Hyperbolic Cosines and Sines Theorems for the Triangle Formed by Arcs of Intersecting Semicircles on Euclidean Plane

Classically, the models of the hyperbolic plane are regarded as based on Euclidean geometry. One starts with a piece of a Euclidean plane, a half-plane, or a circular disk and then, in that half-plane or disk the notions of points, lines, distances, angles are defined as things that could be described in terms of Euclidean geometry [1–3]. The model of the hyperbolic plane is the half-plane model. The underlying space of this model is the upper half-planemodelH in the complex plane C, defined to be


Introduction
Classically, the models of the hyperbolic plane are regarded as based on Euclidean geometry.One starts with a piece of a Euclidean plane, a half-plane, or a circular disk and then, in that half-plane or disk the notions of points, lines, distances, angles are defined as things that could be described in terms of Euclidean geometry [1][2][3].The model of the hyperbolic plane is the half-plane model.The underlying space of this model is the upper half-plane model  in the complex plane , defined to be  = { ∈  : Im () > 0} . ( In coordinates (, ) the line element is defined as The geodesics of this space are semicircles centered on the axis and vertical half-lines.The geometrical properties of the figures on the half-plane are studied by considering quantities invariant under an action of the general Möbius group, which consists of compositions of Möbius transformations and reflections [4].The curvilinear triangle formed by circular arcs of three intersecting semicircles is one of the principal figures of the upper half-plane model .The hyperbolic laws of sines-cosines for that triangle are proved by using properties of the Möbius group and the upper half-plane .
In this paper we suggest another way of construction of proofs of the sines-cosines theorems of the Poincaré model.The curvilinear triangle formed by circular arcs is the figure of the Euclidean plane; consequently, on the Euclidean plane we have to find relationships antecedent to the sines-cosines hyperbolic laws.Therefore, first of all, we establish these relationships by making use of axioms of the Euclidean plane, only.Secondly, we proove that these relationships can be formulated as the hyperbolic sine-cosine theorems.For that purpose we refer to the general complex calculus and within its framework establish a relationship between exponential function and the cross-ratio.In this way the hyperbolic trigonometry emerges on Euclidean plane in a natural way.
The paper is organized as follows.In Section 2 a key formula connecting the hyperbolic calculus with the crossratio is established.By employing the key formula and the Pythagoras theorem the elements of the right-angled triangle are expressed as functions of the hyperbolic trigonometry.In Section 3 we explore Euclidian properties of the curvilinear triangle bounded by circular arcs of three intersecting semicircles.Two main relationships connecting three intersecting semicircles are established.In Section 4, we prove the theorem of cosines for the curvilinear triangle bounded by the circular arcs.In Section 5, the hyperbolic law of sines and second hyperbolic law of cosines are derived on the basis of the main relationships between these semicircles.

Hyperbolic Trigonometry in Euclidean Geometry
where coefficients  0 ,  1 are given by real positive numbers and  2 1 − 4 0 > 0. The following Euler formula holds true: Denote by  1 ,  2 ∈  roots of the quadratic equation (3).The Euler formula (4) is decomposed into two independent equations: from which one may find explicit formulae for -functions.These functions are linear combinations of the exponential functions exp(  ),  = 1, 2. Geometrical interpretation of the general complex algebra is done in [6].Form the following ratio exp where Introduce a pair of variables  1 ,  2 by Then, ( 6) is written as follows: exp This formula we denominate as the key formula and use it in order to introduce hyperbolic trigonometry on Euclidean plane.Notice that the argument of the exponential function is proportional to the difference between the numerator and the denominator,  1 −  2 =  1 −  2 , and this difference does not depend on the parameter .

Elements of
Notice that these two ratios are functions of the same angle .On making use of formula (9) we are able to introduce the hyperbolic trigonometry besides the circular trigonometry.Define the following relationships: From this equation it follows that In this way we arrive to the following interrelations between circular and hyperbolic functions: Now let us introduce the following geometrical motion; namely, change position of the point  along the line  (see, Figure 1).Then, the sides  and  will change while the side  will not change.Since the length  is a constant of this evolution, in agreement with key formula (9) we rewrite (11) as follows: that is, the argument of exponential function is proportional to :  = , where  is a parameter of the evolution.Formulae in (13) are rewritten as From these formulas it is derived that  Install the triangle  in such a way that the side  =  lies on the line , and the side  =  is perpendicular to this line at the point In Figure 1 the line  moves in such a way that cuts line  at points ,   ,   , and so forth.Now, let us recall the problem of parallel lines in geometry [7].Let the ray  tend to a definite limiting position; in Figure 1 this position is given by line  1 .As the point  moves along line  away from point  there are two possibilities to consider.
(1) In Euclidean geometry, the angle between lines  1 and  is equal to right angle.(2) The hypothesis of hyperbolic geometry is that this angle is less than the right angle.
The most fundamental formula of the hyperbolic geometry is the formula connecting the angle of parallelism Π() and the length  of the perpendicular from the given point to the given line.In order to establish those relationships the concept of horocycles, some circles with center and axis at infinity, was introduced [8].The great theorem which enables one to introduce the circular functions, sines, and cosines of an angle is that the geometry of shortest lines (horocycles) traced on horosphere is the same as plane Euclidean geometry.The function connecting the angle of parallelism with the distance  is given by Now, we introduce the value inverse to  by K = 1/ and rewrite (17) in the form Let the point  run away from the point  to infinity.The following two cases can be considered.
(1)  tends to zero, and K tends to infinity; then the angle  will tend to right angle.This is true in Euclidean plane.
(2) Suppose that , and K and the angle  go to some limited values, lim In this way, we have established connection of formula (17) with the main formula of hyperbolic geometry (19).

