/sci/ - Science & Math

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>>11599641
lol this is obviously not homework

>>11599629
you can use the laws of exponent and the Cotes-Euler formula to do it generally for complex numbers, then just take the real part of the expansion and it gives you sin(a+b) and cos(a+b)

>>11599656
Thats not elementary and a stupid way to do it. My method is much simpler and gets to the meat of the issue.

>>11599629
You mean with dot products?

I dont get it. How does no one know of a simple proof of such a common and useful formula?
> exponential function and complex numbers
> drawing complicated unintuitive diagrams and chasing lengths
Seriously, there has to be one (1) person who knows how to prove it in a simple way, right?

>>11599698
Explain.

>>11599673
>Thats not elementary
Its quite elementary and also extremely fast

>>11599703
Somebody already gave you a very simple way to do it

>>11599732
> extremely fast
Well provide a quick proof using your method then that a fifth grader can understand. Be rigorous.

>>11599735
Which one?

>>11599736
>that a fifth grader can understand
That was not part of your OP. Where are you moving the goalpost next?

>>11599746
>elementary
Can you read? Complex exponentials are not elementary compared in the context of basic trig were talking about.

>>11599746
>>11599741
[eqn] e^{ja}e^{jb}=\exp[j(a+b)]\implies\cos a\cos b+j\cos a\sin b+j\cos b\sin a-\sin a\sin b=\cos(a+b)+j\sin(a+b) [/eqn]
set real part=real part and imaginary part=imag. part. Done. Euler's formula is like a two line proof as well.

A child is learning trig. He comes up to you and asks you:why are the trig addition formulas true?
You then start rambling about complex numbers, how actually sine and cosine can be represented by infinite series (good luck proving that they coincide with the childs definition of sine and cosine) , prove that exp multiplies when you add the arguments ( good luck making the child believe you ). By the time you finish explaining to the child why complex arithmetic actually makes sense and is not some black magic fuckery and when you start to put ix in the argument of exp, the child will have completely lost you and fallen asleep.
Again, in the context of trig that this question is based on, your methods is an extreme clusterfuck and not a good way to prove it at all.

>>11599773
Not a simple or an elementary way to show it, in the context of trig. Presupposes too much. It's like using a cannon to shoot a fly. See
>>11599790

>>11599790
>He comes up to you and asks you:why are the trig addition formulas true?
I tell him: "Just wait a couple years and it will make much, much more sense compared to whatever ugly geometric garbage I can use to justify it"

>>11599804
If you cannot come up with a clear way to prove it to a child like I did, you probably have low IQ.
> ugly geometric garbage
Lmao you seriously have no idea that theres a simple way to prove it, do you?

>>11599758
>Can you read?
Yes.
>Complex exponentials are not elementary
Yes, they are. So that's where you moved the goalpost next. Okay.

>>11599790
>A child is learning trig
Is that you? Because that's still not the situation you described. Now you're mad you fucked up in your problem setup and people found easy solutions. lol

>>11599811
>Lmao you seriously have no idea that theres a simple way to prove it, do you?
>simple
no

>>11599790
>He comes up to you and asks you:why are the trig addition formulas true?
You're not going to convince him _why_ they're true with whatever autistic diagram-bashing argument you came up with either. At the very best you'll convince him _that_ they're true.
>It's like using a cannon to shoot a fly.
No, it's like using a flyswatter to swat a fly. While you sit there and screech that it's better to throw a rock at it.

>>11599813
You understand that the word elementary depends on the context were talking about, dont you? In the context of teaching basic trig, complex exponentials are not elementary. God I cannot believe youre so stupid as to not realize this.

I wonder if those posting the complex exponential proof even understand how it works themselves. I challenge you to translate the proof into a language that a fifth grader who is learning trig can understand without looking up the details.

>>11599815
Thats why I posed the challenge. Try and figure it out.
>>11599822
>You're not going to convince him _why_ they're true with whatever autistic diagram-bashing argument you came up with either
What I have is something much better than a diagram bashing argument. There are no diagrams at all in my argument and its completely elementary.

>>11599838
Pretty sure I do. Why are you so asshurt? "Maclaurin expand the sine and cosine function and you will find that the sum of cosine with j time sine is equivalent to the exponential function evaluated at an imaginary argument. also, the exponent of sum is the product of the exponents. QED."
>a hurr durr but what does that mean bloo bloo thas not simple
See >>11599804

>>11599851
> Maclaurin expand
Explain this step. How do you prove that it converges to sin everywhere and not some other function?

>>11599856
no

>>11599861
Cant do it?

Still waiting for someone to present a simple, elementary proof, in the context of basic trig.

>>11599828
That's really just your opinion. I can't think of something more elementary than complex exponentials, except for integers.
Not sure why you're so mad though. You could have just said "oh yeah, I mean even without complex exponential". Your replies show what kind of person you are. Nobody likes you.

>>11599874
Provide a full elementary proof using complex exponentials then that a person whos just learning trig can understand. No one has been able to do it so far. Perhaps you can?

>>11599874
Also its a fact that people who are just learning trig dont know what complex exponentials are.

>>11599873
Still waiting for OP to prove he isn't full of shit.

>>11599889
lmao, yeah, keep defending. You still didn't pose the problem like that.
Also, the easy proof was given to you here by this dirty unwashed electrical engineer: >>11599773

This thread was intended to be a fun challenge thread. I cant believe people are getting so assblassted over their inability to find an elementary proof lmao.
The whole point of the thread as I have already stated is that I already know the proofs by complex exponentiation or diagram chasing, and they all either assume way too much than they need or are plain ugly and unintuitive. Hence the challenge to find a simple, intuitive proof (there is one).
Saying "kid, youre just gonna have to wait a couple of years when you learn analysis till you can understand why this simple formula about triangles is true" is just cope by someone who doesnt understand why the formula holds themselves.

Pretty sure op wants to stack triangles with base length one and use area.

