Introduction
In Vector angles: geometric intuition and algebraic definition, we defined and described the group of Euclidean plane vector angles algebraically, using an equivalence relation on unit vectors. Just as we can measure lengths, we learn at primary school that we can measure angles. Here we present the basics of measuring vector angles, whose mathematical unit is the radian (see also The trigonometric circle: where Pythagoras meets Thales), using real and complex analysis.
1 Measuring angles using the circular exponential
If \(\vec\alpha=[(\vec u,\vec v)]\) is a vector angle, so that $\vec u,\vec v$ are two unit vectors, we know that there exists a unique vector rotation $r$ such that $r(\vec u)=\vec v.$ By definition, the angle $\vec\alpha$ is the angle of the rotation $r$, which is written in the form $r(x,y)=(ax+by,-bx+ay)$ for any vector $(x,y)\in\mathbb R^2$, with $a^2+b^2=1$. Remember that the point $(a,b)$, seen as a unit vector $\vec{u’}$, provides the standard representation of the angle $\vec\alpha=[(\vec i,\vec{u’})]$, with $\vec i=(0,1)$.
Now, since the point $(a,b)$ is thus on the trigonometric circle, it is the image by the circular exponential function $e(t)=\exp(it)$ of a real number $t$. Indeed, for reasons linked to its continuity and to the elementary study of cosine and sine functions, this function from $\mathbb R$ into $S^1$ is in fact surjective. In other words, there exists a real number $t$ such that $a+ib=(a,b)=\exp(it).$ By definition, a real number $t$ such that $\exp(it)=(a,b)$ is a measure of the vector angle $\vec\alpha$.
2 A measure ‘ to within $2\pi$’ according to the definition of $\pi$.
Such a real number $t$ therefore has the following property: we have $\exp(it)=(a,b)=(\cos t,\sin t).$ Now, we can show that there is a smallest real number $t>0$ such that $\cos t=0$; if we denote this number $a$, we can define the number $\pi$ as $2a$. For any real number $t$ we then have $\cos(t+2\pi)=\cos(t)$ and $\sin(t+2\pi)=\sin(t)$ – we say that the number $2\pi$ is the period of the cosine and sine functions. So we can still write $(a,b)=\exp(it)=(\cos(t),\sin(t))=(\cos(t+2\pi),\sin(t+2\pi))=\exp(i(t+2\pi))$!
In other words, if $t$ is a measure of the angle $\vec\alpha$ – obtained by $e(t)$ thanks to the point $(a,b)$ associated with the rotation $r$ with angle $\vec\alpha$ – then $t+2\pi$ is another measure of the same angle. Intuitively, since the function $e(t)$ ‘wraps’ the real line around the trigonometric circle, and the perimeter of the circle is $2\pi$, we don’t change the angle by adding $2\pi$, or an integer multiple of $2\pi$, to its measure. In short, there are several possible measures of a vector angle.
3 The ‘kernel’ of the circular exponential
This can be interpreted through the properties of the circular exponential $e(t)=\exp(it)$. We mentioned earlier that $e(t)$ is a group homomorphism from $(\mathbb R,+)$ into $(S^1,\times)$, transforming the addition of real numbers into the complex multiplication of elements of $S^1$. It can be shown that the period $2\pi$ of $\cos$ and $\sin$ is the smallest real number $t>0$ such that $e(t)=1$, or such that $\exp(it)=1$. It follows that if $t\in\mathbb R$ and $k\in\mathbb Z$, we can write $e(t+2k\pi)=\exp(i(t+2k\pi))=\exp(it).\exp(2ik\pi)=\exp(it).\exp(2i\pi)^k=\exp(it).1^k=\exp(it)$. By studying the subgroups of $(\mathbb R,+)$ we can establish that the real numbers $t$ such that $e(t)=1$ are exactly numbers of the form $t=2\times k\times \pi$, where $k\in\mathbb Z$ is an integer.
