2. Angular measure, trigonometric functions

1. The multiplicative group u

2. Angles

Since the field $\mathbf{R}$ is ordered, we may orient the real number plane $\mathbf{R}^2$ by taking $e_1 \wedge e_2$ as positive bivector ($e_1, e_2$ being the vectors of the canonical basis). In the oriented real number plane $\mathbf{R}^2$ (identified with $\mathbf{C}$ in what follows) we can then define the angle $(\Delta_1, \Delta_2)$ of an arbitrary pair of rays $(\Delta_1, \Delta_2)$ with origin o (*). The set $\mathfrak{A}$ of all angles has the structure of an abelian group (written additively) defined by

so that, in particular, $(\Delta_1, \Delta_1) = 0$ and $(\Delta_2, \Delta_1) = - (\Delta_1, \Delta_2)$.

The flat angle $\varpi$ is the solution $\neq 0$ of the equation $2\theta = 0$ in $\mathfrak{A}$; it is the angle which the negative real semi-axis makes with the positive real semi-axis.

If $z$ is any non-zero complex number, the amplitude (or argument) of $z$, denoted by $\mathrm{Am}(z)$, is the angle which the ray through $z$ with origin o makes with the positive real semi-axis. The mapping $z \to \mathrm{Am}(z)$ is a homomorphism of the multiplicative group $\mathbf{C}^*$ onto the additive group $\mathfrak{A}$, and therefore we have

The angle $\delta = \mathrm{Am}(i)$ is called the positive right angle; it is one of the solutions in $\mathfrak{A}$ of the equation $2\theta = \varpi$, the other one being $- \delta = \delta + \varpi$.

The homomorphism $z \to \mathrm{Am}(z)$, restricted to the subgroup $\mathbf{U}$ of $\mathbf{C}^*$, is an isomorphism of the group structure of $\mathbf{U}$ onto that of $\mathfrak{A} \ (***)$; if we

(*) We know from algebra that an equivalence relation is defined on the set of all pairs $(\Delta_1, \Delta_2)$ of rays with origin o by considering two pairs $(\Delta_1, \Delta_2)$ and $(\Delta'_1, \Delta'_2)$ as equivalent if there exists a rotation which transforms $\Delta_1$ into $\Delta'_1$ and $\Delta_2$ into $\Delta'_2$ simultaneously; the angle of the pair $(\Delta_1, \Delta_2)$, or the angle $\Delta_2$ makes with $\Delta_1$, is then by definition the equivalence class of the pair $(\Delta_1, \Delta_2)$.

(**) This is because every ray with origin o meets the circle $S_1$, since the field $\mathbf{R}$ is Pythagorean.

use this isomorphism to transport the topology of $\mathbf{U}$ to the group $\mathfrak{A}$, the latter group becomes a compact topological group, and the mapping $z \to \mathrm{Am}\,(z)$ of $\mathbf{C}^*$ onto $\mathfrak{A}$ is a strict morphism of the topological group $\mathbf{C}^*$ onto the topological group $\mathfrak{A}$.

Let us denote by $\theta \to f(\theta)$ the isomorphism of $\mathfrak{A}$ onto $\mathbf{U}$ which is the inverse of the isomorphism $z \to \mathrm{Am}\,(z)$ of $\mathbf{U}$ onto $\mathfrak{A}$. By definition $\Re(f(\theta))$ is denoted by $\cos \theta$ and is called the cosine of the angle $\theta$; $\Im(f(\theta))$ is denoted by $\sin \theta$ and is called the sine of the angle $\theta$. These functions are continuous on the topological group $\mathfrak{A}$, and satisfy the following relations (loc. cit.), which are immediate consequences of the definitions above:

By definition, the tangent of an angle $\theta \in \mathfrak{A}$ is defined, whenever $\cos \theta \neq 0$, to be $\sin \theta / \cos \theta$ (loc. cit.) and is denoted by $\tan \theta$; it is a continuous function, which extends by continuity to $\tilde{\mathbf{R}}$ (Chapter VI, § 3, no. 4) by taking the value $\infty$ for the angles $\delta$ and $-\delta$. We have $\tan (\theta + \varpi) = \tan \theta$. The cotangent of $\theta$, denoted by $\cot \theta$, is the element of $\tilde{\mathbf{R}}$ equal to $1 / \tan \theta$.

Note that, if $\mathrm{Am}\,(z) = \theta$, we have $z = |z| (\cos \theta + i \sin \theta)$; this expression is called the trigonometric form of the complex number $z \neq 0$.