>>11599904
>people are getting so assblassted
Only one assblasted here is obviously you lol

>>11599898
I will give my own proof if nobody is able to come up with one. Dont want to spoil the fun too early though.
>>11599899
I asked for an elementary proof. In the context of trig, using complex exponentials is not elementary.
> Also, the easy proof was given to you here by this dirty unwashed electrical engineer
As I said, I challenge you to translate it into an actual rigorous proof in terms understandable to a person learning trig. I bet you dont even know how the proof works yourself without looking it up.

>>11599908
That is not my approach. Stacking triangles in my opinion is not intuitive and an ugly, hard to remember proof.
My proof is simple, elementary and easy to remember because it is based one one simple idea: applying the formula pops out itself.

>>11599851
>Maclaurin expand the sine and cosine function
Even this is more complicated than it needs to be. You don't need to know any calculus, power series, or even what e is.
All you need to know is that complex numbers can be written in polar coordinates, which is geometrically completely obvious. The cosine/sine addition formulas are just the fact that multiplying in rectangular and polar coordinates have to give you the same answer.

>>11599930
Kek no. If you get rid of calculus and what e is, the "proor" becomes circular.

>>11599933
oh no my proor

>>11599930
>The cosine/sine addition formulas are just the fact that multiplying in rectangular and polar coordinates have to give you the same answer.
Prove it.

>>11599939
Typo. Meant to say "proof". It's in quotes because it's not actually a proof.

>>11599904
My father definitely knows how to reinvent all the trigonometric formulas. I think it's all about those triangles in the unit disk. I think now you can do it. Good luck.

f(x) = sin(a+b+x)cos(x) + cos(a+b-x)sin(x)

f'(x) = 0, so this is a constant function. Hence, f(0) = f(b)
QED

>>11599930
>the cosine and sine sum formulas are equivalent to the fact that multiplication of complex numbers in polar coordinates sums angles
NIGGA
THAT'S LITERALLY INDISTINGUISHABLE FROM THE EULER FORMULA THINGY
WHAT THE FUCK

>>11599951
Correction:
f(x) = sin(a+b-x)cos(x) + cos(a+b-x)sin(x)
f'(x) = 0, so this is a constant function. Hence, f(0) = f(b)
QED
Fite me OP

3blue literally just live streamed the proof

>>11599951
Good luck explaining to the child what derivatives are, proving the mean value theorem, product rule and proving that the derivative of sin is cos and the derivative of cos is -sin.

>>11599958
Really? What was the proof? Was it elementary like mine?

>>11599963
Who said anything about children?

Take the dot product of the two unit vectors (sin(a), cos(a)) and (sin(-b), cos(-b)). The result is the cosine of the angle between them. Thus cos(a+b) = cos(a)cos(b)-sin(a)sin(b).

>>11599968
>result is the cosine of the angle between them
Prove it.

>>11599968
That's the same as the Euler formula one.

>>11599974
Lol are you retarded?

>That's the same as the Euler formula one.
"Yes."

>>11599975
No, actual mathematician here. Shame you don't know about correspondence.

>>11599952
It's not indistinguishable, the whole point is that it's strictly weaker. Yeah, you normally write the polar coordinates (r,t) as re^(it), but the introduction of the exponential function with a complex argument brings in a bunch of unnecessary calculus that you don't need because the entire content of the theorem is the purely geometric idea that multiplying by a complex number is a scaling and a rotation.
All that matters in the proof is that you have polar coordinates for C which multiply like (r1,t1)(r2,t2) = (r1r2,t1+t2). You don't need any reference to power series or calculus to show this.

>>11599629
I just proved it ancient greek style, only geometry and used the definitions of sin, cos, similar triangles and some angle equalities. My shape was essentially two right triangles a circle and some lines. I don't think there is a reason to post it, but is it trivial enough?

>>11599968
https://www.youtube.com/watch?v=V3-xCPDzQ1Q

>>11599968
OP here. Good job, this is much closer to what I wanted, provided you finish by justifying why the dot product of two vectors is the cos of the angle between them. Its elementary, so thats good, but its not as good as the proof that I have in mind
My main issue with your proof is that it presupposes that the reader knows about the relation between the dot product and the cosine. Essentially, the dot product here is like a magic hat that the formula falls out of.
On the other hand, my proof is completely natural.

>>11599975
>>11599977
Translate the point in Euclidean space described by (cos(a), sin(a)) into the complex plane.
Fucking idiots.

>>11599982
Im an actual mathematician. These proofs prove the same result but are in no way equivalent as proofs: one is MUCH MUCH more elementary than the other.

>>11599790
OP, I think if you started talking to a kid in any way to how you're doing in this thread he'd probably get annoyed at your smug faggotry and kick you in the shins.

Anyway, a shift of the argument [math] \phi [/math] by an angle [math] \theta [/math] corresponds to a rotation of the unit circle (more specifically, the point [math] (\cos \phi, \sin \phi)^\intercal [/math]). It's rather obvious that a rotation must analytically be implemented as a linear transformation (it should not matter whether we add two vectors and then rotate the result, or first rotate and then add the rotated vectors). This means that
[eqn] (\cos ( \phi + \theta), \sin ( \phi + \theta))^\intercal = R(\theta) \, (\cos \phi, \sin \phi)^\intercal [/eqn]
To find the exact form of [math]R(\theta)[/math], just use the special cases of [math] \phi = 0, \frac{\pi}{2}[/math].

>>11599983
>You don't need any reference to power series or calculus to show this.
Then how do you show this?

>>11599997
THIS is the proof I had in mind, although not presented as neatly as I would like.
GOOD JOB on finding it!

>>11599995
Which one? The one necessitating group structure or the one requiring a metric and scalar product?

>>11599983
Do you even know how to prove the angle sum in polar coordinates thing without using the angle sum formulae?
>>11599997
>using SO(2) to prove trig angle sum
This is what peak autism looks like.

>>11599629
Draw two straight triangles, one over the other such that the seconds' 90 degree angle sits on the first's hypotenuse (the side touching the hypotenuse has the same length as the hypotenuse). If the first's hypotenuse is 1 the second's hypotenuse is the sum of sin (a+b)
There's a similar picture to cos(a+b), can you find it?

>>11600008
This. OP is a faggot retard as usual.

