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=cosαcosβ−sinαsinβ
So we obtain cos(α+β)=cosαcosβ−sinαsinβ, next, try subtraction
sin(α−β)=sin(α+(−β))=sinαcos(−β)+sin(−β)cosα=sinαcosβ−sinβcosα
And also, we have
cos(α−β)=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(α−β)=sinαcosβ−sinβcosαcos(α+β)=cosαcosβ−sinβsinαcos(α−β)=cosαcosβ+sinβsinα
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=cosαcosβ−sinαsinβ
So we obtain cos(α+β)=cosαcosβ−sinαsinβ, next, try subtraction
sin(α−β)=sin(α+(−β))=sinαcos(−β)+sin(−β)cosα=sinαcosβ−sinβcosα
And also, we have
cos(α−β)=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(α−β)=sinαcosβ−sinβcosαcos(α+β)=cosαcosβ−sinβsinαcos(α−β)=cosαcosβ+sinβsinα
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