The Three-Ray (Trihedral Angle) Cosine Identity

★ 7/10trigonometryLaw of Cosinessolid geometryTrigonometric Identities

Problem

figure

Let PAPA, PBPB, PCPC be three rays emanating from a common point PP in space. Denote

∠APC=α,∠BPC=β,∠APB=θ,\angle APC=\alpha,\qquad \angle BPC=\beta,\qquad \angle APB=\theta,

and let φ\varphi be the size of the dihedral angle A-PC-BA\text{-}PC\text{-}B along edge PCPC. Prove that

cos⁡θ=cos⁡αcos⁡β+sin⁡αsin⁡βcos⁡φ.\cos\theta=\cos\alpha\cos\beta+\sin\alpha\sin\beta\cos\varphi.

Answer

Solution

Difficulty7/10
Topicstrigonometry, Law of Cosines, solid geometry, Trigonometric Identities

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