>>11600005
I live for your approval

>>11600008
>he doesn't prove all his special function identities with Lie group representations
pleb

>>11599997
>>11600005
What is a rotation? What is a linear transformation? What is a vector?

OP here: the essence of the proof that the anon gave is exactly that rotations are linear, or the way anon put it, "it doesnt matter if you first rotate and then add or first add then rotate". Knowing this, we can easily find where a vector goes after a rotation by decomposing it into x and y components and using linearity.
Simple, quick, elegant, elementary.

>>11600027
>just consider the homomorphic projection map of the punctured plane multiplicative group onto the unit circle and use the uniqueness of the circle's Lie group structure bro, easy peasy

>>11600036
Although Im open to the possibility that there is an even simpler proof. Will be interested to see more replies.
Already saw a proof that i havent seen before using the mean value theorem, so I guess the thread was a success.

>>11600036
>elementary is what I define as elementary
Pathetic. I'm thoroughly disappointed. If you really think a 5yo would understand that you are 100% autistic.

>>11600049
A fifth grader, not a five year old. A 5yo is too retarded.

>>11600008
>This is what peak autism looks like.
I dunno, I always thought compass-and-straightedge geometry was the pinnacle of autistic pursuits.

>>11600028
>rotation
Look at the ground and start turning on the spot, the way everything looks kind of the same but slanted is what's called a "rotation".
>vector
Draw two lines with right angles between them, we can call them "axes". You can get to any point in the plane by first walking a specific length along one axis, and then a specific length parallel to the other one. Note down those two lengths, they're called "coordinates" of the "vector" of this specific point. With some thinking, we can even add two "vectors" and interpret the result.
>linear transformation
Exactly what I said in that post.

>>11599918
>inb4 OP's argument is circular

>>11600055
So, just like, you know, the Euler identity?
Except that for it we don't need vectors.
You don't have to explain the i.
Way easier and elementary, since no vectors are necessary. Only rotations.

>>11600090
>no vectors are necessary
>proceeds to treat R^2 as a vector space and look at linear maps on it

OP is a faggot as always. His "proof" is more complicated than a diagram and relies on handwaving.

>>11600095
Errr, no, I'm not. That's the whole point. e^ia doesn't live in R^2.

>>11600055
Very formal definition of your terms, which makes your proof highly rigorous. It literally leaves no doubt in the mind of the reader who was not familiar with the terms before. As such we are honoured to tell you that your proof is 100% logically sound, no ambiguity exists anywhere and, most of all, can be presented to a fifth-grader with all the math required, like vectors, rotations and linear tramsformations, being understood by said fifth-grader via your explainations, who would then be able to remember and replicate it when needed.

Inb4 explaining concepts without using math does not constitute a proof.

>>11599629
The one I remember learning was that you draw some seven or so extra lines and then angle chase - it was really disgusting.

>>11599802
you need to be 18+, fuck off

>>11600108
It does not rely on handwaving and is not more complicated than a diagram. It seems like you are a bit too low-IQ to get the idea of the proof just by reading anon's post. Here, let me elucidate it.
1. You realize that cosa, sina are the components of the point in the plane that's rotated by angle a. This is the unit circle definition of sin, cos. Absolutely nothing new.
2. You can add points on the plane by the formula (a,b)+ (c,d)=(a+b,c+d). It's a simple formula and it corresponds to drawing a parallelogram with sides 0-(a,b), 0-(c,d) and looking where the end of the diagonal lands. Again, this is very elementary and doesn't take more than 1 minute to explain.
3. Rotations are nice. They map parallelograms to parallelograms and squares to squares.
4. But that means that if you have a rotation R, then R(x,y)= R(x,0) + R(0,y) since (x,y)=(x,0)+(0,y). This is all about decomposition: it makes rotations much easier to calculate since you only need to rotate points lying on the x and y axes!
5. The point (cos(a+b), sin(a+b)) is the result of rotating the point (cos(a), sin(a)) by angle b. Let R be the rotation by angle b. Then R(1,0)=(cosb, sinb) and R(0,1)=(-sinb, cosb).
Hence we calculate: (cos(a+b), sin(a+b)) = R(cosa, sina)= R(cosa, 0) + R(0, sina) = cos(a) (cos(b), sin(b)) + sin(a) ( -sin(b) , cos(b))=(cosacosb - sinasinb, cosa sinb + sina cosb).
As you can see, the result pops out itself!
Easy, intuitive, rigorous, elegant. The only idea you need to remember is : "Rotations linear, decompose".
Where do you see the handwaving?
>>11600674
Exactly! That's why I posed the challenge. Both the complex analysis proof and the angle chase proof are disgusting and doesn't get to the meat of the issue: rotations of the plane are nice!
>>11600152
Absolutely every fifth grader knows what rotations are and vectors don't take longer than a minute to explain. The complex analysis proof, however, relies heavily on stuff that will go WAY over the fifth-grader's head.

>>11600090
>So, just like, you know, the Euler identity?
>Except that for it we don't need vectors.
>You don't have to explain the i.
>Way easier and elementary, since no vectors are necessary. >Only rotations.
You know you're retarded, right?
If you believe it's way more elementary, PROVE IT. NO ONE HAS BEEN ABLE TO PROVIDE A FULL PROOF using Euler's identity. I'm 100% certain by now that this is because they don't know themselves how to prove it. If you have a full proof using Euler's identity in terms that a (theoretical) fifth grader or a person who's just started learning trig can understand, present it!
NO ONE HAS BEEN ABLE TO DO IT
Why do you keep mentioning the proof if you can't write it out in full!? Maybe if you actually write it out you'll realize it's a complete autistic clusterfuck of a proof that no fifth grader will ever be capable of fully understanding or have the attention span to read through.

>>11601223
That post is 7 hours old, dude. Everybody went home.

>>11601230
Yes I went to sleep and just woke up. Let's get the thread going again!

>>11601209
>The complex analysis proof, however, relies heavily on stuff that will go WAY over the fifth-grader's head.
>stuff
Which one?

>>11601349
The one using the fact that exp(ix) = cosx + isinx for real x.

