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单词 AngleSumIdentity
释义

angle sum identity


It is desired to prove the identities

sin(θ+ϕ)=sinθcosϕ+cosθsinϕ

and

cos(θ+ϕ)=cosθcosϕ-sinθsinϕ

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 degenerateangle AdD=π, so that

adc=θ+ϕ

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

sin(θ+ϕ)=sinϕcosθ+cosϕsinθ

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

cos(θ+ϕ)=sin(θ+ϕ+π/2)=sin(θ)cos(ϕ+π/2)+cos(θ)sin(ϕ+π/2)=sin(θ)sin(ϕ+π)+cos(θ)cos(ϕ)=cosθcosϕ-sinθsinϕ
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更新时间:2025/5/4 17:03:16