Transcription of Spherical Trigonometry
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Spherical Trigonometry
Spherical TrigonometryRob JohnsonWest Hills Institute of Mathematics1 IntroductionThe sides of a Spherical triangle are arcs of great circles. Agreat circle is the intersection of a sphere witha central plane, a plane through the center of that sphere. The angles of a Spherical triangle are measuredin the plane tangent to the sphere at the intersection of the sides forming the avoid conflict with the antipodal triangle, the triangle formed by the same great circles on the oppositeside of the sphere, the sides of a Spherical triangle will be restricted between 0 and radians. The angleswill also be restricted between 0 and radians, so that they remain derive the basic formulas pertaining to a Spherical triangle, we use plane Trigonometry on planes relatedto the Spherical triangle. For example, planes tangent to the sphere at one of the vertices of the triangle,and central planes containing one side of the specified otherwise, when projecting onto a plane tangent to the sphere, the projection will be fromthe center of the sphere.
One of the simplest theorems of Spherical Trigonometry to prove using plane trigonometry is The Spherical Law of Cosines. Theorem 1.1 (The Spherical Law of Cosines): Consider a spherical triangle with sides α, β, and γ,
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