Beginning self-study - one question

[Laura Palmer]

Hi,

I will begin studying calculus this or next week. I'm registered for the online version of U of Wisc's Calculus and Analytic Geometry. It's the standard, non-honors class, so we're using Thomas and Finney. Because I foresee wanting a deeper understanding of the subject, I will also be working through Spivak's Calculus concurrently.

I spent the last few weeks reviewing precalc topics. I'm all set, I just have one specific question: which trig identities should I be able to pull out automatically? I'm highly motivated in studying - and I hate the feeling of coming across something I don't know due to prior laziness - but I also abhor bad/wasted studying time. I'd rather not spend a few days cramming dozens of trig identities into my head if there's really no point. I'd rather spend those days reading the beginning of Spivak, laying some conceptual foundations.

Thanks in advance for your opinions!

 

[transphenomen]

sin^2 + cos^2 = 1

is the most used trig identity by at least an order of magnitude, everything else you can look up later for specific questions.

 

[Laura Palmer]

Thanks. Of course that's the ideal answer! Now I can focus on the interesting stuff...

 

[Buri]

I think the best way to go about this is to learn how to derive them. Rather than actually go about memorizing them. As you go a long you'll find out which ones are necessary and which ones aren't (hopefully while doing problems rather than on a test). And Spivak's Calculus is an excellent choice! I used that for an honours analysis course and its definitely one of the best introductory analysis books out there! Good luck with your self-study! :)

 

[micromass]

There's a difference between trig identities that you should know, and identities that you know exist and be able to use.

The identities that you should know by heart are:
$$\cos^2(x)+\sin^2(x)=1$$
$$\sin(2x)=2\sin(x)\cos(x)$$
$$\cos(2x)=\cos^2(x)-\sin^2(x)$$
$$\tan(x)=\sin(x)/\cos(x)$$
$$1+\tan^2(x)=1/\cos^2(x)$$

Trig identities that you should not know by heart, but just know that they exist:
-formula's that calculate sine, cosine and tangent of a sum: not so important, but sometimes useful.
- formula's that change the product of sines/cosines to sum of sines/cosines (Simpson's formula's): these are very important for some integrals!
- formula's that calculate sin(2x), cos(2x) and tan(2x) in terms of the tangent function. These are very important for other integrals!

There's no use in learning the above three classes, you'll forget them anyway and end up looking them up. But you should know that they exist and be able recognize when to use them.

 

[spamiam]

I like Buri's idea--being able to derive them is better in my opinion. For instance, I can't remember the cos(2x) identity for the life of me, but I can remember the addition formulae by using the complex exponential, [itex] e^{ix} = \cos(x) + i\sin(x) [/tex]:

$$(\cos\alpha + i \sin\alpha)(\cos\beta + i \sin\beta)=e^{i\alpha}e^{i\beta} = e^{i(\alpha + \beta)} = \cos(\alpha + \beta) + i \sin(\alpha + \beta)$$

and then multiplying out the LHS. I can then figure out the double angle formula by setting $\alpha = \beta$.

So, in my opinion, $\sin^2x + \cos^2x = 1$ and the addition formulae are the only identities that are essential to memorize.

 

[micromass]

I like Buri's idea--being able to derive them is better in my opinion. For instance, I can't remember the cos(2x) identity for the life of me, but I can remember the addition formulae by using the complex exponential, [itex] e^{ix} = \cos(x) + i\sin(x) [/tex]:

$$(\cos\alpha + i \sin\alpha)(\cos\beta + i \sin\beta)=e^{i\alpha}e^{i\beta} = e^{i(\alpha + \beta)} = \cos(\alpha + \beta) + i \sin(\alpha + \beta)$$

and then multiplying out the LHS. I can then figure out the double angle formula by setting $\alpha = \beta$.

So, in my opinion, $\sin^2x + \cos^2x = 1$ and the addition formulae are the only identities that are essential to memorize.

One danger with deriving formula's instead of memorizing them is pattern recognition. If you for example end up with the formula

$$\frac{\cos^2(x)-\sin^2(x)}{\sin(2x)}$$

then someone who memorized his formula's will immediately recognize this to be $$cotan(2x)$$ (and simplify the subsequent calculations a lot!). Somebody who can only derive his formula's, will not see the shortcut and make very long calculations instead! This is why I think it is better to have some formula's memorized. Of course, understanding is much better than memorizing, but in mathematics you cannot derive everything (otherwise it would take to long). You really can't do without memorizing some formula's...

 

[Buri]

One danger with deriving formula's instead of memorizing them is pattern recognition. If you for example end up with the formula

$$\frac{\cos^2(x)-\sin^2(x)}{\sin(2x)}$$

then someone who memorized his formula's will immediately recognize this to be $$cotan(2x)$$ (and simplify the subsequent calculations a lot!). Somebody who can only derive his formula's, will not see the shortcut and make very long calculations instead! This is why I think it is better to have some formula's memorized. Of course, understanding is much better than memorizing, but in mathematics you cannot derive everything (otherwise it would take to long). You really can't do without memorizing some formula's...

When I say derive formulas I implicitly assume that the individual is familiar with certain trig identities. Obviously, if you don't even recognize something you're doomed (what are you going to derive?), but memorization isn't necessary for pattern recognition. And besides, memorization in the sense that I'll sit down with a list of trig identities and attempt to commit them to memory will also fail. Using them a lot will lead someone to remember them, but if after not using them for a while, its inevitable that you'll forget them also. I'm a pure mathematics student and so memorization has never been my cup of tea (Pugh style! lol).

 

[Laura Palmer]

Thanks very much for your feedback.

Ok, so a kind of consensus is emerging - or at least a synthesis, in my mind: 1) Definitely don't waste time trying to memorize them all as if I have an exam. I won't retain them anyway. 2) It's important to be familiar with the sum/difference, double/half-angle, and product identities. 3) At that point, either a) leave it at that and derive the identities (based on familiarity) when necessary or b) commit a handful to memory, most notably the sum and double angle identities for example.

Option b) strikes me as a good balance. I'll work with the lot of them for the next day or two and memorize a half-dozen.

Again, thanks for the help!