证明和差化积和积化和差公式

Question

如题所述

Answer 1

解:和差化积:

sina+sinb=2sin[(a+b)/2] cos[(a+b)/2]

设a=x+y, b=x-y, 则

x=(a+b)/2, y= (a-b)/2

sin(x+y)+sin(a-b)=2sinxcosy

=2sin[(a+b)/2]cos[(a-b)/2]

所以sina+sinb=2sin[(a+b)/2] cos[(a+b)/2]

积化和差:

 sinacosb=1/2[sin(a+b)+sin(a-b)]

因为sin(a+b)=sinacosb+cosasinb

sin(a-b)=sinacosb-cosasinb

将以上两式左右两边相加得,

 sin(a+b)+sin(a-b)=2 sinacosb

所以sinacosb=1/2[sin(a+b)+sin(a-b)]

Answer 2

sinx+siny=1/2*sin[(x+y)/2]cos[(x-y)/2]
sinx-siny=1/2*cos[(x+y)/2]sin[(x-y)/2]
cosx+cosy=1/2*cos[(x+y)/2]cos[(x-y)/2]
cosx-cosy=-1/2*sin[(x+y)/2]sin[(x-y)/2]