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The wikipedia page for Elementary Function defines it as follows.

In mathematics, an elementary function is a function of one variable which is the composition of a finite number of arithmetic operations (+ – × ÷), exponentials, logarithms, constants, and solutions of algebraic equations (a generalization of nth roots).

It goes on to say the following about trigonometric functions.

The elementary functions include the trigonometric and hyperbolic functions and their inverses, as they are expressible with complex exponentials and logarithms.

However, the trigonometric function definitions I have seen do not appear to satisfy the 'Elementary Function' requirements. Here are two definitions I've seen alongside my objections on why the definition does not appear to be Elementary.

The relevant section of the Trigonometric Functions wikipedia page includes the following definitions of $\sin$ and $\cos$.

$$\cos x = \operatorname{Re}(e^{i x})$$ $$\sin x = \operatorname{Im}(e^{i x})$$

  • My objection: the definition uses the $\operatorname{Re}$ and $\operatorname{Im}$ operators, which do not appear to be Elementary Functions themselves according to the definition.

Also, the Wolfram Research Functions site lists the following sum as a definition for $\sin$.

$$\sin z = \sum_{k=0}^\infty \frac{(-1)^k z^{2 k + 1}}{(2 k + 1)!}$$

  • My objection: an infinite number of operations are used, which is in violation of the definition.

Is the wikipedia article correct in asserting that the trigonometric functions are indeed Elementary?

If so, how can one construct the trigonometric functions using a finite number of compositions of elementary operations?

A side note:

In addition to the objections listed above, it seems strange that $e$ -- being transcendental and therefore not constructible using finite compositions of the elementary operations -- could be used as a constant in the construction of other elementary operations.

I would be interested in hearing perspective on whether or why transcendental constants are permitted in the construction of Elementary Functions.

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    $\begingroup$ Possible duplicate of What makes elementary functions elementary? $\endgroup$
    – Mr. T
    Oct 26, 2017 at 20:51
  • $\begingroup$ To summarize Mr. T's comment as it relates to your question, the functions $e^x$ and $\log(x)$ are also considered elementary, and all trig functions can be written in terms of these functions. $\endgroup$
    – Carl Schildkraut
    Oct 26, 2017 at 20:57
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    $\begingroup$ It might make things clearer to write $\cos z=\frac{e^{iz}+e^{-iz}}{2}$ and $\sin z=\frac{e^{iz}-e^{-iz}}{2i}$. $\endgroup$
    – carmichael561
    Oct 26, 2017 at 21:00
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    $\begingroup$ The first sentence of a Wikipedia article doesn't necessarily provide a technically precise definition, but rather "should informally define or describe the subject." en.wikipedia.org/wiki/Wikipedia:Manual_of_Style/… $\endgroup$
    – PersonX
    Oct 27, 2017 at 13:12

2 Answers 2

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Let $\mathbb{F}$ be the field of real numbers $\mathbb{R}$ or the field of complex numbers $\mathbb{C}$. Let $\mathcal{P}(\mathbb{F})$ the set of $\mathbb{F}$-valued power functions ()functions of the form $x \mapsto x^{\alpha}$), $\mathcal{EL(\mathbb{F})}$ the set of $\mathbb{F}$-valued exponential and logarithmic functions and $\mathcal{T}(\mathbb{F})$ the set of $\mathbb{F}$-valued trigonometric functions. I think the class of elementary functions as the minimal $\mathcal{E}=(\mathcal{E},+,\cdot, \circ)$ set such that $\mathcal{P}(\mathbb{F}),\mathcal{EL(\mathbb{F})},\mathcal{T}(\mathbb{F}) \subseteq \mathcal{E}$ and $\mathcal{E}$ is closed under addition, substraction, multiplication, division and compsoition of functions (so the set if polynomials $\mathbb{F}[x] \in \mathcal{E}$ and the set of hyperbolic functions $\mathcal{H}(\mathbb{F})$ also lives in $\mathcal{E}$).

If you think of the complex exponential function $\exp:z \to e^z$ as an elementary function then $\sin(z)=\dfrac{e^{iz}-e^{-iz}}{2}$ and $\cos(z)=\dfrac{e^{iz}+e^{-iz}}{2}$, so $\sin, \cos \in \mathcal{E}$

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a) The Functions

There are several different ways to define a given function. What we call an elementary function (according to differential algebra) was characterized by Liouville with means of algebra as obtainable "in a finite number of steps by performing algebraic operations and taking exponentials and logarithms" ([Ritt 1925]). Liouville's method are functions "in finite terms".
[Hardy 1905] contains a decription of the elementary functions in chapter II. Elementary functions and their classification.

Each of the elementary standard functions (sin, cos, tan, cot, sec, csc, sinh, cosh, tanh, coth, sech, csch, arcsin, arccos, arctan, arccot, arcsec, arccsc, arcsinh, arccosh, arctanh, arccoth, arcsech, arccsch) can be brought to this expln-form. See e.g. the Wikipedia articles for the single functions or [Abramowitz/Stegun 1970].

Considering $\sin(z)=\dfrac{e^{iz}-e^{-iz}}{2}$ and $\cos(z)=\dfrac{e^{iz}+e^{-iz}}{2}$, these trigonometric functions belong quite obviously to the set of elementary functions.

b) The Constants

No conditions are set for the constants. Hardy ([Hardy 1905]) writes: "It is hardly necessary to remark that it is in no way involved in the definition of a rational function that these constants should be rational or algebraical* or real $\textit{numbers}$. Thus $$\frac{x^2+x+i\sqrt{2}}{x\sqrt{2}-e}$$ is a rational function."

We could define elementary functions by allowing only elementary numbers or closed-form numbers as constants. But defining elementary functions implicitly or by integration (with integration constants) needs also non-closed-form complex constants.

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[Abramowitz/Stegun 1970] Abramowitz, M.; Stegun, I.: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. National Bureau of Standard 1970
[Hardy 1905] Hardy, G. H.: The Integration of Functions of a Single Variable. Cambridge University Press, 1905
[Ritt 1925] Ritt, J. F.: Elementary functions and their inverses. Trans. Amer. Math. Soc. 27 (1925) (1) 68-90

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