Proposition 3

The function $\mathbf{e}(x)$ has a derivative equal to $2\pi i \mathbf{e}(x)$ at every point of $\mathbf{R}$, where $\pi$ is a constant $> 0$.

Now, th. 1 of III, p. 51, applied to the case where E is the field $\mathbf{C}$ of complex numbers, yields the relation $\mathbf{e}'(x) = \mathbf{e}'(0) \mathbf{e}(x)$; moreover, since $\mathbf{e}(x)$ has constant euclidean norm, $\mathbf{e}'(x)$ is orthogonal to $\mathbf{e}(x)$ (I, p. 7, Example 3); one thus has $\mathbf{e}'(0) = \alpha i$, with $\alpha$ real. Since $\sin_1 x$ is increasing on $[-\frac{1}{4}, \frac{1}{4}]$ its derivative for $x = 0$ is $\geqslant 0$, so $\alpha \geqslant 0$, and since $e(x)$ is not constant, $\alpha > 0$; it is conventional to denote the number $\alpha$ so obtained by $2\pi$.

In §2 (III, p. 23) we shall show how one can calculate arbitrarily close approximations to the number $\pi$.

We thus have the formula

One sees that this formula simplifies when $a = 2\pi$; this is why one uses the circular functions relative to base $2\pi$ exclusively in Analysis; we agree to omit the base in the notation for these functions; unless mentioned expressly to the contrary, the notations $\cos x$, $\sin x$ and $\tan x$ denote $\cos_{2\pi} x$, $\sin_{2\pi} x$ and $\tan_{2\pi} x$ respectively.

With these conventions, and $a = 2\pi$, formula (12) can be written

which is equivalent to

from which one deduces

Besides the three circular functions $\cos x$, $\sin x$ and $\tan x$ one also uses, in numerical work, the three auxiliary functions: cotangent, secant and cosecant, defined by the formulae

Recall (Gen. Top., VIII, p. 109) that the angle corresponding to the base $2\pi$ is called the radian.