sin(x+y)=sinxcosy+sinycosx和cos(x+y)=cosxcosy-sinxsiny的推导过程

Question

sin(x+y)=sinxcosy+sinycosx和cos(x+y)=cosxcosy-sinxsiny的推导过程

Answer 1

设x,y是锐角,作直径AB=1的圆O,C,D是位于AB两侧的圆周上的两点,连结CD,由托勒密定理有
CD?AB=BC?AD＋AC?BD． (*)
(1)设∠CAB=x,∠DAB=y(如图1),则AC=cosx,BC=sinαAD=cosy,BD=siny,CD=sin(x＋y),代入(*)得
sin(x＋y)=sinxcosy＋cosxsiny,(1)(2)设∠CAB=x,∠DBA=y,x≥y,AC=cosx,BC=sinx,AD=siny,BD=cosy,CD=cos(x－y),
代入(*)得
cos(x－y)=cosxcosy＋sinxsiny,(2)
由诱导公式易见(1),(2)对任意角x,y都成立,若用－y替换(1),(2)中的y,则可得
sin(x－y)=sinxcosy－cosxsiny,(3)
cos(x＋y)=cosxcosy－sinxsiny．(4)