>>11601238
Have you ever been diagnosed with autism? Has anyone ever suggested you might be autistic in conversation with them?

>>11601362

>>11601352
How is that going way over their head but not your expression of a point on the circle as a vector with trig functions in it when that is literally the same, except that the Euler one is not a vector (in the sense you're explaining to the child).
Just show them that giving an angle is enough to determine where you are on the circle and that the decomposition gives you x-ness and y-ness of the point. You know, like in your answer.
Because eine angle is enough, there is a function that tells you where you are on a circle. Adding angles leads to the same result as rotating around the first, then the second. Same argument.

OP is absolutely based and not a faggot

>>11601390
>How is that going way over their head but not your expression of a point on the circle as a vector with trig functions in it when that is literally the same, except that the Euler one is not a vector (in the sense you're explaining to the child).
Because it presupposes the knowledge of complex numbers, of the fact that exp(a+b) = exp(a)exp(b) where exp(x)= 1 + x + x^2/2! +...,
of the fact that sin(x) = x - x^3/3! + ... , of the fact that cos(x) = 1 - x^2 /2 + .... (you cannot take them as definitions: you have to prove that these expressions coincide in the context of trig, that the series does indeed converge to the ratio of an opposite side of a right-angled triangle to the hypotenuse).
I challenge you to provide a full proof using Euler's formula in terms that a fifth grader can understand.
No one has been able to do it so far ITT.
>Because eine angle is enough, there is a function that tells you where you are on a circle. Adding angles leads to the same result as rotating around the first, then the second. Same argument.
If you have a full elementary proof of it, please show me! That's why I made this thread.

>>11601408
>Because it presupposes the knowledge of complex numbers
As I outlined in that post, no, it doesn't. You just need to see that rotations can be added, as in your post. Nowhere have I used the complex property of i.
>No one has been able to do it so far ITT.
Because you're changing the goal constantly and only accept answers of your liking.

Wild guess, but are you the retard who sperged out over people saying "belief" in scientific theories was uncalled for in those "continuous vs discrete spacetime" threads?

>>11601476
>As I outlined in that post, no, it doesn't. You just need to see that rotations can be added, as in your post. Nowhere have I used the complex property of i
Can you provide a full elementary proof then? At this point I don't see what you mean.
>Because you're changing the goal constantly and only accept answers of your liking.
I never changed the goal. The goal is to provide an elementary proof of an elementary proposition, in the context of trig. Feel free to provide a proof if you have one.
>>11601478
>retard
No.
> who sperged out over people saying "belief" in scientific theories was uncalled for in those "continuous vs discrete spacetime" threads
You're thinking of someone else, sorry.

>>11601478
Not OP but not-so-wild guess, are you the dumbfuck that sperged out over people saying belief regarding unproven scientific concepts?

>>11601478
>>11601488
Guys can we please not start this again? This thread is for discussing trig formulas. Thanks.

>>11601488
kek, no. Just a bystander. OP just really writes in the same style like that guy creating five threads about it.
>>11601486
>You're thinking of someone else, sorry.
Okay.
>>retard
>No.
Hmmmmmmm, not so sure about that one.

>>11601486
>I never changed the goal.
Yes, you did. Twice. Providing an elemental proof and providing an elemental proof that a fifth-grader can understand are different things.
>At this point I don't see what you mean.
Where does someone need to understand complex numbers in the Euler proof?

>>11601548
Can we just lay off the ad-hominems, it's really not productive. Do you have an elementary proof using complex numbers that you can break down in terms understandable to a person who's just learning trig? If so, I would genuinely, unironically would love to see it. That's the whole reason why I made this thread, after all!
I know of such a proof but it requires proving a lot of unintuitive facts like sin(x)=x-x^3/3! +... and properties of the exponential.
I'm entirely open to the idea that you have a simpler proof. Please provide it!

>>11601553
>Where does someone need to understand complex numbers in the Euler proof?
Please write out the full proof then. No one has done it ITT. Surely if you think it's a simple proof, you are able to write it out in full? I wrote out my proof here:
>>11601209
can you do the same?

>>11599629
Depends on how much you consider linear maps as advanced math.

The naive definition of sin and cos is that [math] (\cos\varphi,\sin\varphi)[/math] are coordinates of the point on unit circle with angle [math]\varphi[/math] from the x-axis. The rotation by [math]\varphi[/math] maps [math](1,0)\mapsto (\cos\varphi,\sin\varphi)[/math] and [math](0,1)\mapsto (-\sin\varphi,\cos\varphi)[/math]. Therefore the matrix is [math]\big(\begin{smallmatrix} \cos \varphi & -\sin \varphi\\ \sin\varphi & \cos\varphi \end{smallmatrix}\big)[/math]. Rotation by [math]\varphi[/math] followed by rotation by [math]\psi[/math] is the same as rotation by [math]\varphi + \psi[/math]. So you write the corresponding matrices and compare the entries. That gives you the formula.

>>11601584
This is the proof I had in mind, except that matrices are a bit of an overkill and are likely to confuse the child: the idea of linearity suffices. I gave my version of the proof here:
>>11601209

>>11601563
It takes a lot of writing and I want to understand what problems you have with the one provided first before opening myself up to petty criticism again.

>>11601595
>what problems you have
It's not elementary: a person who is just learning learning trig will not be able to understand it.

>>11601603
>>Where does someone need to understand complex numbers in the Euler proof?
That was the question.
I already understood that your definition of elementary is different from what others deem elementary.

>>11601604
Holy fucking shit can you just give the proof that e^(ix) = cosx + isinx?
Surely if you consider it to be elementary you are able to prove, it no?
Please do.

