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In spherical trigonometry, the law of cosines (also called the cosine rule for sides) is a theorem relating the sides and angles of spherical triangles, analogous to the ordinary law of cosines from plane trigonometry.
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, and angle Γ opposite γ. To compute γ, we have the formula cos(γ) = cos(α)cos(β) +sin(α)sin(β)cos(Γ) (1.1)
Spherical trigonometry is the branch of spherical geometry that deals with the metrical relationships between the sides and angles of spherical triangles, traditionally expressed using trigonometric functions.
The Spherical Law of Cosines Suppose that a spherical triangle on the unit sphere has side lengths a, b and c, and let C denote the angle adjacent to sides a and b. Then (using radian measure): cos(c) = cos(a) cos(b) + sin(a) sin(b) cos(C).
Jul 1, 2024 · Let a spherical triangle be drawn on the surface of a sphere of radius R, centered at a point O=(0,0,0), with vertices A, B, and C. The vectors from the center of the sphere to the vertices are therefore given by a=OA^->, b=OB^->, and c=OC^->.
This represents essentially the law of cosines for a spherical triangle . The other two versions follow at once by interchanging the a,b,c,A,B,Cs. They are- cos(b)=sin(a)sin(c)cos(B)+cos(a)cos(c) and cos(c)=sin(a)sin(b)cos(C)+cos(a)cos(b) A law of sines law follows from manipulating cos(A) and cos(B) in the above formulas. We find- 2 2 2 2
Mar 26, 2020 · Theorem. Let ABC be a spherical triangle on the surface of a sphere whose center is O . Let the sides a, b, c of ABC be measured by the angles subtended at O, where a, b, c are opposite A, B, C respectively. Then: cosa = cosbcosc + sinbsinccosA. Corollary. cosA = − cosBcosC + sinBsinCcosa. Proof 1.
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