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I want to be a mathematician or computer scientist. I'm going to be a junior in high school, and I skipped precalc/trig to go straight to AP Calc since I've studied a lot of analysis and stuff on my own. My dad wants me to memorize about 30 trig identities (though some of them are very similar) since I'm missing trig. I've gone through and proved all of them, but memorizing them seems like a waste of effort. My dad is a physicist, so he is good at math, but I think he may be wrong here. Can't one just use deMoivre's theorem to get around memorizing the identities?

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    $\begingroup$ Actually, knowing the routes to proving them is more useful. Beyond the definitions, there are only a handful of the basic identities (Pythagorean Identity, "double-angle formulas" for sine and cosine, "angle-addition formula" for sine, maybe a couple more) that it is important to have in your memory, since it isn't too much work to "reconstruct" the others from those. (But any serious practitioner should have derived all of those identities at least once in their life. I feel that it's even good to go back every several years and see if you still know how to "get them".) $\endgroup$ – colormegone Jun 27 '15 at 18:06
  • $\begingroup$ Do you intend to constantly reinvent the wheel? Also, it may be good for Alzheimer's prevention… $\endgroup$ – Bernard Jun 27 '15 at 18:32
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    $\begingroup$ What are the 30 trig identities? Memorizing 30 sounds excessive to me, but as you said it's good to be able to derive them all (which you are able to do already). $\endgroup$ – littleO Jun 27 '15 at 19:33
  • $\begingroup$ In returning to such things as these identities (or other basic and useful ideas) later, one isn't necessarily stepping into the same river again. From what else one has learned since the last time, one can figure out the derivations far faster than before. Also, in the larger context one has gained over time, one may sometimes see new connections to what one has learned that one hadn't thought about before. In any case, I'm not talking about all that much time consumed here. (Naturally, this sort of activity isn't necessary if one doesn't have much use for those particular "wheels".) $\endgroup$ – colormegone Jun 27 '15 at 22:24
  • $\begingroup$ @littleO If you look through the "review" lists in the trig chapters of any number of pre-calculus texts, you can easily get thirty or so of such "basic" relations. I don't think many people try to remember most of them. (When you have to teach them to others, of course, just about all of them unavoidably stick in your head...) $\endgroup$ – colormegone Jun 27 '15 at 22:31
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Let me first note that physicists may be better people to ask this question than mathematicians are.

I think it's worth remembering a few, and knowing how to rederive the others.

The important ones to remember initially are:

(a) The definitions of $\tan$, $\cot$, $\sec$, $\csc$, in terms of $\sin$ and $\cos$, as well as the identity $\cot x = 1/(\tan x)$.

(b) The Pythagorean identity $\sin^2 x + \cos^2 x = 1$, and the corresponding ones relating $\tan$ and $\sec$, then $\cot$ and $\csc$.

(c) The reduction formulas involving trigonometric functions of $-x$, $\pi/2 - x$, $\pi + x$, $2\pi + x$ (and especially the periods of $\sin, \cos, \tan$). These, as well as the geometric reasons for them, should be learned independently of (d), focusing mainly on $\sin$, $\cos$ and $\tan$.

(d) the angle-sum formulas for $\sin$ and $\cos$.

Eventually, from repeated use, you will probably learn the angle-difference formulas for $\sin$ and $\cos$, the double-angle formulas for $\sin$ and $\cos$, the formulas for $\sin^2 x$ and $\cos^2 x$, perhaps the angle-sum and angle-difference formulas for $\tan$. I have never learned the product-to-sum and sum-to-product formulas and re-derive them whenever I need them, though some people might disagree with this approach.

Overall then, you should learn the most important ones first, and then through practice your list of memorized identities will start to look more and more like the one your dad wants you to learn.

It is better to use complex numbers for anything involving $3x$, $4x$, etc., and cubes or higher powers of $\cos$ and $\sin$.

Sometimes in calculus, depending on the level you learn it at, it's important to know $\sin x$ and $\cos x$ in terms of $\tan x/2$. There is a nice geometric way to do this by considering $\tan x/2$ as the slope of the line $l$ from $(-1,0)$ to $A = (\cos x,\sin x)$. $A$ can be found as the point of intersection of $l$ and the perpendicular to $l$ passing through $(1,0)$.

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  • $\begingroup$ I would argue that your suggestion that "physicists may be better people to ask this question than mathematicians are" is only true if you plan to be a physicist, and maybe not even then. $\endgroup$ – joeA Jun 30 '15 at 19:09
  • $\begingroup$ I suppose I meant they would have a better answer to the question "What identities are most often used in practical computations in calculus?" There are a lot of people studying AP calculus who don't know yet whether they will be focusing more on math or physics in university, but for a lot of people the answer will be both. And in physics, they'll need to be more at home with calculating integrals explicitly, etc. $\endgroup$ – Keith Jun 30 '15 at 19:19
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Usually, yes, though I prefer Euler's identity. Pretty much every trig identity can be derived from $$e^{ix}=\cos(x)+i\sin(x).$$ However, it is useful to memorize some of the common ones because they will help you a lot in calculus and beyond to quickly identify when an expression can be simplified. I would start with memorizing the angle addition formulas. From there you can quickly derive the double and half angle formulas as well as some others.

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    $\begingroup$ Yes, I'd always forget the darn things, and then I realized that $(e^{x})^2=(e^{2x})$ and then the "idea" of using Euler's theorem just clicked. $\endgroup$ – Zach466920 Jun 27 '15 at 21:20
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To be honest, the only trig identities you really need are the definitions of the 6 trig functions, and Euler/De Moivre. You can prove almost all the trig identities from $e^{ix}$. Deriving an identity is much easier than just rote memorization, and with repeated deriviation, it eventually becomes memorized.

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In favor of rote memorization, unless all your exams are open-notes you will find that it is very awkward when you have $n$ minutes left to complete $m$ problems and you are busy re-discovering each trig identity that you need to use as you go along. Working out the identity to use in one part of the problem can easily take longer than the time allotted to solve the entire problem.

For other purposes, such as solving real-life math problems (where usually you are in a work environment that provides access to reference books, or even better, to the Internet), completely memorizing all of those formulas by rote is probably unnecessary. What is useful is to be aware that there are identities to deal with certain trigonometric expressions that you may encounter, as a prompt for you either to derive them or to look them up.

If you encounter certain identities very frequently, the time you might spend deriving them or even looking them up on every occasion is wasteful, but in those cases it it likely you will find that you have memorized the formula after using it some number of times without consciously trying to memorize it.

Putting the effort in to know these identities (being able to write them out from memory, but also knowing how they can be derived) is a good investment at this time. But there are options in exactly how you remember the identities; if you cannot simply write out the formula by rote, but you can doodle a small diagram in a couple of seconds (perhaps illustrating part of the derivation of the formula), then write out that formula with that aid, I think that is good enough.

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