The term `angle' in the international system of units
aa r X i v : . [ m a t h . HO ] J a n The term ‘angle’ in the international system of units
Michael P. Krystek
Physikalisch-Technische Bundesanstalt,Bundesallee 100, D-38116 Braunschweig, GermanyE-mail:
Abstract.
The concept of an angle is one that often causes difficultiesin metrology. These are partly caused by a confusing mixture of severalmathematical terms, partly by real mathematical difficulties and finallyby imprecise terminology. The purpose of this publication is to clarifymisunderstandings and to explain why strict terminology is important. It willalso be shown that most misunderstandings regarding the ‘radian’ can be avoidedif some simple rules are obeyed.
Keywords : International System of Units (SI), quantity, quantity value, magnitude,angle, angular magnitude, reference angle, radian.
1. Introduction
The concept of an angle is one that often causes difficulties in metrology. Theseare partly caused by a confusing mix of several mathematical terms, partly by realmathematical difficulties, and finally also by inaccurate terminology.Confusion arises e. g. from using the term ‘angle’ to refer to more or less connected,but nonetheless different mathematical objects, such as a pair of intersecting straightlines, a pair of rays with a common point, a cluster of rays bounded by two rays, acircular arc, a convex part of the plane, a sector of a circle, or a rotation.The mathematical difficulties are mainly due to the fact that statements thatapply to lengths cannot be directly transferred to angular magnitudes, because thestructures of angles and straight lines are different.Inaccurate terminology causes misunderstandings. The term ‘angle’ is used withdifferent meanings. In mathematics, the term ‘plane angle’, for example, refers toa geometric object (a figure, such as a straight-line segment). To geometric objectsat least one magnitude can be assigned (to a straight-line segment, for example, themagnitude called ‘length’). The magnitude assigned to an angle is often referred toas ‘angle’. Moreover, the numerical value associated with this magnitude is usuallyalso called ‘angle’. The usage of the same term with different meanings might beacceptable in daily life, but not in basic documents of metrology.In the following sections, we will deal in a mathematically rigorous way withthe various terms associated with the concept of an angle. The aim is to clarifymisunderstandings and to show why a strict terminology is important. he term ‘angle’ in the international system of units
2. The concept of a plane angle
Since ancient times, several definitions of the term ‘plane angle’ have been given. Butas we are not dealing with the history of mathematics, we go directly to the definitionthat is generally accepted in mathematics today (see Hilbert [1], for example).
Definition 1.
A ‘plane angle’, denoted by ∠ pq , is a geometric figure, formedby a pair of distinct rays p and q , called the ‘sides’ of the angle, that originatefrom a common point O , called the ‘vertex’ of the angle. Note
A plane angle may also be defined by a triple of points, then writtenin the form ∠ P OQ instead of ∠ pq , where O denotes the vertex, P any pointon the side p , and Q any point on the side q . pqPO Q s Figure 1.
A plane angle.
The definition of a plane angle does not require an ordered pair of rays, i. e. itdoes not distinguish between ∠ pq and ∠ qp , or between ∠ P OQ and ∠ QOP . FollowingEuclid [2], this is still common in Euclidean geometry of the plane, in which planeangles are not considered to be signed.The sectors of a circle created by an angle are usually convex figures. Thisproperty guarantees uniqueness by associating to an angle always the shorter of thetwo possible arcs of a circle around its vertex, which is denoted by s in Figure 1. Theconvexity of a sector excludes so-called ‘reflex angles’.Due to the distinctness of the sides, the arc s has always a non-zero length, i. e.the ’zero angle’ does not exist. In contrast to the definition given by Hilbert [1],however, the sides of an angle can lie on the same straight line (then they, of course,have opposite orientation), i. e. the ‘straight angle’ is not excluded.
3. Angular magnitudes
In geometry, equality is expressed by congruency. Two geometrical objects aresaid to be congruent if, and only if, each can be transformed into the other by anisometry. Congruency is an equivalence relation, which provides a partition of a setof geometrical objects into disjoint equivalence classes.To every angle a magnitude can be assigned. All congruent angles, i. e. thosewhich belong to the same equivalence class, have the same magnitude. Thus we canequate the magnitude of all congruent angles with their equivalence class. he term ‘angle’ in the international system of units Definition 2.
The equivalence class [ ∠ pq ] ∼ = := { ∠ uv | ∠ uv ∼ = ∠ pq } , i. e. the set of all angles congruent to ∠ pq , is called their ‘angular magnitude’. Note
The angle denoted here by ∠ pq is called a ‘representative’ of theequivalence class [ ∠ pq ] ∼ = . Any element of an equivalence class can be chosenas its representative.Traditionally, the ‘angular magnitude’ is often referred to just as ‘angle’. There isno harm to do this (except in standardization, where strictness is important), providedit is clear from the context whether reference is made to a geometric object or itsassigned magnitude. Most misunderstandings can be avoided if a clear distinctionbetween angles as geometric objects and their magnitudes is made.All angular magnitudes form a set, having the cardinality of a continuum, onwhich the structure of an ordered commutative semigroup can be established byintroducing a binary operation between the elements of the set. Theorem 1.