The set of integer multiples of $2\pi$, i.e. real numbers of the form $2k\pi$, for $k\in\mathbb Z$, is denoted $2\pi \mathbb Z$. In layman’s terms, we say that the ‘kernel’ of $e(t)$ – i.e. the set of real numbers $t$ such that $\exp(it)=1$ – is the group $(2\pi\mathbb Z,+)$. Two measures of the same angle then differ by exactly one integer multiple of $2\pi$. To explain this precisely, we can define a new equivalence relation between two real numbers $t$ and $s$ by stating that $t$ is equivalent to $s$ if there exists $x\in 2\pi\mathbb Z$ such that $t-s=x$, i.e. if there exists $k\in\mathbb Z$ such that $t=s+2k\pi$. We then say that $t$ and $s$ are congruent modulo $2\pi$.
4 The isomorphism between $\mathbb R/2\mathbb Z$ and the group of angles
As with vector angles, we can add two classes $[t]$ and $[s]$ for this relation according to the equality $[t]+[s]=[t+s]$, and describe the opposite of a class $[t]$ as $[-t]$; the ‘zero’ of the addition is then $[0]$, i.e. the set $2\pi\mathbb Z$…. The set of equivalence classes of real numbers for this relation thus forms a group which we call the quotient group of $(\mathbb R,+)$ by $(2\pi\mathbb Z,+)$ and which we denote $(\mathbb R/2\pi\mathbb Z,+)$. By definition of the kernel of a homomorphism, this group is isomorphic to – i.e. mathematically indistinguishable from – the group $(S^1,\times)$. We thus obtain, by an isomorphism $f:[t]\in\mathbb R/2\pi\mathbb Z\mapsto \exp(it)\in S^1$, a fourth representation of the same essential mathematical object: the trigonometric circle $(S^1,\times)$, the group of vector rotations $(\mathcal R,\circ)$, the group of vector angles $(\mathcal A,+)$.
If we combine $f$ with the isomorphism $g: (a,b)\in S^1\mapsto \vec\alpha\in \mathcal A$, where $\vec\alpha=[(\vec i,\vec u)]$ is the angle of vectors associated with the pair $(\vec i=(1,0),\vec u=(a,b))$, we obtain the isomorphism $h=g\circ f: (\mathbb R/2\pi\mathbb Z,+)\to (\mathcal A,+)$. It is therefore a bijection which associates with the class $[t]$ of a real number $t$ modulo $2\pi$ the angle of vectors of which $t$ is a measure, and of which all the measures are the elements of $[t]$, i.e. the numbers $t+2k\pi$, for $k\in\mathbb Z$. The inverse bijection $(h^{-1}=g\circ f)^{-1}=f^{-1}\circ g^{-1}:\mathcal A\to\mathbb R/2\pi\mathbb Z$ therefore associates all its measures with a vector angle.
5 The principal measure of an angle and the radian
We can thus define ‘The’ measure of a vector angle, as an equivalence class of real numbers modulo $2\pi\mathbb Z$, and each element of this class is a possible measure. Since $h^{-1}$ is also a group homomorphism, it preserves the ‘sum’: if $\vec\alpha$ and $\vec\beta$ are two vector angles, the measure $h^{-1}(\vec\alpha+\vec\beta)$ is the sum $h^{-1}(\vec\alpha)+h^{-1}(\vec\beta)$. Similarly, the measure of $-\vec\alpha$ is $-h^{-1}(\vec\alpha)$, and the measure of the zero angle $\vec 0$ is $[0]$, or $2\pi\mathbb Z$.
Another way of associating ‘a’ measure with an angle $\vec\alpha$ is to choose the measure that lies in the interval $[0,2\pi[$. Indeed, if $h(\vec\alpha)=[t]$, there exists a single real number $s\in [0,2\pi[$ such that $[s]=[t]$, in other words such that $s$ is a measure of $\vec\alpha$. This number, a sort of ‘standard measure’ of the angle, corresponds in fact to the length of the arc of a circle between the point $I=(0,1)$ and the point $(a,b)$ of $S^1$ determined by the angle $\vec\alpha$ and described in the trigonometric orientation. This is why the radian is the ‘mathematical unit of measurement’ for angles, $1$ radian corresponding to the measure of an angle determining an arc of a circle with length $1$.