>>11601590
>except that matrices are a bit of an overkill
>the idea of linearity suffices
this is the same thing, lol. anyone who knows about linear maps knows about matrices.
also you didn't mention anything about "the child" or about starting from absolute scratch in the OP

>>11601633
Is it so hard to understand that the word "elementary" is context-dependent? In the context of an undergraduate mathematics course, sure, the euler proof is elementary.
In the context of basic trig that you learn in high school, it's not elementary and is a total overkill.
In the context of an algebraic geometrist, you can prove that every quadratic equation has a complex solution using the nullstellensatz, but although the methods used are elementary for the algebraic geometrist, the proof itself is not elementary because it concerns quadratic equations and is way above the level that quadratic equations are and what is actually needed.
> anyone who knows about linear maps knows about matrices
Linear maps are easy to explain. Matrices are also not hard, but add unnecessary detail. As I said, your proof is fine. It is in fact elementary because in the context of a person learning trig, with that person's toolbox it can be explained in less than 10 minutes.

>>11599629
,

>all this talk about fifth graders
nigger I didn't touch trig until at least 7th grade, and at that point it was only geometric understandings, soh cah toa stuff
trig as functions was highschool for me, by then I knew about e (explaining exponential functions comes pretty early with Pe^(rt) and such) and frankly explaining how complex numbers multiply (not hard at all, it's literally just one rule) and then transitioning into the proof that uses e^ix=cos(x)+isin(x) would be quick

point out
[math]i=\sqrt{-1}[/math]
[math]e^xe^y=e^{x+y}[/math]
then point out:
[math]-1\cdot -1 =1[/math]
[math]i\cdot i =-1[/math]
[math](\frac{1}{\sqrt{2}}+i\frac{1}{\sqrt{2}})^2=i[/math]
you could just albebraically work through those, then point out that you can always get finer and finer degrees of things where you need to exponentiate more and more to get to 1 as you're taking larger and larger roots of unity.
then point out that if we define
[math]e^{ix}=\cos(x)+i\sin(x)[/math]
(we can leave derivation of this fact to the students calc teachers later in life)
then we can carry over some intuition
[math]e^{\frac{\pi}{2}i}=i[/math]
[math]i\cdot i=e^{\frac{\pi}{2}i}e^{\frac{\pi}{2}i}=e^{\pi i}=\cos(\pi) +isin(\pi)=-1[/math]
and notice that things work out as we expect
then just do what the anon above did
[math]e^{ix}e^{iy}=e^{i(x+y)}=(\cos(x)+i\sin(x))(cos(y)+i\sin(y)=(\cos(x)\cos(y)-\sin(x)\sin(y))+i(\cos(x)\sin(y)+\sin(x)\cos(y))=\cos(x+y)+i\sin(x+y)[/math]
then point out that the i and real parts must correspond since x and y are real and sine and cosine have real outputs for real inputs.

I would've understood this explanation just fine by the time trig functions were used in my education, in fact I would've prefered this expanation since it:
1 sets the foundation for thinking about complex numbers
2 sets the foundation for thinking about them as rotational
3 provides a formula that works, and allows the student to ponder why that may be, possibly leading to further learning

>>11601751
> wecanleavederivationofthisfacttothestudentscalcteacherslaterinlife)thenwecancarryoversomeintuition
So it's not a full proof, nor is it elementary.
I don't believe you when you say e^(ix) = cos(x) + isin(x)

>>11601764
>I don't believe you when you say e^(ix) = cos(x) + isin(x)
nigger
crack out some calc if you wanna be a faggot
[math]e^{x}=\sum_{n=0}^{\infty}\frac{x^n}{n!}[/math]
[math]e^{ix}=\sum_{n=0}^{\infty}\frac{i^nx^n}{n!}[/math]
[math]e^{ix}=\sum_{n=0}^{\infty}\left(-1\right)^{n}\frac{x^{2n}}{\left(2n\right)!}+i\sum_{n=0}^{\infty}\left(-1\right)^{n}\frac{x^{2n+1}}{\left(2n+1\right)!}[/math]
[math]=cos(x)+isin(x)[/math]
you can show that with taylor polynomial construction for the cosine and sine parts, and for the e^x taylor polynomial you could even point out that the rate of change, the derivitave (which isn't easy to explain in terms of limits of secants), of e^x is e^x which is the one of the more formulations of saying e^x has exponential growth.
>So it's not a full proof, nor is it elementary.
the proof is the e^i(x+y) part, the rest is simply some guiding explanation for young lads and retards like yourself. It's as elemetary as it would need to be for a person learning trig to understand it, which is elementary enough. I do not care if some 10 year old can understand it, you'd be an idiot to bother explaining anything more complex than fractions and triangles to them.

die

>>11601781
>with taylor polynomial construction for the cosine and sine parts
This is getting closer to the full proof but it's still not complete. I still don't believe you.
Taylor series for a function don't always converge to the function. Can you prove those series converge to sin and cos?
Also please prove that these are indeed the taylor series for sin and cos.

>>11601620
That is a construction, not a theorem which can be proven. Can you answer the question first?
I'm giving a function depending only on the angle a, that describes position on a circle, e^ia. The righthand side gives the "x-ness and y-ness" of that position, depending only on that angle. You can show that it does using simple trigonometry.

>>11601782
derivatives for sine and cosine
sin-> cosine-> -sine-> -cosine ->loop
at 0
0 -> 1 -> 0 -> -1 ->loop
building taylor polynomials out of these give the above formulas
these converge everywhere. I'm not going to break out convergence theorems for a highschooler. showing they converge on a graphing calculator on desmos or something would be far more helpful and clear for them

d
i
e

>>11601791
Not construction, definition.

>>11601805
>derivatives for sine and cosine
>sin-> cosine-> -sine-> -cosine ->loop
I'm only just learning trig. How do you show this? I've now idea why it should be true!
>these converge everywhere. I'm not going to break out convergence theorems for a highschooler. showing they converge on a graphing calculator on desmos or something would be far more helpful and clear for them
Let's say it converges everywhere (trivial to show by ratio test). How do you know that it converges to sinx and cosx? For all I know, it could converge to anything else!

