Goldner says:

I would just put a formula up and ask “True or False, and why?” (Or, if they know what you mean by it, just say “Prove or disprove.”) Then give them a class or two or three and see what they can do. If they can’t get it or get it too easily, give them more identities.

10. Avery says:

Historically, I am under the impression that the major motivation for these formulas was to fill in trig tables in the pre-calculator days. Right now I have 4 pieces of technology within reach (phone, iPad, computer, & phone) that could quickly give me sin(7.5). I was even able to get sin(7.5) on my nonsmart phone by texting “sin 7.5 degrees” to GOOGL (you can also text sin(7.5) but it’s in radians). While interesting in their own right, this begs the question of whether these identities are foundational (ie should be taught to every student in precalc). Sure, they also help solve some ugly integral problems in calculus, but if that’s the motivation my vote would be to wait until then to talk about these identities.

11. @ Japheth – of course I had to solve it with calculus first because not until I knew what the solution was could I see what it could possibly have to do with the angle sum formulas! You are such a formalist! I’m tellin ya!

@ Dan – Love it. If you had made this comment any earlier than Sunday I would have been a little annoyed that my student didn’t incorporate it into Monday’s lesson. (It fits perfectly with how he opened class anyway.)

@ Avery – With due deference to the fact that I love how you think, I think this stance may be overly dogmatic. Almost every mathematical concept’s historical motivation became outmoded by subsequent developments – this doesn’t bear on the question of whether it’s foundational or not. Shit man, the whole computational side of algebra and calculus can basically be done by WolframAlpha, so if a person thinks that the fact that technology can bail me out on sin 75 means this isn’t legitimate motivation for the angle sum formulas, they aren’t going to let me use an integral as motivation either.

Personally I don’t feel that I understand the proper organization of a precalc course (in particular because whenever I taught calculus I couldn’t escape the feeling that what I really needed my kids to have down cold – the algebra of rational expressions – hadn’t been where the attention went). However, if there is a time in the high school curriculum that is going to be devoted to the theory of trigonometry, then it seems to me that the sum and difference formulas belong there. (And not, for example, in calculus, even though that would be one way of providing motivation, because the kids have plenty of other heavy stuff to think about.) Along with the pythagorean identity $\sin^2+\cos^2=1$, these are the bedrock of the amazing set of relationships that the trig functions all have with each other. (“The rest is commentary…”)

Half of me is leaning toward the idea that it’s natural to do these identities around the time you develop complex numbers. The other half is attached to the drama (which I’ve exploited numerous times in tutoring) of what happens when the kid already knows the formulas, and has changed gears to complex numbers (possibly with time in between), and you innocuously ask her to compute the product (c+si)(C+Si). This necessitates already having developed these identities independently of complex numbers. Either way, though, if we’re going to bother with either complex numbers or trig for their own sake, it seems to me these guys belong at the heart.

Of course this doesn’t solve the motivation problem though 😉

• Joel Patterson says:

Hey Ben. I’m teaching calc BC at CRLS now, and our kids are now well-prepped in the algebra of rational expressions. I find it is the trig stuff I need to remind them on. Thanks for posting the images of the angle sum identities by a fold in paper.

12. Avery says:

I agree that I may have overstated my point earlier. You’re right that historical motivations oftentimes (inevitably?) become outdated (although the opposite can also be true–ideas aren’t developed with any “real world” application in mind and are only later used outside the realm of theoretical mathematics…RSA encryption being the classic example). I also agree that what we teach shouldn’t be driven solely by the question of historical usefulness. That said, I still think we should be asking the question: Why is this important/interesting/cool/exciting? If you and your students can answer this (which may be as simple as “let’s play 16th century mathematician”), explore away.

13. vlorbik says:

thanks for the post; another great thread.

i recently posted some remarks

stuff to any class i introduce to trig-sum “laws”.
thinking of (cos[t], sin[t]) as
an ordered pair, subject to certain manipulations,
just turns out *easier* (for at least some
users; for example, me) than geometric stuff.
if the (x, y) \LeftRightArrow x + i*y
correspondence isn’t yet familiar…
what a shame; let’s start now.
quickly sketch the situation
(multiplying by complexes
adds angles and multiplies lengths, e.g.);
declare by fiat that e^it = cos(t) +i*sin(t)
calculate out
cos(t+s) + i*sin(t+s)
[applying laws-of-exponents
along the way…]

14. Aran says:

@Avery and @Ben: Both of you make great points, but personally I lean toward Avery’s first comment (are they really foundational?). I think it’s important to consider what learning outcomes we want from an entire math course, or even for a full four year curriculum. Once you nail down those learning objectives, then the content just needs to fit into the narrative. Whether they prove these specific formulas or not, they will certainly prove some sort of identity somewhere along the way (if not, I would disagree with your learning objectives). There is plenty of great math to be done in precalculus (and high school math in general), so if you can get the kids motivated and engaged and thinking critical around sum/difference formulas then do it. If not, don’t belabor the point just because there’s a section of the text book devoted to it. Just make sure they know it’s there so they can look it up when (and if) they ever need it.

15. vlorbik says:

the first * in my line -3
should be a +.
thank you and good day.

16. […] a new packet, and the process continued like that for most days. [We did have a bit where we did a paper folding activity, which was pretty […]

17. […] a new packet, and the process continued like that for most days. [We did have a bit where we did a paper folding activity, which was pretty […]

18. Deacon John AIken says:

Here is the most intuitive way to derive a geometric of the sum and difference formula for the sine and the cosine.
http://www.cut-the-knot.org/triangle/SinCosFormula.shtml