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So, I'm studying some math because my math sucks.
I'm going through this calculus textbook (Thomas' Calculus, Early Transcendentals), and I realize I need to study these stupid sin and cos functions, because this textbook assumes I know them by heart already.
So, I need to get me a book on trigonometry. I'm looking for stuff that deals with all those identities and relationships and their proofs (28 of them? something like that). Recommendations please!
Edit: I'm looking for stuff that deals with stuff like
cos(a+b) = cos(a)*cos(b) - sin(a)*sin(b)
sin(a+b) = sin(a)*cos(b) + cos(a)*sin(b)
 
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www.mathworld.com
Also, you can just google "Trigonometry identities" and you should at least get the formulas by themselves.
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The reason I want a textbook is for the longer, drawn-out proofs, preceded by all of the theorems they use. And the exercises. And something I can take to the library and read off a desk.
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I mean, this guy showed me this textbook in Korean that does exactly what I need, but my Korean sucks and I especially don't understand all of the Korean math jargon (new words for everything from angle and curve to plane and coordinate!).
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SOH-sin over hypotenuse
CAH-cos adjacent hypotenuse
TOA-tan over adjacent
easy!
ummm library ftw
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On January 29 2008 11:14 Hypnosis wrote:
SOH-sin over hypotenuse
CAH-cos adjacent hypotenuse
TOA-tan over adjacent
easy!
ummm library ftw
i think he means shit like sin'(x) = cos(x), etc xDDDD
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I'm actually thinking of stuff starting with
cos(a+b) = cos(a)*cos(b) - sin(a)*sin(b)
and
sin(a+b) = sin(a)*cos(b) + cos(a)*sin(b)
and
sin^2 + cos^2 = 1
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Belgium9950 Posts
from a higher math approach (with like mcclaurinseries or whatever the english name is) or the more standard approach?
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My math sucks, so the idea is for me to get to understand these functions and their relationships. I'm not sure what a "higher math approach" would constitute, but it sounds scary.
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Proofs are not really necessary for something not of a higher math approach - you just take them for granted. But as far as I've seen, mathworld.com has some pretty lengthy and extensive proofs, maybe you should give it a try.
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They're all in my textbook, should be the same with yours.
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Melbourne5338 Posts
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On January 29 2008 11:18 paper wrote:Show nested quote +On January 29 2008 11:14 Hypnosis wrote:
SOH-sin over hypotenuse
CAH-cos adjacent hypotenuse
TOA-tan over adjacent
easy!
ummm library ftw
i think he means shit like sin'(x) = cos(x), etc xDDDD
that's calculus not trig...
don't get a trig textbook
they're worthless
and they're like 200 pages long, as opposed to 1000-page calculus textbooks
i'll just type out the identities/proofs... there aren't very many that you need to know that are hard to remember
and, that being said, you don't need to *remember* most of them anyways past pre-calculus
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Korea (South)11598 Posts
i just passed trig by 0.2% WEEEEEEEE!
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okay for sin^2+cos^2=1
the proof is:
imagine a triangle on the circle defined by x^2+y^2=r^2 with base=x and height=y
if you divide out r^2 you get (x/r)^2+(y/r)^2=1
then, if you draw an angle theta, you see that sin(theta)=y/r and cos(theta)=x/r
so sin^2(theta)+cos^2(theta)=1
i don't see why you have to know the proof though
it' should be pretty obvious when you can use it
you use it a lot to simplify (especially in calculus), so it's really useful to know
like if you have sqrt(4sin^2(x)+4cos^2(x)) or something, the answer is just 2
edit: for the proofs of the identities 1+tan^2=sec^2 and 1+cot^2=csc^2, divide x^2+y^2=r^2 by x^2 or y^2
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okay for sin(a+b) = sin(a)cos(b) + cos(a)sin(b) and cos(a+b) = cos(a)cos(b) - sin(a)sin(b)
i don't remember how to do the proof, but i remember it being seriously gay, and you honestly won't need to know how to do it, nor will it help your understanding of trig at all (i think)
i think you just have to memorize these
okay the tan(a+b) = (tan(a)+tan(b) / (1 - tan(a)tan(b)) isn't that bad to prove
just remember tan(a+b) = sin(a+b) / cos(a+b)
then use the previous identities and simplify by dividing out cos(a)cos(b) (i think...)
edit: for when you use them...
if you have something like sin(15)cos(30)+cos(15)sin(30), you can easily use the identity to get sin(15+30)=sqrt(2)/2 (yes it's degrees not radians)
or solve sin(x+30)=sqrt(3)/2 or something like that...
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dont forget double angles and half angles
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double angles:
sin(2a) = 2sin(a)cos(a)
cos(2a) = cos^2(a) - sin^(a) = 1 - 2sin^2(a) = 2cos^(a) - 1
tan(2a) = 2tan(a) / (1 - tan^2(a))
proofs for the double angle formulas: just set b = a
so, for example, instead of sin(a+b), you have sin(a+a) = sin(2a)
and plug it into the identity
sin(a+b) = sin(a)cos(a)+cos(a)sin(a) = 2sin(a)cos(a)
you should be able to prove the rest 
you need to know double angles for calculus (integrating some trig functions)
subtraction "formulas" are basically the same... use -b instead of b
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For the sin(a+b) and cos(a+b) you can prove using a well constructed triangle but it is a total bitch as our teacher showed us 
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A friend took the time to prove the first two to me (using rotation of a point around the origin), and told me that there are at least a dozen more useful ones.
I'm looking for something that not only lists these identities, but also proves them and provides exercises for me to do.
I'm thinking the Mathworld articles are pretty good, I'll definitely be looking at those a lot. Thanks to the people who pointed them out.
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okay i guess that means i can stop...
honestly, trig is mostly memorization as opposed to comprehension (which calculus is 100x cooler)
in that class, just remember doing unit circle, identities, vectors, graphs, law of sines/cosines
you can honestly learn all the trig you'll need to know in less than a month
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If your finding a book, then it depends on what level of math are you planning to learn. Trigonometry is a very large topic.. when you mentioned you didnt know the basics of it, then i would assume your learning it in school instead of university. Thomas' Calculus, Early Transcendentals is meant for university students who have basics in maths, if you find it too hard then i would recomend the school's textbooks.
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If you want an excellent textbook with many difficult problems and proofs I would highly suggest "Trigonometry" by Gelfand and Saul. Gelfand is one of the 20th century's great mathematicians. It is a dover book, so very cheap. I found it user friendly and was able to read the whole thing without any teacher input. It does not assume that you already know any trigonometry or more advanced mathematics. After reading it once I found myself able to solve AIME and Olympiad trigonometry problems. This is one of my top three favorite math books in my collection of more 50.
I CANNOT RECOMMEND THIS BOOK ENOUGH AND IT SOUNDS EXACTLY LIKE WHAT YOU ARE LOOKING FOR!
http://www.amazon.com/Trigonometry-I-M-Gelfand/dp/0817639144/ref=pd_bbs_3?ie=UTF8&s=books&qid=1201610925&sr=8-3
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The book I mentioned above really changed my life...
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After a recommendation like that (and lack of other recommendations), I think I'll go for it. Thanks a bunch.
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EPT