Let A be a set of angular magnitudes and let ⊕ : A × A 7→ A be a binary operation, defined by [ ∠ pr ] ∼ = ⊕ [ ∠ rq ] ∼ = = [ ∠ pq ] ∼ = , then ( A , ⊕ ) is an ordered, commutative and cancellative semigroup. Note 1
The operation ⊕ is called ‘addition of angular magnitudes’. Thisaddition is done modulo the magnitude of a straight angle, denoted by ̟ .Its result is called ‘sum of angular magnitudes’. Note 2
The meaning of the expression [ ∠ pr ] ∼ = ⊕ [ ∠ rq ] ∼ = = [ ∠ pq ] ∼ = is thefollowing: Take an angle ∠ pr of equivalence class [ ∠ pr ] ∼ = and an angle ∠ rq ofequivalence class [ ∠ rq ] ∼ = , both lying in the same plane and having a commonvertex O , and join them at their common side r , such that r lies between theother two sides p and q , respectively. The result is an angle ∠ pq , which is inthe equivalence class [ ∠ pq ] ∼ = . Note 3
The expression [ ∠ pr ] ∼ = ⊕ [ ∠ rq ] ∼ = = [ ∠ pq ] ∼ = may be written in theform α ⊕ β = γ . However, it should be kept in mind, that the Greek lettersdo not denote angles, but angular magnitudes.The ‘addition of angular magnitudes’ is often called just ‘addition of angles’. Thisshould be avoided, since the two terms have a different meaning.
4. Angular measures
Angular magnitudes are not numbers, but mathematical objects of a different kind.However, the binary operation ⊕ introduced in Theorem 1 shares some similaritieswith the addition of real numbers. Therefore, in order to assign real numbers toangular magnitudes, there is a strong temptation to look for a morphism between thesemigroup ( A , ⊕ ) and the additive group of the positive real numbers ( R + , +) . Butthis would not be in accordance with the fact that it is not magnitudes but rather theratio of magnitudes of the same kind that reflects physical reality. he term ‘angle’ in the international system of units ‡ between a ratio of two magnitudes of the same kind and aratio of two numbers, i. e. instead of assigning real numbers directly to magnitudes,we rather assign them to their ratios.It is known since ancient times that there is a relationship between an angle andan arc of a circle whose centre is located at the vertex of the angle subtending thisarc. The Babylonians used this knowledge in a pragmatic way to define an angularmeasure by a uniform subdivision of the circumference of the circle. This measureis still in use today. But it was the ancient Greeks who first proved the relationshipbetween circular arcs and angles.In equal circles, according to Euclid [2] ( Elements , Book III, Prop. 26 and27), angles whose vertices are subtended by equal arcs are equal and vice versa.Furthermore (
Elements , Book VI, Prop. 33 § ), the ratio of the angular magnitudes isequal to the ratio of the lengths of the arcs subtended by the vertices of the respectiveangles, i. e. the proportion α : β :: s α : s β (1)is valid, where α and β denote angular magnitudes and s α and s β refer tothe corresponding arc lengths. In addition Archimedes of Syracuse has shown( Measurement of a Circle , Prop. 3), that for any circle the ratio of its circumferenceto its diameter is a constant number, which we today denote by π . k Hence we haveanother proportion c : r :: 2 π : 1 , (2)where c denotes the circumference of the circle, r its radius and the real number π isa metric constant of the circle in Euclidean geometry.If we now set in proportion (1) s β = c and β = 2 ̟ , i. e. the central angle of afull circle, and use proportion (2) we obtain the proportion π α : ̟ :: s α : r . (3)Since both the left and the right side of this proportion are ratios of magnitudes ofthe same kind, we can equate each side with the same real number. This yields thetwo equations s α = ϕ α r (4)and α = ϕ α π ̟ , (5)where ϕ α denotes a positive real number, called ‘angular measure’. Note that ϕ α depends neither on an “angular unit” nor on the choice of a length unit, because theangular measure is of dimension number (for details see [5]).It is easy to verify that the angular measure takes values in the interval ]0 , π ] ,where the value π corresponds to the central angle of a full circle. ‡ A proportion establishes an equivalence relation between two ratios. A ratio is a relation andshould not be confused with a quotient, which is a binary operation defined for numbers. § There is an inaccuracy in the proof of this theorem, since the sum of the angles can exceed tworight angles. This contradicts the restriction to angles less than or equal to two right angles as givenby Euclid. However, the proof can be amended (see e.g. Legendre [3] and Crelle [4]). k In fact Archimedes only gave an upper and a lower limit for π , but his algorithm offers the possibilityto approximate this number with any desired accuracy. he term ‘angle’ in the international system of units
5. Reference angles and the circle graduation
