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Is there a more efficient method of trig mastery than rote memorization?

i find myself loosing it in 1st semester calculus, mainly because people are using trigonometric identities i never heard of before.

Are those usually explained in any way? They're listed in the front of the book, and that's it. We also never did those in Highschool (maybe the people here did, I'm not from here).

I could go ahead and memorize all of those, but I think that's stupid. So where did you learn those or is it just something I'll have to figure out myself?

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marked as duplicate by Giuseppe Negro, Hagen von Eitzen, Ittay Weiss, Davide Giraudo, Cameron Buie Jan 6 '13 at 15:45

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

I am having a hard time figuring out what the question is. Do you want to learn trigonometric identities? Which ones? Is this a poll of how other people learned them? – Jonas Meyer Jan 6 '13 at 9:53
Most of the trigonometric identities are related to one another and you can often learn a couple of key ones from which the others are derived. – Michael Albanese Jan 6 '13 at 9:54
@JonasMeyer both, kinda. I'm just wondering what I missed. Why everyone uses them and I never actually... heard of it before. – foaly Jan 6 '13 at 10:36
I went through exactly that. I never took trig at all and then started drowning very quickly when I took calc. After a bunch of us asked the teacher, he devoted a lecture and showed us how to start with the complex definition $\sin(x)=\frac{e^{ix}-e^{-ix}}{2i}$ and $\cos(x)=\frac{e^{ix}+e^{-ix}}{2}$ and derive the entire trig textbook from this and a decade later I was doing the same with my students. – Fixed Point Jan 6 '13 at 11:12
Practice deriving a few identities by hand. You'll see you don't need much practice. After two/three times you will just remember the identities. Start with the most well known ones like $\sin^2(x)+\cos^2(x)=1$ and the sine and cosine double angle and addition formulas. Understand those simple derivations (it is just absolutely simple algebra with complex numbers) and I guarantee you when others mention identities you haven't seen before, you will "see" them too. – Fixed Point Jan 6 '13 at 11:16
up vote 6 down vote accepted

I started out with just these three trigonometric identities:
1] Expansion of $\sin(A + B)$
2] Expansion of $\cos(A + B)$
3] $\sin^2(A) + \cos^2(A) = 1$

Almost every identity I know today can be pretty much easily derived just by using a combination of the three I listed above [1].

Having said that, I used to solve a whole lot of problems with trigonometry and some identities just stuck in my mind (without my conscious need of memorizing). This has helped a lot since it allowed me to identify situations where I could use trigonometry especially in calculus.

So, I would suggest that you need not worry about not knowing a lot of trigonometric identities. On the other hand, do practice a whole bunch of problems wherever you can find them. There is no substitute for practice in mathematics.

[1] Giussepe Negro has given an interesting piece of info regarding these three identities below in the comments section. Do check it out!

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+1. Indeed you can prove that the identities 1], 2], 3] listed above, together with the initial condition $\sin 0=0,\ \cos 0=1$, completely characterize the sine and cosine functions. So any trigonometric identity must be a consequence of those things. – Giuseppe Negro Jan 6 '13 at 11:14
@GiuseppeNegro Wow, I did not know that! That is a very interesting piece of info (+1). Thank you! – TenaliRaman Jan 6 '13 at 12:37
I agree that this is very interesting. The precise statement is that the function $$\theta\in \mathbb{R} \mapsto e^{i \theta}\in \mathbb{S}^1\subset \mathbb{C}$$ is the unique continuous group homomorphism of the group $(\mathbb{R}, +)$ into the group $(\mathbb{S}^1, \cdot)$ such that $e^{i 2\pi}=1$. All other such homomorphisms have the form $\theta \mapsto e^{i\lambda\theta}$ for some $\lambda\in \mathbb{R}$. Since $e^{i\theta}=\cos(\theta)+i\sin(\theta)$, this shows that sine and cosine are essentially determined by their homomorphism properties, that is, by properties 1, 2, 3 above. – Giuseppe Negro Jan 6 '13 at 12:45
@GiuseppeNegro Wow, that is neat! – TenaliRaman Jan 6 '13 at 13:01

The Wikipedia page is a good place to start, it gives proofs for most of the basic ones:

Proofs of trigonometric identities

Once you finish that, the page with the list of trigonometric identities has for most of them a proof sketch:

List of trigonometric identities

In general they are worth memorizing - you don't want to have to prove them on the exam.

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In my opinion, the best option is to derive them.And after using these identities many times, you'll automatically remember, so there is no point in memorizing. Now, deriving trig identities is not a difficult thing. They can be easily done using the euler formula. See this link for example.

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