>>11601811
die
>I'm only just learning trig. How do you show this? I've now idea why it should be true!
I already explained you can explain derivatives quickly as a limiting process of secants
just demonstrate that this holds for the graphs of sine and cosine
alternatively there are simple graphs using the unit circle definition of sine and cosine that make this more clear for the particularly unconvinced student
die
>Let's say it converges everywhere (trivial to show by ratio test). How do you know that it converges to sinx and cosx? For all I know, it could converge to anything else!
you constructed them from the derivatives of sine and cosine, they're going to converge to sine and cosine if they converge. what the fuck are you even asking?
die

>>11601822
>I already explained you can explain derivatives quickly as a limiting process of secants
>just demonstrate that this holds for the graphs of sine and cosine
>alternatively there are simple graphs using the unit circle definition of sine and cosine that make this more clear for the particularly unconvinced student
Sure, I know what derivatives are, but how do you prove that d/dx(sin(x)) = cos(x) and d/dx(cosx)=-sinx? I don't believe you.
>you constructed them from the derivatives of sine and cosine, they're going to converge to sine and cosine if they converge
Prove it. Protip: what you said is not even true, so you can't prove it.
You don't even seem to understand your own proof LMAO. This is hilarious.

>>11601843
>Prove it. Protip: what you said is not even true
news to me
a taylor series has a radius of convergence, is defined by a function, what other function is it converging to within that radius?
I know you can have smooth functions that do not have their taylor series equal them anywhere, but there radii of convergence are 0

>>11601850
> I know you can have smooth functions that do not have their taylor series equal them anywhere, but there radii of convergence are 0
Not necessarily. Take the common example
f(x) = 0 if x<=0 and f(x)=e^-1/x for x>0.
At x=0, all the derivatives are zero so the taylor series is just 0, hence has radius of convergence infinity.

OP you're being a faggot.

you've asked for an "elementary proof". literally anything is or isn't elementary in the right context, but you never provided the context. and honestly, now it seems that your context is that you have one proof which you consider optimal and everything different is either not elementary enough, or too elementary.

it's great that you've figured something out on your own, and all. but you're being a faggot.

>>11601858
Do you think you can provide a full proof that e^ix = cosx + isinx?
No one has been able to prove it so far. It seems to me they don't even understand why it's true.

>>11601855
interesting
most of what I learned about analytic functions and taylor series past the trivial level was in the complex plane where this is impossible

>>11601871
Yeah, holomorphic functions are much nicer in that sense.

>>11601858
This. OP, stop the delusions.

>>11601861
Not him, but see
>>11601791
Are you going to ignore this? You cannot prove a definition.

>>11602000
If you have a full proof through euler's formula, please provide it. No one has been able to do it so far.
>>11602003
His proof is circular.
>I'm giving a function depending only on the angle a, that describes position on a circle, e^ia. The righthand side gives the "x-ness and y-ness" of that position, depending only on that angle. You can show that it does using simple trigonometry.
And using this definition of e^ia, how do you prove that e^(ia) * e^(ib) = e^(i(a+b))?

>>11602011
OP stop this, you're being a massive faggot. you want the euler's formula proof only so you can say that it assumes too much for a fifth grader and therefore your proof wins.

>>11599703
>I dont get it. How does no one know of a simple proof of such a common and useful formula?
it's not that people don't know your proof. literally anyone who knows a tiny bit of pure math does. the things is that people are not fucking mind readers. you don't want ""an"" elementary proof. you want EXACTLY the one proof you have in mind, including arbitrary specifics like: you can use linear maps, but matrices are too much, you can use vectors, but complex numbers are too much etc.

>>11602011
>how do you prove that e^(ia) * e^(ib) = e^(i(a+b))?
/thread

>>11602039
>OP stop this, you're being a massive faggot. you want the euler's formula proof only so you can say that it assumes too much for a fifth grader and therefore your proof wins.
I don't care about what's the simplest proof anymore. At this point I just want to see if anyone on this board (besides me) actually knows why Euler's formula holds true. No one has been able to provide proof for it. Some have tried but got stuck.
Now I'm just exposing that pseuds on /sci/ use formulas and don't even know why they're true lmao.

>>11602064
>At this point I just want to see if anyone on this board (besides me) actually knows why Euler's formula holds true
LHS and RHS are both solutions to the same initial value problem, hence they're equal.
/thread ?

>>11602064
>why Euler's formula holds true
Could you please repeat which of Euler's formulae you mean? Sorry I have not followed the whole thread. And there is an awful lot of those to pick from.

>>11602076
The formula e^(ix) = cosx + isinx
>>11602073
Explain what you mean in detail. Provide proofs.

>>11602080
>Explain what you mean in detail. Provide proofs.
you know very well what I mean, what's the point

>>11602091
No I genuinely don't. Perhaps your proof will be different. I imagine you will need to prove that lim sinx/x = 1. There are many different ways to do it. I'm curious what anons will come up with.

>>11602011
>His proof is circular.
So you're going to ignore that Eulers formula is a definition for complex exponentials and hence cannot be proven. Noted. Thanks for playing.

>>11602104
>Eulers formula is a definition for complex exponentials and hence cannot be proven
Typically euler's function e^x is defined by the series exp(z)= 1+ z + z^2/2 + ...
Then it's a theorem that exp(ix)= cosx + isinx for real x. There is a proof of it.
Of course, you can do it in a nonstandard way and define e^ix to be cosx + isinx. But then for your proof of trig formulas to work, you need to prove that e^ix * e^iy = e^i(x+y) . And this is equivalent to proving the angle addition formulas so we're back to square one and you haven't achieved anything. Surely you realize this?

>>11602011
>using this definition of e^ia, how do you prove that e^(ia) * e^(ib) = e^(i(a+b))?
Aha, finally you were able to answer the question.
Answer is: the same way you did. Since it describes a position on the circle, adding angles must lead to the same result as doing two rotations about the summands.

>>11602080
>The formula e^(ix) = cosx + isinx
Thanks. But it looks like a definition to me. If I prove it by saying it is the definition, surely you will accuse me of being circular.
Have you got another identity to offer as a definition of e^(ix) where x is real?

>>11601811
>How do you know that it converges to sinx and cosx? For all I know, it could converge to anything else!
taylor's remainder theorem, no?

>>11602111
>Typically euler's function e^x is defined by
>Then it's a theorem that
No, that is just a different notation for literally the same thing. You even used that fact in the beginning.

Also, since complex numbers are constructed to be a superset of the reals, the identity for real numbers follows from that.

