The WitCHfest is coming to an end. Our final WitCH is, once again, from Cambridge’s Specialist Mathematics 3 & 4 (2019). The section establishes the compound angle formulas, the first proof of which is our WitCH.
Update (25/08/19)
Similar to our parallel WitCH, it is difficult to know whether to focus on specific clunkiness or intrinsic absurdity, but we’ll first get the clunkiness out of the way:
- John comments that using x and y for angles within the unit circle is irksome. It is more accurately described as idiotic.
- The 2π*k is unnecessary and distracting, since the only possible values of k are 0 and -1. Moreover, by symmetry it is sufficient to prove the identity for x > y, and so one can simply assume that x = y + α.
- The spacing for the arguments of cos and sin are very strange, making the vector equations difficult to read.
- The angle θ is confusing, and is not incorporated in the proof in any meaningful manner.
- Having two cases is ugly and confusing and was easily avoidable by an(other) appeal to trig symmetry.
In summary, the proof could have been much more elegant and readable if the writers had bothered to make the effort, and in particular by making the initial assumption that y ≤ x ≤ y + π, relegating other cases to trig symmetry.
Now, to the general absurdity.
It is difficult for a textbook writer (or a teacher) to know what to do about mathematical proofs. Given that the VCAA doesn’t give a shit about proof, the natural temptation is to pay lip service or less to mathematical rigour. Why include a proof that almost no one will read? Commenters on this blog are better placed to answer that question, but our opinion is that there is still a place for such proofs in school texts, even if only for the very few students who will appreciate them.
The marginalisation of proof, however, means that a writer (or teacher) must have a compelling reason for including a proof, and for the manner in which that proof is presented. (This is also true in universities where, all too often, slovenly lecturers present incomprehensible crap as if it is deep truth.) Which brings us to the above proof. Specialist 34 students should have already seen a proof of the compound angle formulas in Specialist 12, and there are much nicer proofs than that above (see below). So, what is the purpose of the above proof?
As RF notes, the writers are evidently trying to demonstrate the power of the students’ new toy, the dot product. It is a poor choice, however, and the writers in any case have made a mess of the demonstration. Whatever elegance the dot product might have offered has been obliterated by the ham-fisted approach. Cambridge’s proof can do nothing but convince students that “proof” is an incomprehensible and pointless ritual. As such, the inclusion of the proof is worse than having included no proof at all.
This is doubly shameful, since there is no shortage of very nice proofs of the compound angle formulas. Indeed, the proof in Cambridge’s Specialist 12 text, though not that pretty, is standard and is to be preferred. But the Wikipedia proof is much more elegant. And here’s a lovely proof of the formula for sin(A + B) from Roger Nelson’s Proof Without Words:
To make the proof work, just note that
x cos(A) = z = y cos(B)
Now write the area of the big triangle in two different ways, and you’re done. A truly memorable proof. That is, a proof with a purpose.
Apart from the question of why vectors are actually required… why the need for two different cases?
Thanks, RF. No, there is no need for two cases, at least not in the laborious and thus obfuscating manner done here. You are also correct, there is no “need” for vectors; the Cambridge Specialist12 gives a non-vector and more standard proof. I think the purpose of using vectors here is to illustrate the use of the dot product. The question is, is the above proof a good illustration?
I will confess to preferring a geometric proof of this identity. Vectors may work, but seem to complicate and confuse (at least in the way presented here) the beauty that can exist in these trigonometric identities.
I agree.
I will make just the one comment for now:
It really irks me that x and y are used as labels for the coordinate axes *and* to represent angles.
Thanks, John. I hand’t noticed but, yes, that choice of notation is a real Laurel and Hardyism.
Slightly off-topic: I don’t understand this school geometry tradition of defining points in the form “Q(a,b)”. (It clearly is a definition, because afterwards the point is called just “Q”.) I would strongly prefer to define points like any other mathematical object, writing “Q = (a,b)” or “Q := (a,b)”. Then the logic is clear: There’s an object Q, and it’s defined to be the pair (a,b) — which can be interpreted as a point in the plane — of real numbers. If one does not want to identify points with real number tuples, then one could use a map POINT from the set of tuples to the set of points and write something like “Q := POINT(a,b)”.
Can anyone explain the “Q(a,b)” tradition? Where does it come from? Wouldn’t it be didactically and logically preferable to replace it with the standard notation used to define arbitrary mathematical objects?
Hi, CW. I guess it’s a little slick in style, but I’m not sure it’s that uncommon. For example if I consider the function f(x) = sin(x), is that any different? Is there some active confusion that might result from such defining-naming?
Marty,
Seems to be a lot of words defining the two definitions of the dot product.
