Question

已知asinα bcosα=m,bsinα-acosα=n,求证a^2+b^2=m^2+n^2

Answer 1

你的题目有问题，应该是"
已知asinα+bcosα=m,bsinα-acosα=n,求证a^2+b^2=m^2+n^2"才对

证明：m^2=(asinα+bcosα)^2=(asinα)^2+2absinαcosα+(bcosα)^2
n^2=(bsinα-acosα)^2=(bsinα)^2-2absinαcosα+(acosα)^2
m^2+n^2=(asinα)^2+(bcosα)^2+(bsinα)^2+(acosα)^2

=a^2*(sinα^2+cosα^2)+b^2(sinα^2+cosα^2)
因为
sinα^2+cosα^2=1，所以a^2*(sinα^2+cosα^2)+b^2(sinα^2+cosα^2)=a^2+b^2
即a^2+b^2=m^2+n^2