YOU CAN'T USE EULER'S IDENTITY TO PROVE THE ANGLE SUM IDENTITIES! | Tricky Parts of Calculus, Ep. 4

[@Jaeghead]

I agree that a geometric interpretation of sine and cosine is useful, but I think it's better to first define the exponential function through its series and then define cosine and sine as real and imaginary parts of exp(ix) for real x (or more generally as linear combinations of exp(z) and exp(-z) for complex z). This gives us Euler's identity without any further proof and makes the "illegitimate" proof in your video noncircular again. Of course one would still need to establish the properties of the exponential function like you did at 3:18, but this is useful and necessary anyway and we would still get a geometric interpretation of the trig functions through the properties of the exponential function in the complex plane (and the other proofs including 2pi-periodicity are relatively straightforward with this definition). I understand that you don't want your proofs/definitions to be "unnatural and unmotivated", but I find it equally problematic to use undefined objects such as "triangles", "angles" and possibly even "side length (of a triangle)" in the definition of trig functions (unless the students only need a working understanding of trigonometry, but to a mathematician you have to present self-contained definitions and proofs).

[@DanielRubin1]

For those who prefer to start from power series, my challenge to you is to establish the periodicity of sine and cosine, and connect the period to the circumference and area of a circle. It's doable, but I think that seeing what's involved might convince you of the virtue of the geometric approach. A more difficult challenge is to come up with some way people might have written down these power series without the geometry. Also, triangles, angles, and lengths are perfectly well-defined quantities; check out the first couple of episodes of this series. These are the kinds of quantities that people deal with most directly. A good mathematical theory should address them.

[@Jaeghead]

@@DanielRubin1 >my challenge to you is to establish the periodicity of sine and cosine I don't see the problem with establishing periodicity once you have defined pi (using the IVT, taking twice the smallest positive zero of cosine), it follows directly from the properties of the exponential function >and connect the period to the circumference and area of a circle I don't think it's absolutely necessary to connect them at this point, but for example the connection to the circumference can be made as a limit of lengths of polygonal paths where the length of each line segment is |exp(i2pik/n) - exp(i2pik/(n-1))|, and that limit can be calculated easily once you have shown how to get the derivative of sine. The connection to the area definitely should be made later when the students have an appropriate definition of area (but again it could be made easily using the exponential function if necessary). >It's doable, but I think that seeing what's involved might convince you of the virtue of the geometric approach Well that's more or less exactly how it's taught where I'm from, so I have seen what is involved and it didn't seem too bad to me. >Also, triangles, angles, and lengths are perfectly well-defined quantities; check out the first couple of episodes of this series I wasn't saying that those terms are generally undefined, but that the students would not have yet seen them at that point (and even if they have I find it unnecessary to rely on them for definitions), so I was suggesting defining trig functions without referencing triangles and angles (later on it's useful to establish the connection, but not as a definition).

[@DanielRubin1]

It's not obvious from the Taylor series definition of cosine centered at 0 that the function becomes negative and so the Intermediate Value Theorem says there's a first positive zero. You'd have to pick a nice value (between pi/2 and 3pi/2), evaluate the first few terms to a negative result, and use the Lagrange error estimate to bound the remainder to show that the whole series is negative. The point is that I don't think you can avoid the Mean Value Theorem, on which the Lagrange Error Estimate depends. An alternate approach is that differentiating term-by-term gives d/dx(cos x) = -sin x and d/dx(sin x) = cos x, and so both satisfy y'' + y = 0. Therefore cos has negative second derivative whenever cos is positive, so it's graph lies below any tangent line here, which shows it must have a zero (this also uses the Mean Value Theorem). For the periodicity, I think you need the theory of linear ODE so the periodicity will follow from satisfying y'' + y = 0 with the same initial conditions later. This also needs the angle addition formulas, which also come from the differential equation. I think the way you propose to relate pi to the circumference assumes what you want to prove, that (cos, sin) parameterizes the unit circle by arc-length. I would use the inverse function theorem to get d/dx( arcsin x) = 1/Sqrt(1-x^2) to calculate the length of a quarter of the circle.

[@Jaeghead]

@@DanielRubin1 >It's not obvious from the Taylor series definition of cosine centered at 0 that the function becomes negative I agree, but you don't have to use the MVT to get a zero. For example you can show that for a fixed x in [0, 2], the sequence (x^n/n!) is decreasing (starting at n = 1) so by the properties of alternating series cos(x) is always between consecutive partial sums of its series, and we get 1 - x²/2 I think the way you propose to relate pi to the circumference assumes what you want to prove I don't see where I'm assuming that, obviously one should prove the possibility of writing complex numbers in exponential form (using IVT) before using them to calculate arc lengths, but once you have that the calculation is easy. Of course there would still be the question of whether a different partition would lead to the same length, but I don't think it's necessary to address this at that point.

[@azimuth4850]

This series is hilarious, looking forward to studying this in more detail when I get around to more advanced calc. Thanks again man.