In practical measurement of angles, a circle graduation is used instead of the angularmeasure. The main reason is, that a sectioning of the angular measure leads toirrational values, which cannot be subject to measurement.If we set in equation ̟ = p ǫ , where ǫ denotes a certain fraction of the centralangle of a full circle, denoted by ̟ , we obtain α = ϕ α p π ǫ . (6)In angle measurement technology, the angular magnitude ǫ is called a ‘reference angle’.The reference angle is often imprecisely called angular unit, but in contrast to lengthmeasurement, no unit is required for angle measurement, as was clearly shown inthe previous section. Angle measurement can be accomplished without calibration.It is based on the principle of circle closure, a natural conservation law for planeangles, known since the time of Euclid, expressing the fact that the sum of the angularmeasures around any point in a plane is equal to π .If we now express the angular magnitude α by its numerical value ϕ α with respectto a reference angle, we obtain ϕ α = 2 π p { α } , (7)where both { α } and p denote the numerical value of an angular magnitude withrespect to an arbitrary reference angle. The special value p is assigned to the centralangle of a full circle ̟ . Note that the numerical values { α } and p change, of course,with a change of the reference angle, while their quotient always remains constant,independent of the selected reference angle.Equation (4) relates the length of a circular arc s α , subtended by the vertex ofan angle with an angular magnitude α , to the radius r of that arc. The angularmeasure ϕ α occurring in this equation is a constant of proportionality whose valuecan be calculated by using equation (7). This equation depends on the choice of areference angle, but it is valid for any reference angle. The reference angle is implicitlyfixed by the choice of the real number p . Note that this factor is also a constant ofproportionality which in this case relates the angular measure ϕ α to the numericalvalue { α } assigned to an angular magnitude α . It has to be emphasized that both ϕ α and π /p are pure real numbers.Equation (7) shows that the angular measure ϕ α and the numerical value { α } associated with an angular magnitude α are generally not identical, but onlyproportional. However, it would be more natural, if these two values were identical.This can be achieved, if we in particular choose p = 2 π , whereby equation (7) issimplified to ϕ α = { α } . (8)In this case the numerical value { α } is related to the reference angle ‘radian’. This caneasily be seen by combining equations (8) and (4) and subsequently setting { α } = 1 ,which yields s α = r , i. e. the arc is measured by its radius, as it is required by thedefinition of the radian.In everyday life and in technical applications, the reference angle ’degree’, withsymbol ◦ , is usually used, which corresponds to the value p = 360 . he term ‘angle’ in the international system of units
6. Trigonometric functions
Let an angle with magnitude α be given. If a circle with radius r is drawn aboutthe vertex of this angle, it is intersected by the sides of this angle in two points. Wedenote the length of the arc between these two points as before by s α and the lengthof the chord subtending this arc by cord( α ) ¶ . Using modern calculus, we find thatthe length of the arc is related to the length of the chord by s α r = F (cid:18) cord( α )2 r (cid:19) , (9)with the function F : [0 , → R , x x Z d t √ − t . (10)The domain of F needs to be restricted to the interval [0 , , because a chord can neverbecome greater than the diameter of the same circle. F is monotonically increasing and maps the interval [0 , to the interval [0 , π / .Thus there exists an inverse function which we will denote by S . Combining equations(7) and (9) and applying the inverse function S yields cord( α )2 r = S (cid:18) π p · { α } (cid:19) . (11)If the considered angle is bisected, two similar right-angled triangles are obtainedin which the side opposite to the vertex of the angle is equal to half of the chord andwhose hypotenuses are equal to the radius of the circle. In trigonometry the ratioof these two lengths is called the sine of the related angle. Thus the function S is acertain version of the sine function restricted to the domain [0 , π / . By real analyticcontinuation of S we can define the function Sin p : R → R , { α } 7→ sin (cid:18) π p { α } (cid:19) . (12)Furthermore, by using the Pythagorean theorem we can also introduce the function Cos p : R → R , { α } 7→ cos (cid:18) π p { α } (cid:19) , (13)such that (Cos p x ) + (Sin p x ) = 1 (14)is valid. The functions Sin p and Cos p can be considered as a generalisation of the sineand cosine function, respectively. They are continuous on R and all known algebraicrules for trigonometric functions apply. A generalisation of all other trigonometricfunctions can be derived from them in the usual way.The function terms Sin p and Cos p are related by Euler’s identity [6] exp (cid:18) i 2 π p { α } (cid:19) = cos (cid:18) π p { α } (cid:19) + i sin (cid:18) π p { α } (cid:19) . (15) ¶ This was the trigonometric function of the ancient Greek, which is best known from the tableof chords in Book I, chapter 11, of the μαθηματική σύνταξις (also known as Almagest) written byClaudius Ptolemy during the 2 nd century. he term ‘angle’ in the international system of units exp maps the real numbers R to the unit circle S in the complex plane.It is therefore relevant for the definition of the so-called ‘phase angle’.We can also introduce the functions Arcsin p : [ − , → R , x p π arcsin x (16)and Arccos p : [ − , → R , x p π arccos x , (17)which are generalisations of the inverse functions of Sin p and Cos p , respectively. Itsresult is a real number of the interval [0 , p ] , which is a numerical value assigned toan angular magnitude. This numerical value is called ‘principal measure’ of an anglerelative to p . It allows a classification of angles as given in Table 1. Table 1.