Instead I would use Euler’s Formula to prove this
In contrast to above there is an elegant proof of this formula in the comments 238
https://math.stackexchange.com/questions/3510/how-to-prove-eulers-formula-ei-varphi-cos-varphi-i-sin-varphi
Steve R
That’s a pretty proof, though possibly circular, depending how you establish the properties of complex exponentials. But it’s also trivial to find non-complex proofs that are way more elegant than the mess above.
One example of a proof and graphical explanation without too many words using the Euler formula
Is in this us blog
https://betterexplained.com/articles/easy-trig-identities-with-eulers-formula/
I haven’t looked at all of the supposed proofs of Euler’s identity in the stackexchange link, but a quick sampling turns up any number that are no proof at all. For instance, they might define a new quantity by a Taylor series and then label it “e^{ix}”. If established notation (such as an exponential) is employed in a -definition-, then it’s necessary to show that the established notation is meaningful in the new context. That is, this power notation must be shown to really have the properties that we know -real- exponentials have. I doubt that even a small number (if any) of the “proofs” on the stackexchange page show that the new quantity e^{ix} e^{iy} will equal e^{i(x+y)}. At least one proof on the site just assumes that d/dx e^{ix} = i e^{ix} -before- it establishes what e^{ix} is going to be; that is, it uses property of the Euler identity to “prove” the identity. Such non-proofs of the Euler identity are common on the internet, where the shortest ones get voted up on sites like stackexchange by unsophisticated readers. They are also common in signal-processing books, which make heavy use of the identity. The authors of these books are often clearly out of their mathematical depth.
Thanks, Don. I think that’s a little harsh, at least for comments on stack exchange, where it is not entirely clear what people are taking as definition or accepted knowledge. But of course you are correct, that if trig properties are used to determine the properties of the complex exponential, then it is not proving anything to then derive trig properties from exponential properties.
Don,
You make a valid point as 238 in the link relies on deriv of e^ix being ie^ix
Which should be demonstrated first
https://www.quora.com/What-is-the-derivative-of-e-ix
Steve R
Hello Steve and Marty,
Yes, the link does just that. For future readers of this blog, the relevant part might no longer be searchable as “238” because that’s only its current number of up-votes. This high number demonstrates how many readers of that stack exchange page wish to–and have the ability to–vote the answer up, even though it’s wrong. (Or perhaps 10,000 people voted it down, and 10,238 voted it up!) This, I think, is a real problem with stack exchange. I don’t know about the maths stack exchange, but as a mathematical physicist and judging by what I see on the physics stack exchange, that site has become a repository for anyone with just enough learning to be dangerous, to convince others and themselves that their answers are fully correct and insightful. Often they don’t know what they are doing, but get voted up by readers who haven’t thought things through.
And since questions and their answers tend to get locked when the slightest bit of noise enters, wrong/naive answers get preserved, with no recourse for later readers to correct anything. I think this is a sad road down which pseudo-academic discussions are heading. As for my comment on signal processing books: I’d say that these books’ authors usually have engineering backgrounds, and are not concerned with mathematical rigor. It would be fine if they admitted that, but the problem is that they come across as suggesting that their scant or convoluted discussions are fully rigorous.
Don
Thanks, Don. I don’t frequent maths stack exchange, but I’ve usually been impressed when I’ve gone there. There are clearly some very smart, knowledgeable and generous posters. But, I’m not sure it works as well for those with a weaker mathematical background. I haven’t spotted unchallenged wrong answers, but there are a fair few non sequiturs.
Don,
Thought you might be interested in another physicists empirical approach to algebra from 1963 Cal Tech lectures
http://www.feynmanlectures.caltech.edu/I_22.html
Steve R
PS I found his summary of how log tables were originally calculated to several figures of interest too
That’s very funny. I have a different chunk of Feynman with me for a holiday project. Thanks, Steve. I had forgotten that excellent lecture. Of course there’s tons to nail down, but in the main Feynman is upfront about what requires nailing.
Thanks for the link Steve.
I first read that chapter about 30 years ago; I have all three volumes in paper copy, and have worked minutely through every one of his lectures. But I read it again yesterday. It packs a huge amount of information, and it’s great to see how various things can be calculated from such a lean starting point. Of course, at the time of Feynman’s lectures, students used tables all the time. So did I at high school. I wonder what a modern generation of students makes of that lecture, who might think that logs and sines are just a natural by-product of putting a voltage across a piece of silicon?
Don
As far as I can see (through limited research) regarding the P(a,b) notation is that textbooks started using it after the exam papers started using the notation.
Will take a lot more time than I have to get more specific though.