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Sunday, September 27, 2015

Some trigonometry formula (I)

In this post, we assume a few formula and derive some interesting trigonometry formula:

Assumed equations:

sin(α+β)=sinαcosβ+sinβcosα
sin(x)=sin(x)
cos(x)=cos(x)
cos(α+π2)=sin(α)
sin(α+π2)=cos(α)

Let's get started:

cos(α+β)=sin(α+β+π2)=sinαcos(β+π2)+sin(β+π2)cosα=sinαsinβ+cosβcosα=cosαcosβsinαsinβ

So we obtain cos(α+β)=cosαcosβsinαsinβ, next, try subtraction

sin(αβ)=sin(α+(β))=sinαcos(β)+sin(β)cosα=sinαcosβsinβcosα

And also, we have

cos(αβ)=cos(α+(β))=cosαcos(β)sin(β)sinα=cosαcosβ+sinβcosα

So we obtain the subtraction formula as well, as a short summary, we have

sin(α+β)=sinαcosβ+sinβcosαsin(αβ)=sinαcosβsinβcosαcos(α+β)=cosαcosβsinβsinαcos(αβ)=cosαcosβ+sinβsinα

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