>>11602125
>Also, since complex numbers are constructed to be a superset of the reals, the identity for real numbers follows from that.
not OP, but some real identities break in the complex numbers

[math]1=\sqrt{1}[/math]
[math]=\sqrt{-1\cdot-1}[/math]
[math]=i\cdot i[/math]
[math]=-1[/math]
this fails since it is not always true in the complex numbers that
[math]\sqrt{xy}=\sqrt{x}\cdot\sqrt{y}[/math]

>>11602137
Your very first equation is wrong. Also, roots of negative numbers arent defined, exactly to make the decomposition of roots possible, since otherwise roots aren't functions.

>>11602150
>Also
Anon, you win today's prize for having at least one sentence fragment not completely wrong. Congratulations!

>>11602115
>Answer is: the same way you did. Since it describes a position on the circle, adding angles must lead to the same result as doing two rotations about the summands
Can you write out the proof if full? I honestly have no idea what you mean.
>>11602116
>But it looks like a definition to me. If I prove it by saying it is the definition, surely you will accuse me of being circular.
You can use it as a definition, but then you cannot use the property that e^(ix) * e^(iy) = e^(i(x+y))
>Have you got another identity to offer as a definition of e^(ix) where x is real?
Sure, the common mathematical definition is
e^ix = 1 + ix + (ix)^2/2 + (ix)^3 / 6 + ....
>>11602117
I can see a way to prove it using it. But first you would need to calculate the derivatives of sin and cos. How would you do it?
>>11602125
What?

>>11602150
>Your very first equation is wrong
1^2=1 tho?
> Also, roots of negative numbers arent defined
[math]\sqrt{-1\cdot -1}[/math]
is the root of a positive number

>>11602160
You do know there is a difference between
roots of x^2 = a
and
sqrt(a)
right?

>is the root of a positive number
Yes, but not the roots in your next step, so they're meaningless.

>How would you do it?
do we need to? for taylor's theorem you only need the absolute highest and lowest derivative on the range, you can see rather quickly on a graph of sine and cosine that this is never higher than one. This makes sense if you think about the unit circle definitions. moving around a circle at a given velocity, speed is maximal in the y direction at (1,0) and (-1,0) at 0 and pi. etc
sine(ax) can be interpreted as moving around a circle with velocity a. at 0 all that velocity is upward. with a=1 our derivative is 1.

>>11602173
replying to >>11602159

>>11602173
Ok I can see that you understand what's going on, but how would you prove that the derivative of sin is cos and d/dx cos =-sin though?

>>11602166
>Yes, but not the roots in your next step, so they're meaningless.
we assume we're working with complex numbers
[math]i=\sqrt{-1}[/math] is a definition
the step breaking
[math]\sqrt{-1\cdot -1}[/math]
into
[math]i\cdot i[/math]
is faulty when working with complex numbers, but that's what I was trying to demonstrate in the first place

>>11602180
>but how would you prove that the derivative of sin is cos and d/dx cos =-sin though?
globally?
this can be done geometrically in a similar manner
https://www.desmos.com/calculator/eqpwtp9dhd
just note it's the same as the main point going around the circle, just one turn ahead
cos is sine one turn ahead, therefore cos is sin'
repeat with similar argumentation for cos' being -sin

>>11602195
Yeah but that's not a proof.

>>11602197
the tangent is the defining part of what makes a derivative
this skips the need for limit process of secants and jumps directly to the tangent line, since the formula for those is known for circles
what exactly more do you want?

>>11602095
this depends on how you define cos and sin. I like the naive high school definition as coordinates on the unit circle, but at the same time I don't consider this to be fully rigorous. so I fix this using ODEs. parametrization of a uniform circular motion clearly needs to satisfy that the tangent vector is perpendicular to the radius vector. therefore the coordinates must satisfy

x' = -y
y' = x

up to multiplication by a constant. the solution starting at (1,0) is the definition of cos(t) and sin(t). this approach emphasizes the relation to circular motions, that's what's important.

on the other hand, exp(it) is defined as the solution to

z' = i*z
z(0) = 1

but this is also an ODE for a circular motion: it literally says that the tangent vector to the solution curve is the radius vector multiplied by i, therefore rotated by 90 degrees.

the equations are the same, therefore the solutions are the same.

>>11602104
that's not true. e^it can be defined in many ways, euler's formula is just one of them.

>>11602208
neat

>>11602183
>[math]i=\sqrt{-1}[/math] is a definition
No, the definition is i^2 = -1
>is faulty when working with complex numbers
Yes, and also for reals. So?

I know there are many theorems in complex numbers that do not extend to the reals, but your example isn't in of them.

>>11602208
>that's not true. e^it can be defined in many ways, euler's formula is just one of them.
Can you show one which isn't equivalent to Euler's formula? Expressing all the terms as series isn't a different definition, it's just a different notation, just like 0.999... and 1 are different notations for the same number.

>>11602222
>Yes, and also for reals. So?
in the reals there is no way to break that at all
a square root of a negative isn't allowed

>>11602228
of course they're all equivalent. but there's a difference between defining the symbol exp(it) literally to be cos(t)+i*sin(t), and defining exp(it) as a power series and showing that it's the same as the power series of cos(t)+i*sin(t).

>>11602234
I don't get what you want to say. You've just shown you cannot break them in the complexes, and also not in the reals.
Your claim was that this somehow shows Euler's formula being true for complexes doesn't imply it being true for reals. I say it does because there are only functions involved.

>>11602263
>Your claim was that this somehow shows Euler's formula being true for complexes doesn't imply it being true for reals
other way around

>>11602264
Ohhh, okay, yeah, remembered that incorrectly. But still, your argument doesn't work, because it already doesn't work in the reals.

>>11602271
>it already doesn't work in the reals
the only reason you can't do that in the reals is since if you have a standing square root of a negative you're not in the reals

>>11602207
Tangent of a graph of a function. You have a tangent of the circle. Explain how they are related.

>>11602298
circles and sine waves are closely related
the y values of a point moving along a circle forms a sine wave
the change in y of tangent to a cirlce at a point (following that tangent in the direction of the point's motion) is the derivative of sine at that point

>>11602289
Doesn't change anything. Find a better counter-example.