3. Angular measure

By Theorem 1 of no. 1, the topological group $\mathfrak{A}$ of angles is isomorphic to $\mathbf{T}$. Every strict morphism of $\mathbf{R}$ onto $\mathfrak{A}$ can be obtained by composing the isomorphism $z \to \mathrm{Am}\,(z)$ of $\mathbf{U}$ onto $\mathfrak{A}$ with a strict morphism of $\mathbf{R}$ onto $\mathbf{U}$; if we put $\hat{x}(x) = \mathrm{Am}\,(e(x))$, every strict morphism of $\mathbf{R}$ onto $\mathfrak{A}$ is therefore of the form $x \to g(x/a)$ ($a \neq 0$). Given a real number $a > 0$, fixed once and for all, every angle $\theta$ corresponds, by the homomorphism $x \to \hat{x}(x/a)$, to a class of real numbers mod $a$ (i.e., an element of $\mathbf{Z}/a\mathbf{Z}$) which is called the measure of $\theta$ relative to the base $a$; by abuse of language, every real number in this class is also called a measure of $\theta$; the angle $\hat{x}(x/a)$ is called the angle with measure $x$ (relative to the base $a$. If $x$ is a measure of $\theta$ and $x'$ a measure of $\theta'$ (relative to the same base) then $x + x'$ is a measure of $\theta + \theta'$, and $-x$ is a measure of $\theta$. The principal measure of an angle (relative to the base $a$) is that one of its measures which lies in the interval $[0, a[$.

Choice of a base $a$. We restrict ourselves always to bases $a > 1$. To each $a > 1$ corresponds an angle $\omega = \frac{\pi}{a}$ whose principal measure is 1, and which is called the unit of angular measure relative to the base $a$; conversely, to each angle $\omega \neq 0$ there corresponds a unique $a > 1$ such that $\frac{\pi}{a} = \omega$, so that knowledge of the unit of angular measure determines the base $a > 1$ entirely.

In numerical calculations one usually takes either $a = 360$ or $a = 400$; the corresponding unit of angular measure is called the degree ($a = 360$) or the grade ($a = 400$).

In analysis, and indeed in all branches of mathematics where numerical calculation is not involved, the base $a$ defined by the condition

is universally used; this base is denoted by $2\pi$. The corresponding unit of angular measure is called a radian, and the measure is called radian measure; with the definition of $e^z$ for complex $z$ mentioned earlier, we have $e(x) = e^{2\pi ix}$ for all $x \in \mathbf{R}$.

Once the base $a$ has been chosen, when one speaks of an angle one usually means a measure of this angle relative to the base $a$; this abuse of language has no drawback provided (as is always the case when numerical calculations are not involved) the base $a$ remains fixed throughout, and provided that one remembers that two real numbers which are congruent mod $a$ correspond to the same angle.

For example, what is usually understood by the amplitude of a complex number $z \neq 0$ is a radian measure of this angle, determined by conventions which will depend on the question under consideration; once these conventions have been made, the measure of the amplitude thus chosen is denoted by Am $(z)$.

4. Trigonometric functions

If we compose the functions $\cos \theta, \sin \theta, \tan \theta, \cot \theta$ (defined on $\mathscr{A}$) with the homomorphism $x \to \frac{\pi}{a}(x/a)$ of $\mathbf{R}$ onto $\mathscr{A}$, the functions

so obtained are called respectively the cosine, sine, tangent and cotangent of the number $x$ relative to the base $a$, and are written $\cos_a x$, $\sin_a x$, $\tan_a x$, $\cot_a x$. The mapping $x \to \cos_a x + i \sin_a x$ is the composition of $\theta \to \cos \theta + i \sin \theta$ and $x \to \tilde{\sigma}(x/a)$, so that, from the definition of $\cos \theta$ and $\sin \theta$ in no. 2, we have the identity

which is equivalent to

and also, by (4), equivalent to

Hence the identities

The only trigonometric functions which arise in branches of mathematics where numerical calculation is not involved are those relative to the base $2\pi$ referred to above; these functions are denoted simply by $\cos x$, $\sin x$, $\tan x$, $\cot x$ in place of $\cos_{2\pi} x$, $\sin_{2\pi} x$, $\tan_{2\pi} x$, $\cot_{2\pi} x$. For the purposes of numerical calculation, there are tables of the trigonometric functions corresponding to the bases $a = 360$ and $a = 400$; and the formulae (6) allow us to deduce the values of the trigonometric functions relative to any other base.