Classification of angles by their principal measures name of the angle principal measure zero angle acute angle i , p h right angle p obtuse angle i p , p h straight angle p reflex angle i p , p h perigon p The functions
Sin p and Cos p are periodic with the ‘principal period’ p . If inparticular the principal period p = 2 π is chosen, i. e. if the radian is used as referenceangle, we obtain by using equation (8) Sin π α = sin ϕ α and Cos π α = cos ϕ α . These are the sine and cosine functions commonly used in trigonometry. Thecorresponding inverse functions are
Arcsin π α = arcsin ϕ α and Arccos π α = arccos ϕ α . An addition we obtain for p = 2 π from equations (15) and (8) the relation e i ϕ α = cos ϕ α + i sin ϕ α , which is important for the ‘phase angle’. The phase angle used in mathematics andtheoretical physics is equal to the angular measure ϕ α .It turns out that the argument of all commonly used trigonometric functions,as with all other transcendental functions, is not an angle, but rather an angularmeasure, i. e. a pure number. It is therefore neither necessary nor correct to add ‘rad’to their argument, since this particular reference angle is already implicitly part of thedefinition of these functions. The results of the commonly used inverse trigonometricfunctions are real numbers, which are, of course, related to the reference angle radian,which is implied by the definition of these functions as well. EFERENCES
7. Conclusion
It has been shown that most misunderstandings regarding the ‘radian’ can be avoidedif some simple rules are obeyed. Although the discussion has been confined to theplane angle of Euclidean geometry, all conclusions apply equally well to the conceptsof ‘angle of rotation’ and ‘phase angle’, which have not been discussed here, in orderto concentrate on the essential points.When dealing with the radian,the wrong equation α = sr (cid:0) instead of the correct equation s = ϕ α r (cid:1) is usually used in the literature. From this wrong equation then the conclusion is drawnthat the radian is a “derived unit” which is equal to the number one. Unfortunately,this statement also appears in the current SI brochure, where moreover ‘rad’ isexpressed by the quotient m / m, in order to emphasize that it is “derived”. But thesestatements are both not justified at all.If any value associated with an angular magnitude is reported, both the numericalvalue and the corresponding symbol of the reference angle must always be stated,because the numerical value depends on the chosen reference angle. In case of asemicircle, for example, the value associated with the angular magnitude has to bereported by π rad, not simply by π , i.e. the symbol ‘rad’ must not be omitted. On theother hand, c = π r must always be written for the arc of a semicircle with radius r ,i.e. in this case it is mandatory to omit the symbol ‘rad’, because the angular measurehas to be used here, which is a pure number.When dealing with trigonometric functions, wrong expressions, such as sin α instead of the correct expression sin ϕ α are usually used in the literature. From thisincorrect expressions then the conclusion is drawn that the symbol ‘rad’ has to beappended to the numerical value of their argument. But this is wrong. The argumentof the trigonometric functions is always an angular measure, i. e. it is neither necessarynor correct to add the symbol ‘rad’. References [1] Hilbert D.
Foundations of Geometry; translated by E. J. Townsend . Chicago: TheOpen Court Publishing Company, 1902.[2] Euclid.
The Thirteen Books of Euclid’s Elements; translated by T. L. Heath .Cambridge: University Press, 1968.[3] Legendre A. M.
Elements of Geometry and Trigonometry; translated byD. Brewster . New York: James Ryan, 1828.[4] Crelle A. L.
Elemente der Geometrie . Berlin: G. Reimer, 1826.[5] Krystek M. “The term ‘dimension’ in the international system of units”. In:
Metrologia
52 (2015), pp. 297–300.[6] Euler L.