>>11602312
>>11602298
I just realized that this leaves sin'(0) undefined (infinite slope)
better said, you can note that we always move around the circle at a constant rate with our point, so the changes in speed of cos and sine as we move around must have a magnitude 1
ie
[math]\cos'(x)^2+\sin'(x)^2=1[/math]
then use the tangent line to find the ratio between the sin' and cos' and their signs

>>11602351
>you can note that we always move around the circle at a constant rate with our point
Prove it. I believe the rate is a little bit slower when the angle is small.

>>11602364
>I believe the rate is a little bit slower when the angle is small.
why?
(sin(ax),cos(ax))
defines a point along a circle and we're moving along that circle at rate a as x increases. I'm pretty sure that's just getting into how these functions are defined, isn't it?

>>11602364
slightly more rigorously we can say that point's speed is constant since we can just pick two new perpendicular axis and describe the point in reference to the sine and cosine relative to those axis and things work out the same, since we've essentially just rotated our diagram. so the point's overall speed can't be changing, if it did when we turned our diagram something would be different.

>>11602388
>two new perpendicular axis and describe the point in reference to the sine and cosine relative to those axis and things work out the same
Aren't you presupposing the angle addition formula here?

>>11602417
more so a graphical intuition that axis are arbitrary and we can define sine and cosine off any two perpendicular axis and all that changes is phase
this does not, on its own net a formula for capturing this shift from one set of axis to another in the way the angle addition formula can

>>11601209
>4. But that means that if you have a rotation R, then R(x,y)= R(x,0) + R(0,y) since (x,y)=(x,0)+(0,y).
This is handwaving. Proving it is essentially as "complex" as proving the addition formula.

>R(cosa, 0) + R(0, sina) = cos(a) (cos(b), sin(b)) + sin(a) ( -sin(b) , cos(b))
You didn't explain why you're allowed to scale by cos(a) and sin(a). Handwaving.

A fifth grader may understand most of the individual points, but not the important ones. It's not a rigorous proof and diagrams are much simpler. This is a somewhat intuitive justification, but not a proof.

>>11601209
>Let R be the rotation by angle b. Then R(1,0)=(cosb, sinb) and R(0,1)=(-sinb, cosb)
where is this coming from

>>11602433
>This is handwaving. Proving it is essentially as "complex" as proving the addition formula
I don't think so. This boils down to the fact that in cartesian coordinates, the parallelogram rule just asks you to add the individual coordinates. This is very easy to see intuitively but can also be easily proven by drawing a diagram and checking a few cases.
>You didn't explain why you're allowed to scale by cos(a) and sin(a). Handwaving.
You are correct, I did not write out every single detail. You can scale because rotations preserve proportions of lengths. Every child will realize this as an obvious fact.
>It's not a rigorous proof and diagrams are much simpler
The issue with diagrams is that there are many ways to draw the diagrams and it is essentially diagram chasing. My proof is simpler because the idea behind it is to use linearity. The formula pops right out itself.
>>11602443
Alas, this is perhaps my most worrisome part.
R(1,0)=(cosb, sinb) by the definition of the trig functions (coordinates of the point on the unit circle rotated by angle b).
R(0,1)=(-sinb, cosb) can be found also by linearity: to see how rotations by 90 degrees work it's enough to check them on components. We see that (1,0) gets mapped to (0,1) and (0,1) to (-1, 0). Now apply linearity.

>>11599629
Start by using a circumscribed quadrilateral. Apply trigonometry to calculate the cosines wrt to the two angles that occur when angle A is split into two by diagonal AC. Gg

>>11602470
>to see how rotations by 90 degrees work it's enough to check them on components. We see that (1,0) gets mapped to (0,1) and (0,1) to (-1, 0). Now apply linearity.
this is hardly intuitive then
you'd have to think through that every time you wanted to use this to remind yourself of the correct formula

kek, turns out OP doesn't know as much math as he thought he would and is delusional for his arguments. Good thread.

>>11602470
>This boils down to the fact that in cartesian coordinates, the parallelogram rule just asks you to add the individual coordinates. This is very easy to see intuitively but can also be easily proven by drawing a diagram and checking a few cases.
LOL, so now your "proof" requires diagrams.

>>11602544
Why even bother when this particular formula can be proven geometrically fairly easily. Like why bother with university level stuff. Sure you can tell a kid that there is a deeper meaning to it but other than that

>>11602571
Okay now you're also a schizo? Because that doesn't relate to my post at all and sounds like we're all after you.

>>11602584
I am just trying to understand OP. I assume he wanted his dopamine rush and tried to show off

>>11602544
>kek, turns out OP doesn't know as much math as he thought he would and is delusional for his arguments
What makes you say that?
>>11602590
> assume he wanted his dopamine rush and tried to show off
Partly the dopamine rush and wanting to show off, partly sharing something cool I discovered with others.

>>11602611
>sharing something cool I discovered with others
That's cool and all, but why did you present it like a fucking imbecile and changing circumstances instead of just saying "what's the coolest proof of ... you can think of? I think I have found a pretty elegant one." and having some fun?

>>11602680
But I did have a lot of fun!

>>11602701
By ruining the experience for everybody else.
There have been many interesting concepts discussed here, but it wasn't fun to anyone but you. Wasted thread.

>>11602712
U mad bro?

>>11602738
No, just disappointed. This had potential.

>>11602816
Go browse the IQ threads then, moron. This was objectively the best thread (that's not a general) that /sci/ had in weeks!

>199/27

>>11602833
>moron
Who's mad yo?

And no, definitely not. There were much better threads. OP just trolled subtly enough to keep more people around.

>>11602856
>There were much better threads
Name two.

>>11602351
interesting to note
[math]\cos'(x)^2+\sin'(x)^2=1[/math]
immediately implies that (if cos' and sin' are real, and they are) cos' and sin' must be some combination of sine and cosine or their negatives for some angle. cos and sine having squares that sum to one is essentially their definition is you use the description that sine and cosine define points on a unit circle