Rotational Motion of a Line
Tangent to the Semicircle.The concept of the circular angle in Euclidean plane is intimately related to the figure of a circle and to motion of a point along the circumference.The hyperbolic angle is also related to the circle because of a motion along the circumference coherence with the motion along the hyperbola [9].Consider semicircle C (Figure 2) with end-points and the center on -axis.Denote by  the center and by  1 ,  2 the end-points of the semicircle.Through end-points of the semicircle,  1 ,  2 erect the lines parallel to vertical axis, axis.Draw a line tangent to the semicircle at the point ; this line crosses -axis at the point  and intersects with the vertical lines at points  1 and  2 .Draw a line parallel to -axis from the center  which crosses C at the top point .
Denote by  radius of the circle, so that  =  =  and  =  1  2 .Denote by  the angle ∠.The triangle  is a right triangle, so that Consider rotational motion of the line tangent to the semicircle at the point C. In Figure 2, two positions of this line are given by lines    2 and  1 .When the point runs from end point  1 to the top-point , the point  runs along axis from point  1 to infinity.During this motion the triangle formed by the line tangent to the semicircle, the -axis and the radius of the circle remains to be right angled.This is exactly the case considered in Section 2.2, consequently, we can apply formula (13).According to (13) we write Since  1 ∼  1  1 , we have On the basis of obtained formulae the following relationships between circular and hyperbolic trigonometric functions are established: If we make the point  tend to the top of semicircle , the hyperbolic angle  will tend to zero.When the point  tends to end-point  1 , the hyperbolic angle tends to infinity.Thus, the hyperbolic angle is measured from the point , the top of the semicircle.Consider two different positions of the tangent line, corresponding to two positions  1 ,  2 , with hyperbolic angles  2 and  1 .According to key formula (6) we write exp The hyperbolic angle  2 corresponds to the arc ⌣  2 , and the hyperbolic angle  1 corresponds to the arc ⌣  1 .Then we suppose that the difference in the hyperbolic angles  2 − 1 will correspond to the arc It is seen, in the right-hand side we have the cross-ratio.Now, let recall definition of the distance between two points of the geodesic line in the Poincaré model.Let ,  ∈ , and let  1 and  1 be end points of a geodesic line passing through  and .Then the distance between these points is defined by formula where is the cross-ratio.Comparing (29) with ( 27) we come to the following correspondence: Notice, however, in our construction   ,  = 1, 2 are projections of  1 ,  3 on -axis.The semicircle C is the geodesic line, and  1 ,  2 are end-points of the geodesic line.

Hyperbolic Cosine-Sine Functions of Arcs of the Semicircle.
The key formula (6) admits to define hyperbolic trigonometric functions of the arcs originated from the top  of the semicircle C. In order to determine trigonometric functions of the arcs with arbitrary end-points on the semicircle we have to use formula (27).Consider arc ⌣  1  2 is defined in the first quadrant of the semicircle with end-points at  1 and  2 .
The arc ⌣  1  2 can be presented as difference of two arcs, both originated from the top of the semicircle: Denote by  1 ,  2 the angles formed by radiuses  1 and  2 with -axis, correspondingly, where  is a center of the circle.Denote the hyperbolic angle corresponding to the arcs ⌣   ,  = 1, 2 by (  ),  = 1, 2. Then the functions cosh  ( 1 ) , cosh  ( 2 ) , are expressed via circular trigonometric functions according to formulae (25): Then, by taking into account additional formulae cosh  ( we get hyperbolic trigonometric functions of the arcs with arbitrary end-points on the semicircle expressed via periodic trigonometric functions: Compare with analogous formula from Poincaré model ( [3], formula (1.2.6)) given by cosh  (, ) Two complex numbers are related to the radius of the semicircle and the angles  1 ,  2 by Then, cosh  (

