Lemma 1

Let $u$ be a direct similitude of $E$; all the bivectors of the form $x \wedge u(x)$ belong to the same closed half-line of $\overset{2}{\wedge} E$.

The case where $u$ is a homothety is trivial. In the contrary case one has $x \wedge u(x) \neq 0$ for all $x \neq 0$; let $x, y$ be two vectors of $E$ ($x \neq 0, y \neq 0$); there exists $\varphi \in S^+$ such that $y = \varphi(x)$, whence $y \wedge u(y) = \varphi(x) \wedge u \varphi(x) = \varphi(x) \wedge \varphi(u(x)) = (\det \varphi)(x \wedge u(x))$; taking an orthonormal basis of $E$, one sees that $\det \varphi$ is positive (Proposition 1 b)); whence our assertion.

This being so, among the two generators $\omega$ of $\mathbf{A}(\Phi)$ such that $\omega^2 = -1$, there exists one and only one such that the bivector $x \wedge \omega(x)$ is positive for all $x \in E$. It is this generator that we shall choose in order to define the functions $c_w, s_w$ and $t_w$ (No. 2). Let $h$ and $h'$ be the canonical bijections defined above from the group $\mathfrak{A}_0$ of angles of lines onto $S^+/H$ and from the group $\mathfrak{A}$ of angles of half-lines onto $O^+$. The composite mappings $t_w \circ h$ of $\mathfrak{A}_0$ into the projective field $\tilde{\mathfrak{A}}$, $c_w \circ h'$ and $s_w \circ h'$ of $\mathfrak{A}$ into the field $A$ are denoted respectively by $\operatorname{tg}$, $\cos$ and $\sin$, and are called the tangent, cosine and sine functions. The mapping $\varphi \to 1/\operatorname{tg} \varphi$ of $\mathfrak{A}_0$ into $\tilde{\mathfrak{A}}$ is denoted by $\operatorname{cotg}$ and is called the cotangent function. One says that the sine, tangent and cotangent functions are the trigonometric functions. The composite mappings $\operatorname{tg} \circ p$ and $\operatorname{cotg} \circ p$, where $p$ denotes the canonical homomorphism of $\mathfrak{A}$ onto $\mathfrak{A}_0$ (Remark 2 above), are again denoted by $\operatorname{tg}$ and $\operatorname{cotg}$ by abuse of language.

Formulas (2), (8), (3), (4), (5) and (7) of No. 2 give, since one has here $\delta = -1$

for $\varphi, \varphi'$ in $\mathfrak{A}_0$;

for $\theta, \theta'$ in $\mathfrak{A}$. On the other hand one has, by definition or as an easy consequence of the preceding formulas:

for $\theta \in \mathfrak{A}$.

Given two non-zero vectors $x, y$ of $E$, the angle of these two vectors (taken in this order) is defined to be, and is denoted by $\widehat{(x, y)}$, the angle of the half-lines to which they belong. For every vector $x$ of $E$ the length of $x$ is defined to be, and is denoted by $|x|$, the element $\Phi(x, x)^{1/2}$ of $A$.