The relations recalled earlier between the cosines and sines of angles evidently give rise to the same relations between the cosines and the sines of the numbers which measure these angles; in particular, we have

The functions $\cos_a x$ and $\sin_a x$ are continuous on $\mathbf{R}$, and are periodic with period $a$; moreover, $a$ is a principal period of these functions, for the relation $\cos_a x = \cos_a y$ implies that either $\sin_a x = \sin_a y$ or

so that either

and similarly

is equivalent to either $x \equiv y \pmod{a}$ or $x + y \equiv \frac{1}{2} a \pmod{a}$.

It follows from this that $\cos_a x$ never takes the same value twice in the interval $[0, \frac{1}{2} a]$; hence, when restricted to this interval, it is a bijective mapping of this interval onto the interval $[-1, +1]$. Since $\cos_a 0 = 1$ and $\cos_a (\frac{1}{2} a) = -1$, $x \to \cos_a x$ is a strictly decreasing mapping of $[0, \frac{1}{2} a]$ onto $[-1, 1]$ (Chapter IV, § 2, no. 6, Theorem 5 and Remark). We have $\cos_a x = 0$ for $x = a/4$, $\cos_a x > 0$ for $0 \leq x < a/4$, $\cos_a x < 0$ for $a/4 < x \leq a/2$. Since $\cos_a (-x) = \cos_a x$ we can deduce how $\cos_a x$ varies in the interval $[- \frac{1}{2} a, 0]$, and hence throughout the whole of $\mathbf{R}$ by periodicity (Fig. 8). Since $\sin_a x = -\cos_a(x + a/4)$ we can also deduce how $\sin_a x$ varies in $\mathbf{R}$ (Fig. 8).

!Graph showing y = sin_a x and y = cos_a x

Figure 8.

The function $\tan_a x$ is a continuous mapping of $\mathbf{R}$ onto $\tilde{\mathbf{R}}$; it takes the value $\infty$ for the values $\frac{1}{4} a + \frac{1}{2} k a \ (k \in \mathbf{Z})$. Since $\frac{1}{2} a$ is a period of $\tan_2 x$, it is a principal period. In the interval $[0, \frac{1}{4} a]$, $\sin_a x$ increases from 0 to 1, $\cos_a x$ decreases from 1 to 0, and therefore $\tan_a x$ is strictly increasing in $[0, \frac{1}{4} a[$ and maps this interval onto $[0, +\infty[$; it follows that $\tan_a x$ is strictly increasing in the interval

!A graph showing curves labeled y and x, with axes marked -a/4, 0, a/4, a/2, 3a/4

Figure 9.

]— $\frac{1}{4} a, + \frac{1}{4} a$[. and is a homeomorphism of this interval onto $\mathbf{R}$ (Fig. 9).

5. Angular sectors

Given two distinct closed rays $\Delta_1, \Delta_2$ with origin o, let $x$ be the principal measure of the angle $(\widehat{\Delta_1, \Delta_2})$ (relative to a base $a$, chosen once and for all). The union of the closed (resp. open) rays $\Delta$ with origin o which are such that the principal measure $y$ of the angle $(\widehat{\Delta_1, \Delta})$ satisfies $0 \leq y \leq x$ (resp. $0 < y < x$) is the closed (resp. open) angular sector S with origin $\Delta_1$ and extremity $\Delta_2$, as defined in algebra. For by means of a rotation we can always reduce to the case where S does not contain the ray through the point — 1. If $\alpha$ and $\beta$ are then the angles which $\Delta_1$ and $\Delta_2$ respectively make with the positive real semi-axis, then the closed angular sector S is the union of the closed half-lines $\Delta$ which make an angle $\theta$ with the positive real semi-axis such that $\tan \frac{1}{2} \alpha \leq \tan \frac{1}{2} \theta \leq \tan \frac{1}{2} \beta$. Now if $u, v, t$ are the measures of $\alpha, \beta, \theta$ respectively which lie in the interval ]— $\frac{1}{2} a, + \frac{1}{2} a$[, these inequalities are equivalent to $\tan_a \frac{1}{2} u \leq \tan_a \frac{1}{2} t \leq \tan_a \frac{1}{2} v$; and since $\tan_a x$ is an increasing function in the interval ]— $\frac{1}{4} a, + \frac{1}{4} a$[, they are also equivalent to $u \leq t \leq v$, or to $0 \leq t - u \leq v - u$; since $x = v - u, y = t - u$, the result is proved for closed angular sectors, and the proof for open ones is similar.

