The Cosine Rule Ii for a Spherical Triangle on the Dual Unit Sphere S

In this work, we proved the Cosine Rule II for a spherical triangle on the dual unit sphere S 2 % .


INTRODUCTION
Dual numbers had been introduced by W. K. Clifford (1845-1879) as a tool for his geometrical investigations.After him, E. Study used dual numbers and dual vectors in his research on line geometry and kinematics (see [3]).He devoted special attention to the representation of directed line by dual vectors and defined the mapping that is said with his name.There exists one to one correspondence between the vectors of dual unit sphere S 2 % and the directed lines of the space 3  R of lines (E.Study's mapping).
In plane geometry it is studied points, lines, triangles, etc.On the sphere, there are points, but there are no straight lines, at least not in the usual sense.However, straight lines in the plane are characterized by the fact that they are the shortest paths between points.The curves on the sphere with same property are the great circles.Thus it is natural to use great circles as replacements for lines.
The Sine Rule I and Cosine Rule I for the dual and real spherical trigonometries have been well known for a long time (see, [1], [4], [5]).In this study, we prove the Cosine Rule II for a spherical triangle on the dual unit sphere S 2 % ., where λ and * λ are real numbers and ε stands for the dual unit which is subject to the rules: 2 0, 0, 0 0 0, 1 1 .

DUAL NUMBERS AND DUAL VECTORS
We denote the set of dual numbers by D : Equality

Moreover, if
Now let f be a differentiable function.Then the Maclaurin series generated by The norm of x of a dual number Then the formula (3) allows us to write . Let D 3 be the set of all triples of dual numbers: Then a ~ is said to be dual unit vector if the vectors a and * a satisfy the following equations * a ,a 1, a, a 0.
The set of all dual unit vectors is called the dual unit sphere, and is denoted by S 2 % (for more details, see [3], [5]).

THE COSINE RULE II FOR A DUAL SPHERICAL TRIANGLE
In this section we prove the Cosine Rule II for dual spherical triangles.
Since we can write In corollary 3.3, the real part is known as the Sine Rule for a spherical triangle.
Corollary 3.4.The real and dual parts of the formulas ( 5), ( 6) and ( 7) are given by In corollary 3.4, the real parts give the Cosine Rule I for a spherical triangle.
Now we state and prove the correspondence of Cosine Rule II for hyperbolic spherical trigonometry given in [1]

Definition 2 . 1 .
A dual number has the form

.Then D 3 × is the known cross product in 3 R . Lemma 2 . 2 .
The elements of D 3 are called dual vectors.A dual vector a ~ may be expressed in the form becomes a unitary D -module with these operations.It is called D -module or dual space.The inner product of two dual vectors known inner product of the vectors a and b in the 3-dimentional vector space 3 R .The cross product of two dual vectors a a a Let a, b , c, d ∈

Theorem 2 . 4 2 %
(E. Study's Mapping) The dual unit vectors of the dual unit sphere S are in one to one correspondence with the directed lines of the 3-space 3 R lines [3].

2 %
Let A, B % % and C ~ be three points on dual unit sphere S given These points together with the great-circle-arcs AB, BC, CA given similar definitions for the other sides b ~ and c ~ of ABC

Corollary 3 . 3 .
is obvious that a, b % % and c ~ are the dual unit vectors having the same sense as b The real and dual part of the formula (4) are given by , We note that if λ and β are two nonzero elements of a ring R such that 0 λβ = , then λ and β are divisors of 0 (or 0 divisors).