Memorizing trigonometric identities

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adjurovich
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So I am studying precalculus along with some basic calculus (I am not very patient but I feel relatively confident about my precalculus knowledge). Do you think there’s any real use of memorizing all identities for tangent and cotangent?
 
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  • #2
No. It is worth much much more to learn how to derive them should the need arise.
 
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  • #3
adjurovich said:
So I am studying precalculus along with some basic calculus (I am not very patient but I feel relatively confident about my precalculus knowledge). Do you think there’s any real use of memorizing all identities for tangent and cotangent?
Adding to what @Orodruin has said I'd like to mention that things have changed. You had to rush to the library when you forgot something in the past, but nowadays a reach into your pocket is sufficient if you want to look up a formula! However, you do not want to look up even the simplest formulas that you frequently need, so things like Thales, ##\sin^2+\cos^2=1,## or the definition of the three main trig functions on the right triangle are worth learning by heart, but not e.g. Weierstraß's half-angle substitution.
 
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  • #4
fresh_42 said:
Adding to what @Orodruin has said I'd like to mention that things have changed. You had to rush to the library when you forgot something in the past, but nowadays a reach into your pocket is sufficient if you want to look up a formula! However, you do not want to look up even the simplest formulas that you frequently need, so things like Thales, ##\sin^2+\cos^2=1,## or the definition of the three main trig functions on the right triangle are worth learning by heart, but not e.g. Weierstraß's half-angle substitution.
You’re right. Thanks for answering!
 
  • #5
Generally, you’ll get pretty far just starting from ##e^{ix} = \cos x + i \sin x##.
 
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  • #6
Memorize, no. Remember, yes.
 
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  • #7
adjurovich said:
So I am studying precalculus along with some basic calculus (I am not very patient but I feel relatively confident about my precalculus knowledge). Do you think there’s any real use of memorizing all identities for tangent and cotangent?
I'm of the the more you memorize the better school. If you aren't at least familiar with some identity then how will you recognize it as useful when the opportunity comes up? How will you be able to do things other math people can't do? Being really good at math is hard work.

But then again, maybe you have better things to do than trying to be the next Julian Schwinger or Alexander Grothendieck. If you aren't that ambitious and aren't one of those people who sucks up math as though it were nothing then just do what almost everyone else does and concentrate on learning what you believe will be on the test. If you are getting your start later in life -- like 18 years old -- then your chances of stardom are already very remote so what the heck. Me, I only know the most basic trig identities, but then I quit out of a PhD program so maybe that wasn't ideal.
 
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  • #8
Vanadium 50 said:
Memorize, no. Remember, yes.
When I say memorize I don’t really mean writing down the identities 50 times and repeating them in my head like a song, but learning some patterns how to “remember” them? I think only an idiot would learn maths (or any other subject except maybe law) by literally memorizing.
 
  • #9
adjurovich said:
When I say memorize I don’t really mean writing down the identities 50 times and repeating them in my head like a song, but learning some patterns how to “remember” them? I think only an idiot would learn maths (or any other subject except maybe law) by literally memorizing.
The hidden truth behind that advice written as a pun is the following: In order to look up a formula you have to know that there is a formula. The Weierstraß substitution (other name: tangent half-angle substitution) is
$$
\tan \left(\dfrac{x}{2}\right)=t.
$$
This results in formulas for ##\sin x\, , \,\cos x## and ##\tan x## as polynomials of ##t.## I do not memorize those polynomials, but I remember that they exist and that it is one possible way to solve integrals with trig functions because the substitution turns them into polynomials that are usually easier to solve. In case I attempt an integration by that method, I look up the polynomials - simply because it is faster than deriving them again from the defining equation.
 
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  • #10
As a teacher of this stuff, I found it quite useful to memorize these three:
cos^2 + sin^2 = 1,
sin(2t) = 2sin(t)cos(t),
cos(2t) = cos^2(t)-sin^2(t).
These may be wrong after several decades, but this is what pops up in memory.

Then I get others by dividing the first one by cos^2 or sin^2, adding the first and third, and generalizing the last two to get sin(s+t) and cos(s+t). That's all I ever needed, and in a pinch, I know post#5. I have almost never in my entire life used the tangent half angle substitution, except as an experiment. Of course you can't use something you don't know.
 
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  • #11
fresh_42 said:
things like Thales, ##\sin^2+\cos^2=1,## or the definition of the three main trig functions on the right triangle are worth learning by heart,
I didn't even know Thales' Theorem. Surely you meant to include the inverses of the three trig functions. The hyperbolic trig functions are useful in special relativity, which is hyperbolic. I think this characteristic is what makes it seem so weird.
 
  • #12
well if you know Post #5, you can solve for cos(x) = (1/2)(e^ix + e^-ix), and sin(x) = (1/2i)(e^ix - e^-ix), [I think], and then you can remember the hyperbolic sine and cosine by omitting the i's, if memory serves.

and Thales follows from Pythagoras, which most people know, although I once knew a wealthy builder who had no idea why a carpenter uses a (3,4,5) triangle to measure a right angle.
 

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