Relationships between Elements of the Triangle Bounded
by Arcs of the Intersecting Semicircles.In Figure 3 three intersecting semicircles with centers installed on horizontal axis at the points   ,   , and   are presented.Intersections of the semicircles form triangle  bounded by the arcs ⌣ , ⌣ , ⌣ .For each arc we can put in correspondence the hyperbolic angle.Denote the hyperbolic angles by , , , where , ,  consecutively will correspond to the arcs ⌣ , ⌣ , ⌣ .Connect vertices of the triangle with centers of the circle by corresponding radiuses.Denote by   ,   ,   ,  = 1, 2 the angles bounded by the radiuses and -axis, where  1 >  2 > 0,  1 >  2 > 0,  1 >  2 > 0. By making use of (34a) define hyperbolic cosine-sine functions corresponding to bounding segments: The usual notion of the angle is used that is, the angle between two curves is defined as an angle between their tangent lines.Let the angles , ,  be angles at the verteices , , , correspondingly.For these angles we can define thier proper cosine and sine functions.The angles of the triangle  , ,  are closely related to angles  1 ,  2 ,  1 ,  2 ,  1 ,  2 .
From Figure 3 we find the following relationships between them: sin sin Denote distances between centers by The theorem of sines employed for triangles     ,     ,     gives six relations of type From these relations the first set of main relationships follows.
Relation 1.Consider From the draught in Figure 3 it is seen that where Hence, From vertices of  erect lines perpendicular to horizontal line intersect with -axis at points ℎ  , ℎ  , ℎ  , correspondingly.From Figure 3 we find that By equating (47) with (48) we arrive to another main relationship between radii and angles.

Hyperbolic Law of Cosines I for the Triangle Formed by Intersection of Three Semicircles
The main aim of this section is to prove the hyperbolic law theorem of cosines I for the triangle  formed by arcs of intersecting semicircles with centers installed on -axis (Figure 3).This law is given by the following set of equations: Theorem 3 (theorem of cosines I).The following equation for elements of the triangle  formed by arcs of three intersecting semicircles holds true where cosh , cosh , cosh , sinh , sinh , cos  are defined by formulae (51a), (51b), (51c)-(52).
Proof.Square both sides of Relation 2 to obtain and evaluate this equality by taking into account formulae (51a), (51b), (51c)-(52).First of all, evaluate the left-hand side of this equation as follows: where Use (63) and write (62) as follows: Equation ( 61) takes the following form: Journal of Mathematics 7 The underlined term passes to the right-hand side of the equation.Then in the left-hand side remains the expression Dividing both sides of the obtained equation by V 3  3 , we get according to Relation 1 These relations make true the following equation On making use this formula in the right-hand side of (67) we come to the following equation: Evaluate now the underlined term, which we firstly write as follows: From the second equation of (51c) we have By making use of (71) in (70), evaluate (70) as follows: Replace the underlined term of (69) by (72), and pass expression containing sines and cosines to the left-hand side of the obtained equation.As a result, we come to the following equation: By applying the elementary algebra one may show that (see, [10]) Thus, in the right-hand side of (73) we have which according to formulae (52) is nothing else than By using it in (73) we arrive to the following equation:

Hyperbolic Laws of Sines and Cosines II
The main task of this section is to prove hyperbolic law (theorem) of sines, which is given by the formulae and the hyperbolic law (theorem) of cosines II given by the formulae cos  = − cos  cos  − sin  sin  cosh . (81)

Hyperbolic Theorem of Sines and Its Geometrical Interpretation on Euclidean Plane
Theorem 4. The ratios of projections of the sides of triangle  on -axis to corresponding distances between centers of the semicircles are equal to each other.

Journal of Mathematics
Proof.Projections of the sides of  are given by formulae Distances between centers of the circles have been defined as (see, ( 46) and ( 47)) Relation 1 given by the set of equations raises to the second power, Then, square distances  2  , ,  = , ,  and use (86).We get Combine (86) with (87); in this way we come to the following system of equations: From these equations the cosines of the angles  1 , In the sequel come back to designations introduced in Section 3. In these designations (96) is written as follows: Theorem 5.The sides and the angles of triangle  satisfy (78).

Figure 1 :
Figure 1: Motion of the hypothenuse of the right angled triangle.
Right-Angled Triangle as Functions of a Hyperbolic Trigonometry.Let  be a right-angled triangle with right angle at . Denote the sides by , , the hypotenuse by , and the angles opposite to , ,  by , , , correspondingly.Traditionally interrelations between angles and sides of a triangle are described by the trigonometry via periodical sine-cosine functions.The periodical functions of the circular angles are defined via the ratios sin

Figure 2 :
Figure 2: Semicircle and lines tangent to the semicircle.