# Rigorous proof of the Taylor expansions of sin $x$ and cos $x$ revisited

I asked this question a while ago. I exchanged comments with a member(mixedmath) about the rigorous proofs that $\lim_{x\rightarrow 0} \frac{\sin x}{x} = 1$ and the addition formula for $\sin x$. He referred to the wikipedia article. However, I'm not sure if the proofs using pictures are rigorous enough. The proofs take it for granted that what an angle(measured by radian) is. IMO, a straightforward and yet rigorous definition of an angle is that as an arc length of the unit circle. This definition involves the limit or the sup of suitable sums of lengths of line segments. I don't see how this definition incorporates into the proofs. Simply put, are the proofs of the wikipedia article rigorous?

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It depends on what you want to accept as rigorous. The proofs in wikipedia are indeed "proofs by picture." They are not however, the real foundation of trigonometric functions. If you want to start from scratch, I highly recommend you look at the prologue of Real and Complex Analysis by Walter Rudin. As well as being a complete answer to your question, the prologue is in my opinion one of the most lucid and elegant pieces of writing in all of mathematics. I will not copy it here but merely reiterate the important details:

1) The function $e^x$ is defined as

$$e^x:=1+x+\frac{x^2}{2!}+\frac{x^3}{3!} + \cdots$$

To be clear, there are no definitions of $e=\lim_{n\rightarrow\infty}(1+1/n)^n$, etc. This is a definition of a function and the value of $e$ is fixed by plugging in $x=1$.

2) It is mentioned the series converges absolutely for all $z$ (using nothing more than basic series tests from calculus). Now we are sure we are dealing with a real and sensible function.

3) By using binomial identities, one shows $\exp(x+y)=\exp(x)\exp(y)$, and also stuff like $de^x/dx=e^x$, etc. This all follows by term-wise manipulation of the series definition, all of which is made rigorous again through the theory of infinite, absolutely convergent series.

4) He defines $\cos$ and $\sin$ functions as the real and complex part of $e^{ix}$. He then shows that there exists a number, $\pi$ such that $e^{2\pi i}=1$ and that $e^x$ is periodic with period $2\pi i$. All of this is again made "rigorous" with no geometrical mention to the meaning of $\pi$, just through analysis techniques of monotonic functions and various observations of sign changes. Again I'm skipping details.

5) After all is said and done, it is shown that $\cos$ and $\sin$ obey all the usual geometric definitions we are accustomed to. Perhaps here is where the real magic happens as we infer their geometrical meanings by again proving that the sin of a radian is equal to the ratio of appropriate sides, etc. To reiterate, the definitions are not reliant on any geometrical properties of the $\sin$ and $\cos$ we already know about from high school trigonometry.

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There are several ways to lay the rigorous foundation of trigonometric functions. The Rudin's method is one of them. My question is whether the wikipedia's method(proofs by pictures) can be one of them or not. Not small number of people seem to believe so as one of the answers to my previous question(math.stackexchange.com/questions/185356/…) shows. – Makoto Kato Sep 16 '12 at 5:47

Maybe you could start with the differential equation $$y'' + y = 0$$ and the initial conditions $$y(0) = 0, y'(0) = 1.$$ That has a unique solution $s:\mathbb{R} \to \mathbb{R}$ (of course, to be totally rigorous you'd have to prove existence and uniqueness of solutions of linear ODEs, and probably that the general solution to an $n$th order linear ODE is in fact a $n$-dimensional vector space, but that is not too hard).

With initial conditions $y(0) = 1, y'(0) = 0$, you'll get the (unique) solution $c:\mathbb{R} \to \mathbb{R}$. Then, by some moderately clever manipulations you can prove that $s' = c, c' = -s, s^2 + c^2 = 1$, etc.

Incidentally, you could use the same approach to define the exponential function, by means of the ODE $y' = y$ with initial condition $y(0) = 1$.

If you try to obtain power series solutions to these ODEs, you'll get precisely the Taylor series of $\sin$, $\cos$ and $\exp$.

I prefer the ODE method of introducing these elementary transcendental functions because they are motivated by important and well known physical processes - simple harmonic motion and population growth (or radioactive decay), respectively.

The definition of $\exp$ by means of the series is $100\%$ rigorous but totally unmotivated. This kind of thing always puts me off a little, and I imagine it can be baffling for anyone learning calculus or analysis for the first time. Another example: the definition of natural logarithm by means of the integral of $1/x$. Where does that come from? From the need to have a primitive of $x^n$ also when $n = -1$?

$\log$ should be introduced as the inverse of $\exp$, once it is proved that $\exp$ is a differentiable bijection from $\mathbb{R}$ to $\mathbb{R}^+$.

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I would consider that proof as essentially rigorous. You always have to rely on something, and I would say that the existence of the arc-length for a circle can be taken as granted. Once arc-length exists, you define the measure of an angle as a quotient. Your question is slightly "scaring", since you are essentially asking whether the so-called "synthetic geometry" is rigorous. Historically, this is the first piece of rigorous mathematics for mankind. Of course most proofs are based on drawings, and teachers usually repeat that "you must not draw particular cases".

As an instructor, I'd rather teach sines and cosines from an analytic viewpoint, but I understand that this is a bad approach for students: they already know sines and cosines, and they get confused to see them as power series or solutions to differential equations.

So, as long as the proof is consistent with "greek geometry", I think it may be a rigorous proof.

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