angle sum identity

It is desired to prove the identities




Consider the figure

where we have

  • AadCcb

  • BbaDdc

  • ad=dc=1.

Also, everything is Euclidean, and in particular, the interior anglesMathworldPlanetmath of any triangle sum to π.

Call Aad=θ and baB=ϕ. From the triangle , we have Ada=π2-θ and Ddc=π2-ϕ, while the degenerate angle AdD=π, so that


We have, therefore, that the area of the pink parallelogramMathworldPlanetmath is sin(θ+ϕ). On the other hand, we can rearrange things thus:

In this figure we see an equal pink area, but it is composed of two pieces, of areas sinϕcosθ and cosϕsinθ. Adding, we have


which gives us the first. From definitions, it then also follows that sin(θ+π/2)=cos(θ), and sin(θ+π)=-sin(θ). Writing

Title angle sum identity
Canonical name AngleSumIdentity
Date of creation 2013-03-22 12:50:36
Last modified on 2013-03-22 12:50:36
Owner mathcam (2727)
Last modified by mathcam (2727)
Numerical id 14
Author mathcam (2727)
Entry type Theorem
Classification msc 51-00
Related topic ProofOfDeMoivreIdentity
Related topic DoubleAngleIdentity