Memorizing Trig Identities

Question

I adjunct for a local community college teaching College Algebra and College Trigonometry. Every year, the community college math department insists on students memorizing each of the trig identities. I can understand why students should have the pythagorean identities memorized and the unit circle memorized. But, why should a student have to memorize double angle, half angle, product to sum formulas, etc? I'm trying to get a better grasp on why the college requires this and will not allow students to have a notecard for these. I don't remember the product to sum formula being a pre-requisite for Calculus or any higher level courses. Does anyone have any insight on this, or is it just an odd request from the college?

Answer 1

I agree this memorization is not necessary.

If students understand how the trigonometric functions are defined (unit circle) and know several basic identities, everything else can be derived. I think the pythagorean identity and the sine and cosine of sum of angles are sufficient.

$\sin^2 \alpha + \cos^2 \alpha = 1$

$\sin(\alpha + \beta) = \sin(\alpha) \cos(\beta) + \cos(\alpha) \sin(\beta)$

$\cos(\alpha + \beta) = \cos(\alpha) \cos(\beta) - \sin(\alpha) \sin(\beta)$

Math is not about memorization. If students are taught the basics and learn how to derive whatever they need, everybody will be happier.

Answer 2

If you want to integrate $\sin^m x$ or $\cos^m x$ for even $m$, you need to reduce the powers, which requires some trigonometric identities beyond pythagorean identities (in this case, $\sin^2 \theta = \frac{1 - \cos 2\theta}{2}$). If the students are struggling with the application of trigonometric identities at this stage, then they are in trouble when they are asked to perform a "simplification" like the following before the calculus even begins:

$\sin^4 \theta = \left(\sin^2\theta\right)^2 = \left(\frac{1-\cos2\theta}{2}\right)^2 = \frac{1 - 2\cos2\theta + \cos^22\theta}{4} = \frac{1 - 2\cos2\theta + \frac{1+\cos4\theta}2}{4}$

A very very large number of students in my calculus classes do not get the "$4\theta$" at the end there, and the ones who are having trouble applying the identities in this way certainly never get the factor of $\frac14$ that comes from integrating such a thing, because they are buried in arcane symbols.

So I can imagine, at least for "double angle" or "half angle" formulas, that this line of reasoning would justify students committing these things to memory. If our calculus students see this "power-reducing" identity as "easy stuff I learned a long time ago," then they can much more easily reach the goals I've set out above.

That said, I don't know where to draw the line on memorization. But I can easily support drawing the line a little further out than just pythagorean identities and the unit circle.

Answer 3

If you have lots of time then you can surely derive most of the identities from the basics but basically it slows down your learning when these identites are required. Also not memorising these will give you a huge disadvantage in exams.

Another reason: sometimes it is required to know the form of the result of applying an identity quickly. Like while doing integration probably you will want to know that $\sin(x)\cos(y)=\text{some constant}*\text{sum of trig functions}$. You cannot just keep deriving identities and checking weather they will work or not. In other words you must know which identity to use and when and how.

Lastly I want to tell you my approach for memorising these:

(i) Everyday take one identity to be memorised
(ii) Derive it once in the beginning
(iii) Solve many questions related to it
(iv) If you forget it in between, rederive it
(v) repeat from step (iii) until it is memorised.

The advantage of this approach is that it will give you a solid understanding of the identity and also you will know where and when to apply it.

In fact I had never lerned the identities in the trig class .. i somehow got away with it. But when i went further and studies coordinate geometry, conic sections, integration, inverse trigonometry, etc. then I realised the importance of memorising them.

Answer 4

Knowledge is power. The more trigonometry you can derive without needing to consult Google etc the better you can do on tests and, more importantly, the faster you can follow later conversations in calculus and differential equations and so forth. Here I assume the student embraces the idea that mathematical knowledge does not exist in isolation. Every new idea should be contrasted and compared and fit into the tapestry of their existing knowledge. In other words, I encourage the radical idea that college students should be scholarly. I know, this is at odds with the ever encroaching idea that we should make college as easy as possible for customers (students). Of course, if we allow students to look up every basic fact of trigonometry then it does make the course friendlier, but, when the course is about trigonometry it is (in my opinion) natural to expect the students to learn all the basic trig. identities.

Let me discuss here a method to derive such identities, brevity being the soul of wit, draw your own conclusions.

I would like to see us teach how $e^{i\theta} = \cos \theta + i \sin \theta$ naturally encodes just about every trig. identity you run into as a mere consequence of algebra. For example, $$ e^{ia}e^{ib} = e^{i(a+b)} $$ is tantamount to the adding angles formulas for sine and cosine. It's not hard to derive $\cos x = \frac{1}{2}(e^{ix}+e^{-ix})$ and $\sin x = \frac{1}{2i}(e^{ix}-e^{-ix})$ and use them to calculate things like: \begin{align} \sin(a)\cos(b) &= \frac{1}{2i}(e^{ia}-e^{-ia})\frac{1}{2}(e^{ib}+e^{-ib}) \\ &= \frac{1}{2i}\frac{1}{2}(e^{i(a+b)}-e^{-i(a+b)}+e^{i(a-b)}-e^{-i(a-b)}) \\ &= \frac{1}{2}\underbrace{\frac{1}{2i}(e^{i(a+b)}-e^{-i(a+b)})}_{\sin(a+b)} + \frac{1}{2}\underbrace{\frac{1}{2i}(e^{i(a-b)}-e^{-i(a-b)})}_{\sin(a-b)} \end{align} Therefore, $\sin(a)\cos(b) = \frac{1}{2}\sin(a+b)+ \frac{1}{2}\sin(a-b)$. Of course, this is just a token example, there is so much more you can do if you learn this way of thinking.