A closed angular sector is a closed set in $\mathbf{R}^2$, and the open angular sector with the same origin and the same extremity is its interior in $\mathbf{R}^2$

Figure 10.

(Chapter VI, § 2, no. 3, Proposition 3). The angle $(\widehat{\Delta_1, \Delta_2})$, with principal measure $x$, is called the angle of the sector S; S is said to be salient if $x < \frac{1}{2} a$, flat (or a closed half-plane) if $x = \frac{1}{2} a$; re-entrant if $x > \frac{1}{2} a$. A salient angular sector is acute if $x < \frac{1}{4} a$, right if $x = \frac{1}{4} a$, obtuse if $x > \frac{1}{4} a$. The bisector of the sector S is the ray $\Delta$ which makes an angle $y = \frac{1}{2} x$ with $\Delta_1$.

Two distinct closed rays $\Delta_1, \Delta_2$ determine two closed angular sectors; their union is the real plane $\mathbf{R}^2$, and their intersection is $\Delta_1 \cup \Delta_2$.

6. Crosses

We have also defined in algebra the cross of a pair of lines in a two-dimensional vector space over a maximal ordered field (*). This definition applies in particular to the real plane $\mathbf{R}^2$. The set $\mathscr{A}_0$ of all crosses has the structure of an abelian group (written additively) defined by

so that, in particular, $(\overline{D_1, D_1}) = 0$ and $(\overline{D_2, D_1}) = - (\overline{D_1, D_2})$.

The right cross $\delta_0$ is the solution $\neq 0$ of the equation $2\theta = 0$ in $\mathscr{A}_0$; it is the cross which the imaginary axis makes with the real axis.

(*) We recall that an equivalence relation is defined on the set of all non-isotropic pairs of lines $(D_1, D_2)$ by considering two pairs of lines $(D_1, D_2)$ and $(D'_1, D'_2)$ to be equivalent if there exists a direct similitude which transforms $D_1$ into $D'_1$ and $D_2$ into $D'_2$ simultaneously; the cross of the pair $(D_1, D_2)$ is then the equivalence class of the pair.

We define a canonical homomorphism $\varphi$ of the group $\mathfrak{A}$ of angles onto the group $\mathfrak{A}_0$ of crosses by making correspond to the angle which a ray $\Delta$ makes with the positive real semi-axis, the cross which the line D containing $\Delta$ makes with the real axis. A cross $\theta_0$ is the image under $\varphi$ of two angles $\theta$ and $\theta + \omega$; thus $\mathfrak{A}_0$ is isomorphic to the quotient of $\mathfrak{A}$ by the subgroup $\{0, \omega\}$. If we transport to $\mathfrak{A}_0$ the topology of the quotient group $\mathfrak{A}/\{0, \omega\}$ by means of the bijective homomorphism associated with $\varphi$, then $\mathfrak{A}_0$ becomes a compact topological group and $\varphi$ a strict morphism of $\mathfrak{A}$ onto $\mathfrak{A}_0$.

If we compose the homomorphism $\varphi$ of $\mathfrak{A}$ onto $\mathfrak{A}_0$ with the homomorphism $x \to \tilde{\sigma}(x/a)$ of $\mathbf{R}$ onto $\mathfrak{A}$, we have a homomorphism $x \to \tilde{\sigma}_0(x/a)$ of $\mathbf{R}$ onto $\mathfrak{A}_0$; every cross $\theta_0 \in \mathfrak{A}_0$ corresponds, under this homomorphism, to a class of real numbers mod $\frac{1}{2} a$, which is called the measure of the cross $\theta_0$ (relative to the base $a$); by abuse of language, every number in this class is called a measure of $\theta_0$, and that one which belongs to the interval $[0, \frac{1}{2} a[$ is called the principal measure of $\theta_0$; the cross $\tilde{\sigma}_0(x/a)$ is the cross of measure $x$. Every measure of $\theta_0$ is also a measure of one of the two angles $\theta, \theta + \omega$ whose image under the homomorphism $\varphi$ is $\theta_0$.

Here again, once the base $a$ has been chosen, when we speak of a cross we generally mean, by abuse of language, a measure of this cross relative to the base $a$.

Exercises

See the exercises for § 2.