Are the trigonometric functions really Elementary Functions?

Question

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?

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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.

Answer 1

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 $\sin, \cos \in \mathcal{E}$

Answer 2

Yeah, the trigonometric functions are elementary functions. By Euler’s formula one can write $$e^{i\theta}=\cos(\theta)+i\sin(\theta).$$ So you can right $$\sin(\theta)=\frac{1}{2}(e^{i\theta}-e^{-i\theta})$$ and $$\cos(\theta)=\frac{1}{2}(e^{i\theta}+e^{-i\theta}).$$ The exponential function (even the complex one) is an elementary functions, and since linear combinations of elementary functions are elementary functions, sine and cosine (and all the other trigonometric functions) are indeed elementary functions.