[@toyeshjayaswal1645]

My favorite proof for the addition formulas I know goes as follows: take a triangle and draw its height, it splits the angle a+b into a and b, using that the area of a triangle is ABsin(theta)/2 (A,B being the side lengths with theta the angle between them, it's fairly easy to show this too), you get two expressions for the area of the triangle, the side lengths cancel and you immediately get the sin addition formula sure, this only works if a+b is less than 180, but you can justify the other case using properties of sin or cheating a bit and using negative angles for the geometry

[@toddtrimble2555]

If you want a heuristic argument more closely tied to geometry as opposed to formal manipulations, then one way to go is to use e^{it} = limit_{n --> \infinity} (1 + it/n)^n, letting students observe (e.g., with the help of Mathematica) the behavior as n increases, along the lines of the animation shown here: en.wikipedia.org/wiki/Euler%27s_identity#Imaginary_exponents. As to the question of periodicity, one may observe that e^{it} lies on the unit circle, and that the norm of its derivative equals 1, i.e., a particle moving along the circular trajectory moves with unit speed, and hence we are dealing with the arc length parametrization around the unit circle. In any case, I think it's overreaching to suggest that instructors who choose to pursue such lines of thought are (by necessity) not thinking clearly or are necessarily using "circular" (ha ha) reasoning, although I'm not doubting you can find such instances on KZbin. It really all depends how you set things up and what sort of audience you have. Properly preparing students so that they can understand Euler's formula viscerally and geometrically (not just formally) is not only something that can be done, but a wonderful and illuminating adventure. Even in a standard Calculus II classroom in the US, where one isn't pretending to prove every last claim, Euler's formula and what you can formally do with it can be a real eye-opener. Seeing it for the first time was a defining moment in my life.

[@gokucrafter9456]

Yep, I agree with how trig should be taught, and learned through geometric arguments. But for mathematical rigour, developing from the power series is possible, no matter how tedious it is.

[@ingiford175]

Fortunately in HS, I had a class on Trig before calculus and did not see Euler's until later in the Calc class next year.

[@nuckyducky]

Bruh, 2:22-3:14, life changing.

[@nuckyducky]

7 months later and I'm still rewatching this magic.

[@APaleDot]

I don't think you need to go into linear algebra and matrices to use the rotation method. You can just define a a complex number that does rotation called Rot(θ) = cos(θ) + isin(θ). Showing this does in fact rotate by θ is not hard. And then Rot(A)Rot(B) = Rot(A+B) follows from the geometric properties of rotations, which is precisely the exponential property needed to make the original proof work.

[@hacker2ish]

The proof I know is directly from the geometric configuration of the 2 radii with angles a and a+b relative to the x axis, the problem I have is that the configuration changes materially when the angles are more than pi/2 or b is more than pi/2 and so on. It becomes like 4 or 8 cases where it's kind of hard to apply exactly the same method. So I kind of hoped to see a way where all these cases can be unified in a single geometrical argument

[@ulissemini5492]

12:02 this is how 3blue1brown does it in his lockdown math lectures. I find this is the most intuitive view especially since it lets you see complex multiplication is rotation without converting to polar form. I think you should have mentioned this earlier as it's the most direct and elegant proof of the identities, you see the result directly instead of having to prove it with geometry

[@hacker2ish]

The way to establish that complex multiplication is rotation is through the angle sum identities

[@ulissemini5492]

​@@hacker2ish false, you need only show complex multiplication rotates the basis vectors (1 and i) the rest follows from complex multiplication being a linear transformation. let z = (cos t + i sin t), clearly z*1 = z hence z rotates the first basis vector by 't' radians. now z*i = (-sin t + i cos t) is a 90 degree rotation of z and (from basic geometry) the i basis vector is also rotated by 't' radians. at this point the linear transformation is uniquely determined by how it acts on the basis vectors, hence it must be a rotation by 't' radians. now you can *prove* the angle sum identities using complex multiplication. (PS: there isn't "The way to establish X", there are many ways)

[@ulissemini5492]

@@hacker2ish btw this is exactly what he does at 12:02, he just uses pointlessly ugly notation. matrices aren't needed, you can work with the linear transformations directly (this is the way Axler does LA btw)

[@costakeith9048]

Yes, but before you can get to that point, you need to prove that C is a vector space over R and that complex multiplication is a linear transformation; yes, this can be done, but I don't know that 'elegant' is the word I'd use to describe it, more of a brute force approach.

[@Abby-mz6nq]

I disagree with the philosophy implicit in this argument. The goal of teaching mathematics is to make a good mathematician or someone who can use math as a tool to solve their particular problems. That doesn't necessarily mean that everything should be taught only when its rigorous proof can be completely understood. Introducing someone to Euler's Identity could motivate their curiosity. You can point out this issue with the proof. That's probably a good idea. It seems a little much to give someone a sword only if they can craft a sword themselves. And Feynman gives a nice explanation of Euler's identity, that doesn't use calculus, in his lectures.

[@hacker2ish]

How can someone be a good mathematician if they don't have the tools to think for themselves.

[@manavdahra207]

Mathematics or not, any science for that matter should not have circular reasoning while presenting any arguments, I don't see how we use mathematics (as a tool or forefront of research) makes any difference. Circular reasoning is very much like religion to me.

[@seandouglas7526]

I love your content!

[@DanielRubin1]

Thanks!

[@scotthennebeul1705]

I hate commenting on KZbin but if one wishes to start with that identity and some rule of exponentials then one can. Thus the so called angle sum identities are yielded without question. Depends on what one is trying to accomplish. I shall comment no further other then to say motivation is indeed lacking in most courses.

[@Abby-mz6nq]

You can define e^ix = cos(x) + i sin(x). That's not circular.

[@justanotherman1114]

Then to define the power rule of exponentials you would need the formulas for sine and cosine.