SPHERICAL TRIGONOMETRY

1 SPHERICALTRIGONOMETRY THERE IS NO ROYAL ROAD TOGEOMETRY EUCLIDBYKELLY LYNCHS aint Mary s College of CaliforniaMoraga2016 AbstractThe purpose of this paper is to derive various trigonometric formulas for spher-ical triangles. The subject of SPHERICAL TRIGONOMETRY has many navigational and astro-nomical has been developing and evolving for many centuries. Its uses arevast and continue to affect our every day lives. The study of the sphere in particular hasits own unique story, and has two major turning points. This study first began with thepush of astronomy and was developed in depth by the Greeks.

2 There is speculation thatmathematical discoveries about the sphere were made as early as the second century,but there is no proof for this.[1] The major transition for the understanding of sphericalgeometry was the work of Menelaus of Alexandra. In his workSphericathe authordelves deeply into the properties of the sphere and the calculations related to lengthsand measures of a sphere s arcs. For a long time the equations he discovered, such asthe measure of circumference of a sphere and the measure of arc lengths, were acceptedand no further study was next major motivation for learning SPHERICAL TRIGONOMETRY was religiousmatters; the religion of Islam requires that the direction of Mecca is always knownfor daily prayer.

3 Menelaus findings were further developed during the Islamic En-lightenment period. There is some debate as to the discovery of the Law of Sines forspherical triangles. Possible sources of this discovery stem from the debate over thetwo Muslim scientists, Ab u Nasr or Ab u l-Waf a . Who ever was responsible for theprogress in the Law of Sines allowed for a more concise proof to be developed later;as well as leading to other theorems and identities on SPHERICAL TRIGONOMETRY . Anothermajor name in geometry is Euclid. He made a big impact later in the third century;though the system of SPHERICAL TRIGONOMETRY does not incorporate parallel lines, Eu-clidean geometry gave some insight to SPHERICAL behavior.

4 In his workElementsEuclidpublished equations which help lead us to the Pythagorean Theorem and the Law ofCosines. Though mathematicians brought insight to this area of study, many influenceson SPHERICAL TRIGONOMETRY also came from the field of science. Further discovery aboutthe behavior of arcs and angles became prominent in the late Renaissance period. JohnNapier, a Scottish scientist who lived around the 17th century, was the first to work withright SPHERICAL triangles and the basic identities of these shapes. Using Napier s Rules,the law of cosines for spheres was discovered.[1]Definition a sphere, agreat circleis the intersection of the sphere with aplane passing though the center, or origin, of the sphere.

5 [1]Example CircleThe solid line illustrates a great circle which is the intersection of the plane Xand the sphere A. The plane in the is the horizontal plane; however, the planecan have any orientation that bisects the sphere. The circle created by this intersectionwill have radius equal to the length of the radius of the sphere. It also follows that thelength of the circumference of the great circle will equal2 rwhereris the length ofthe radius of the sphere. This is relevant because it enables us to calculate the length ofa circular segment by considering the relation between the inner angle and the radius ofthe sphere.

6 Therefore, for any sphere and any angle, the length of one arc segment willjust equalr where is the measure of the angle in radians. This is true because eachcircular segment is just fraction of the entire circumference. We can show this relationof the angle and arc length in the example arc measure (whereais the length of the arc)This is a general example, but we can apply specific values as in the example onthe following Measure of ArcIn this example we cut our a segment of great circle by relating an inner this example letR= =.93759radians, thus( )(.93759) = ,which is the measure of the length of the arc.

7 This example was created in a dynamicmathematics software program which gave these related next definition required in understanding a SPHERICAL triangle is that of LuneA lune is a part of the sphere which is captured between two great circles.[1]This definition is relevant because it started the ability to capture shapes on asphere. This definition itself is not extremely significant, but it is through this shapewhich we can form other shapes on the sphere. The shape of a lune can also be seen inthe next LuneSpherical triangles can be defined in terms of SPHERICAL TriangleAspherical triangleis the intersection of three distinct lunes.

8 [1]In the figure above we can consider that there are two lunes which are the onopposite sides of the sphere, it is natural that another lune bisecting these two will beneeded. A simpler way to think about a SPHERICAL triangle is the shape captured throughthe intersection of three great circles. It is also interesting to consider that any twounique points, which are not diametric on the spheres surface, lie on a great circle. Thismakes sense because a triangle would have three vertices and therefore three differentgreat circles which go through two different points. Since the SPHERICAL triangle s edgesare curved, it is clear that the equations, sides, area, and general properties will bedifferent from that of a planar triangle.

9 We also know that in some way we should beable to relate the functions that we do discover to the radius of the sphere, because itseems natural that the formulas would change in relation to the sphere s knowthat the behavior of a SPHERICAL triangle will be very different from the planar triangles,there are even some definitions which can illustrate this for TriangleDefinition SPHERICAL Excessis the amount by which the sum of the angles (inthe SPHERICAL plane only) exceed180 .This definition tells us about the behavior of the sphere and its edges. We knowthat the length of the edges on a SPHERICAL triangle will be greater the edges on a corre-sponding planar triangle, since they are curved.

10 This definition allows for a sphericaltriangle to have multiple right triangle with three right anglesDefinition [3] SPHERICAL Coordinatesis a coordinate system in three dimentions. The coordinatevalues stated below requirerto be the length of the radius to the pointPon the value the angle between the z-axis, and the vector from the origin to pointP, and the angle between the x-axis, and the same vector as in the figure Then we cansay thex,y,zcoordinates are defined:(r cos( )sin( ),rsin( )sin( ),r cos( ))Example Coordinates in thexyzplaneTheorem SPHERICAL Pythagorean Theorem[3]For a right triangle,ABCon a sphere of radiusr, with right angle at vertexCandsides lengtha,b,cis defined.