What are Different Approaches to Introduce the Elementary Functions?

Motivation

We all get familiar with elementary functions in high-school or college. However, as the system of learning is not that much integrated we have learned them in different ways and the connections between these ways are not clarified mostly by teachers. Once I read the calculus book by Apostol, I just found out that one can define these functions in a treatise systematic way only analytically. The approach used in the book with some minor changes is like this

$1.$ Firstly, introduce the natural logarithm function by $\ln(x)=\int_{1}^{x}\frac{1}{t}dt$ for $x>0$. Accordingly, one defines the logarithm function by $\log_{b}x=\frac{\ln(x)}{\ln(b)}$ for $b>0$, $b \ne 1$ and $x>0$.

$2.$ Then introduce the natural exponential function as the inverse of natural logarithm $\exp(x)=\ln^{-1}(x)$. Afterwards, introduce the exponential function $a^x=\exp(x\ln(a))$ for $a>0$ and real $x$. Interchanging $x$ and $a$, one can introduce the power function $x^a=\exp(a\ln(x))$ for $x \gt 0$ and real $a$.

$3.$ Next, define hyperbolic functions $\cosh(x)$ and $\sinh(x)$ by using exponential function

$$\matrix{
{\cosh (x) = {{\exp (x) + \exp ( – x)} \over 2}} \hfill & {\sinh (x) = {{\exp (x) – \exp ( – x)} \over 2}} \hfill \cr
} $$

and then defining the other hyperbolic functions. Consequently, one can define the inverse-hyperbolic functions.

$4.$ Finally, the author gives three ways for introducing the trigonometric functions.

$\qquad 4.1-$ Introduces the $\sin x$ and $\cos x$ functions by the following properties

\begin{align*}{}
\text{(a)}\,\,& \text{The domain of $\sin x$ and $\cos x$ is $\mathbb R$} \\
\text{(b)}\,\,& \cos 0 = \sin \frac{\pi}{2}=0,\, \cos \pi=-1 \\
\text{(c)}\,\,& \cos (y-x)= \cos y \cos x + \sin y \sin x \\
\text{(d)}\,\,& \text{For $0 \le x \le \frac{\pi}{2}$ we have $0 \le \cos x \le \frac{\sin x}{x} \le \frac{1}{\cos x}$}
\end{align*}

$\qquad 4.2-$ Using formal geometric definitions employing the unit circle.

$\qquad 4.3-$ Introducing $\sin x$ and $\cos x$ functions by their Taylor series.

and then defining the other trigonometric ones and the inverse-trigonometric functions.

In my point of view, the approach is good but it seems a little disconnected as the relation between the trigonometric and exponential functions is not illustrated as the author insisted to stay in the real domain when introducing these functions. Also, exponential and power functions are just defined for positive real numbers $a$ and $x$ while they can be extended to negative ones.


Questions

$1.$ How many other approaches are used for this purpose? Are there many or just a few? Is there some list for this?

$2.$ Would you please explain just one of the other heuristic ways to introduce the elementary functions analytically with appropriate details?


Notes

  • Historical remarks are welcome as they provide a good motivation.

  • Answers which connect more advanced (not too elementary) mathematical concepts to the development of elementary functions are really welcome. As nice example of this is the answer by Aloizio Macedo given below.

  • It is hard to choose the best answer between these nice answers so I decided to choose none. I just gave the bounties to the ones that are more compatible with the studies from high-school. However, please feel free to add new answers including your own ideas or what you may think that is interesting so we can have a valuable list of different approaches recorded here. This can serve as a nice guide for future readers.


Useful Links

  • Here is a link to a paper by W. F. Eberlein suggested in the comments. The paper deals with introducing the trigonometric functions in a systematic way.

  • There are six pdfs created by Paramanand Singh who has an answer below. It discusses some approaches for introducing logarithmic, exponential and circular functions. I have combined them all into one pdf which can be downloaded from here. I am sure that it will be useful.

Answer

There are two canonical group structures in $\mathbb{R}$: $(\mathbb{R},+)$ and $(\mathbb{R}_{>0}, \cdot)$.

We search for the isomorphisms between the structures.

The identity is an automorphism on $(\mathbb{R},+)$ and the exponential is an isomorphism from $(\mathbb{R},+)$ to $(\mathbb{R}_{>0}, \cdot)$.

Furthermore, they are the only continuous such isomorphisms, once you fix a value on $1$.

So, we get:

The identity $id$ is the only continuous automorphism on $(\mathbb{R},+)$ such that $id(1)=1$ and the exponential $\exp$ is the only continuous isomorphism from $(\mathbb{R},+)$ to $(\mathbb{R}_{>0}, \cdot)$ such that $\exp(1)=e$.

From these, all other elementary functions follow.


Summarizing, in order to obtain the elementary functions, you only need the algebraically (and analytic, since we must suppose continuity) interesting ones.


Expanding a bit, if you don’t want to be allowed to consider exponentiation to complex numbers, reaching $\sin$ and $\cos$ from $\exp$ and the identity may be troublesome. I will therefore provide another way of introducing $\sin$ and $\cos$. Ironically, it involves “complex” ideas.

Consider $C^{\infty}(\mathbb{R})$, and $X: C^{\infty}(\mathbb{R}) \rightarrow C^{\infty}(\mathbb{R})$ given by
$$f \mapsto f’.$$
Consider also the identity function $I$ on $C^{\infty}(\mathbb{R})$. We have that $e^{x}$ and $e^{-x}$ are the two “moral” solutions (more precisely, they form a basis for the solutions) of
$$X^2-I=0.$$
It is natural to search for the solutions of
$$X^2+I=0.$$
(Seems familiar?) We then have that the solutions with appropriate initial conditions are $\sin$ and $\cos$.

Attribution
Source : Link , Question Author : Hosein Rahnama , Answer Author : Aloizio Macedo

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