Plane Geometry - Bruce E. Shapiro

318 SECTION 53. SPHERICAL GEOMETRYProof. Define the x − axis to pass through vertex A and place C on theequator in the xy plane.Let α be the central angle determined by ˜BC. Hence a = αr.Let β be the central angle determined by ÃC. Hence b = βr.Let γ be the central angle determined by ÃB. Hence c = γr.Then the coordinates of the three vertices in R 3 are⎫A = (r, 0, 0)B = (r cos β sin(π/2 − α), ⎪⎬r sin β sin(π/2 − α),cos(π/2 − α) ⎪⎭C = (r cos β, r sin β, 0)(53.3)HenceSince |A| = |B| = r,cos γ = A · B|A||B|cos γ = cos β sin(π/2 − α) = cos β cos αSubstituting the equations a = αr, b = βr, and c = γr gives the desiredresult.For the law of cosines and law of sines we assume that in △ABC the sidesopposite vertices A, B and C have central angles a, b, and c, respectively(their arc lengths are ar, br, and cr).Theorem 53.12 (Spherical Law of Sines) 1sin asin A = sin bsin B = sin csin CTheorem 53.13 (Spherical Laws of Cosines) 2cos a = cos b cos c + sin b sin c cos Acos b = cos c cos a + sin c sin a cos Bcos c = cos a cos b + sin a sin b cos C1 Attributed to Abul Wafa Buzjana (940-998)2 Often attributed to Jamshid al-Kashi (1380-1429) who put it into its present form,although it was likely already know much earlier.« CC BY-NC-ND 3.0. Revised: 18 Nov 2012