My Modern Physics professor (Stephen Reynolds) mentioned that he would like to show us imaginary exponentials, but, he was either forbidden or discouraged (I forget, it's been a few years) from doing it, so, instead he'd just use real trigonometry. In retrospect, I really wish he had ignored whoever had given that advice. Imaginary exponentials make trigonometry into algebra.

I know that insightful application of the adding angles identities for sine and cosine can also derive very many things, but, it seems to me that technique is far more clever than the one I outline here. To use the adding angles formulas you have to come towards your goal in a sort-of sideways fashion. For example, $$ \cos(x+x) = \cos x \cos x - \sin x \sin x = \cos^2 x-(1- \cos^2 x) $$ gives us a path to derive $\cos^2 x = \frac{1}{2}(1- \cos (2x))$. In contrast, to derive this with imaginary exponentials I just begin with my target $\cos^2 (x)$, do algebra, and find $\cos^2 x = \frac{1}{2}(1- \cos (2x))$.

So, why are we afraid of imaginary exponentials? I think the fear is real, but, is it rational? I wish we would all take Gauss' advice and get over this terminology "imaginary". Accept $\mathbb{C}$ as an integral and important part of basic school mathematics.

Answer 5

If you see it filtering out students differentially by race or gender, you could discuss it in your department and college as an equity issue. I have taught trig dozens of times, and still do not have those memorized. (I can derive them easily.)

Math courses being used to filter students out is quite problematic, and equity concerns are one way we are making math courses fairer.

Answer 6

Students cannot be forced to memorize trig identities unless they are tested orally. Some of the students will and should learn to quickly derive some of them.
The ability to derive identities quickly as needed is a valuable skill. But tell them honestly that they can answer the exam questions correctly by either memorization or derivation of identities.

Students who study calculus will benefit from recognizing parts of identities. This is especially evident in learning integration techniques.

I teach mathematics. I am not in any of the other professions that use trigonometry, so I don't know which particular identities are needed in those fields But I do know that many use trigonometry in their jobs. I assume that they have need for some identities.

Answer 7

I think that it's an odd request from the college, for several reasons:

• I cannot remember any situation where I've needed these formulas, except perhaps for one or two integration exercises. (And then it's again the question: How many people are still solving their integrals manually in practice?)

• If I need them, I can find them in any math formula collection.

• If I need them and if I don't have a math formula collection available, I can easily derive them from the identities $e^{i\theta} = \cos\theta + i \sin\theta$ and $\sin^2 \theta + \cos^2 \theta = 1$. (These are the two formulas that one should memorize!)

The formulas may be more important for some people working in physics or electrical engineering, but these people need complex numbers anyhow.

I really believe that there are more important things that students should learn in mathematics, say, mathematical proof techniques.

Answer 8

Disclaimer:

This answer shares the same ideals as James S. Cook's answer.

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Nowadays, I must be honest that I usually don't memorize any of the trig identities, save for possibly the Pythagorean identity because it appears so often. Instead I focus on remembering:

$$e^{i\theta}=\cos(\theta)+i\sin(\theta)\\\cos(\theta)=\Re(e^{\pm i\theta})=\frac{e^{i\theta}+e^{-i\theta}}2\\\sin(\theta)=\pm\Im(e^{\pm i\theta})=\frac{e^{i\theta}-e^{-i\theta}}{2i}$$

From these three identities alone, one can derive not only all of the standard identities, but many other niche identities that are not obvious at first glance. Common examples include:

• The general formulas for $\cos^n(\theta)$ and $\sin^n(\theta)$ in terms of multiple angles via the binomial theorem.

• The general formulas for $\cos(n\theta)$ and $\sin(n\theta)$ as factoring problems aside from the usual "repeatedly applying sums of angles".

• Sums of trig functions over angles in arithmetic progression via geometric series.

Furthermore, the procedure is straightforward algebra. Consider the example identity, supposing we don't get division by zero:

$$\frac{\cos[(n+1)\theta]-\cos(n\theta)}{\cos(\theta)-1}=\frac{\sin[(n+\frac12)\theta]}{\sin(\frac12\theta)}$$

I'd wager that trying to prove this identity with trig identities is a very troublesome task. However, with the complex exponentials, this problem is as easy as simply multiplying both sides out. You don't have to pick out which identities to use or how to use them because everything flows through the algebra you've built up.

Answer 9

1. I think there may be some differences in the right approach for a community college student versus for a math grad student. We are talking about very different intrinsic skill levels. Maybe the smarter students can derive more and the du...less smart students can derive less.

2. Not sure exactly how memorization is enforced. Verbal recital? If it is on problems, what is to stop from just fast deriving it (for the kids that can).

3. Myself, I found some identities not worth remembering (any of the squared ones, can pretty much be derived from sinsq plus cossq. But I did find a few identities, that it just seemed better to memorize (maybe the sum of sins and cosine angles). From that, you can get to double angles or differences of angles (just a different case of what you sum). And I had seen the deriviation of the initial identity. But the deriviation didn't stick as much, so I just committed the sum angles to memory.

4. I actually don't think it is the end of the world to memorize a page of trig identities. It comes pretty natural if you do it in concert with heavy drill to use the identities. And the student can kind of figure out per (3), which ones to memorize and which to re-derive during exams. I guess, basically I am more to the derive bias than the memorize, but it is not an all or nothing. And the key in either case, is heavy drill.