HYPERBOLIC SINE AND COSINE RULES FOR GEODESIC TRIANGLES ON THE HYPERBOLIC UNIT SPHERE

In this work, we proved the sine and cosine rules for a geodesic triangle on the hyperbolic unit sphere 2 0 H by means of timelike unit vectors. We also obtained some useful results. KeywordsGeodesic triangle, Lorentzian space, Timelike vector.


INTRODUCTION
In plane Lorentzian geometry it is studied points, timelike, spacelike and lightlike lines, triangles, etc [4].On the hyperbolic sphere, there are points, but there are no straight lines, at least not in the usual sense.However, straight timelike lines in the Lorentzian plane are characterized by the fact that they are the shortest paths between points.The curves on the hyperbolic sphere with the same property are hyperbolic circles.Thus it is natural to use these circles as replacements for timelike lines.
The formulas for the sine and cosine rules are given for the Euclidean sphere 2 S [2, 3,6] and hyperbolic sphere [5].In this study, we obtained formulas related with the spacelike angles and hyperbolic angles corresponding to the sides of geodesic triangles on hyperbolic unit sphere 2 0 H .

BASIC CONCEPTS
In this section, we give a brief summary of the theory of Lorentzian concepts.Let Lorentzian 3-space 3  L be the vector space 3 IR provide with Lorentzian inner product < , > given by A vector There are two components of the hyperbolic unit sphere

THE SINE AND COSINE RULES FOR HYPERBOLIC GEODESIC TRIANGLES
Let A and B be two future pointing timelike unit vectors in Lorentzian 3-space , 3 L let A and B be the magnitudes (lengths) of these vectors and let θ be the hyperbolic angle between A and B .Then the Lorentzian inner product of A and B is defined by The Lorentzian cross product of vectors A and B is defined by where n is a spacelike unit vector in the direction of K and θ is the length of the vector K. Thus the vectors B A, and K constitute the right-system.In the trihedral OABC ( Figure 2.1), let j i, and k be the timelike unit vectors in the direction of the vectors OB OC, and OA , respectively.Thus When we apply the definitions to calculate the pair wise inner products and cross products of the vectors j i, and k , we find is the volume of the parallel piped (parallelogram-sided box) determined by j i, and k .Now we will state and prove the hyperbolic sine rule, cosine rule I and cosine rule II for a hyperbolic geodesic triangle.(3) On the other hand, from the cross product of the vectors j and i we obtain The Lorentzian inner product of k and i j × gives ., sin sinh sinh (4) In a similar way, we get Further, each of the equations ( 4), ( 5) and (6) gives the volume of the parallel piped determined by the unit vectors j i, and k .Thus, we have .
Proof: For brevity, let Y X , and Z be b a cosh , cosh and , cosh c respectively.Then the cosine rule I yields On the other hand, since , 1 sin cos We note that D is positive and symmetric in Y X , and Z .Then we obtain , If we write the formulas ( 14)-(18) in the right side of the formula (13), then the equality is satisfied:

SOME INEQUALITIES BETWEEN THE SIDES AND THE HYPERBOLIC ANGLES
In this chapter, we give some inequalities related with the angles and sides of a geodesic hyperbolic triangle.

Lemma 2 . 1 .
(The Hyperbolic Sine Rule) Let ABC be a hyperbolic geodesic triangle on the hyperbolic unit sphere 2 0 H . Then the hyperbolic sine rule is given by Consider the trihedral OABC in Figure2.1.Draw the tangent lines at the vertex A to the sides AB and AC .Let AB T and AC T be two unit vectors on these tangent lines.Let N be the point of intersection of the line OB and the tangent line drawn the side AB .Then we have the relationON AN OA = +(1) between the vectors determined by the sides of the right triangle OAN (Figure2.2

Lemma 2 . 2 (
The Hyperbolic Cosine Rule I) Let ABC be a geodesic triangle on the hyperbolic unit sphere 2 0 H . Then the hyperbolic cosine rule I is given by The inner product of the vectors j and i is equal to .cosh , On the other hand, from the equations (2) and (3) we get ,

3 (
and the spacelike tangent vectors AB T and AC T are perpendicular to k , that is, The Hyperbolic Cosine Rule II).Let ABC be a geodesic triangle on the hyperbolic unit sphere 2 0 H . Then the hyperbolic cosine rule II is given by

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By the same way, we can give the similar formulas for b

Lemma 3 . 1 .Lemma 3 . 2 .Let
Let β α, and γ be the angles of a hyperbolic geodesic triangle ABC .Let β α, and γ be the angles of a hyperbolic geodesic triangle ABC .Inequalities related with the hyperbolic angles and sides can be given with the aid of the hyperbolic cosine rules.If two sides of a hyperbolic geodesic triangle ABC are equal to each other, then the opposite angles are equal each other, and conversely.Proof: From the equations (10) and (11), we have ABC be a geodesic triangle on the hyperbolic unit sphere 2 0 H . Then from the hyperbolic cosine rule I we have )increasing function for positive x , then we get an inequality related with the sides of the geodesic triangle as follows: