已知函数f(x)=sinx/2cosx/2+cos^2(x/2)-1/2

问题描述:

已知函数f(x)=sinx/2cosx/2+cos^2(x/2)-1/2
f(x)=sin(x/2)cos(x/2)+cos^2(x/2)-(1/2)
=(1/2)×sinx+(1/2)×(cosx+1)-(1/2)
=(1/2)×(sinx+cosx)
=(√2/2)×sin(x+(π/4))
(1)若f(a)=√2/4=(√2/2)×sin(a+(π/4))
则sin(a+(π/4))=1/2,a+(π/4)=2kπ+(π/2)±((2π)/3);
又a∈(0,π),则a=π/12或(5π)/12;
为什么(1/2)×(sinx+cosx)
=(√2/2)×sin(x+(π/4))
为什么还要乘以(1/2)*√2*(1/2)啊

这个简单(1/2)×(sinx+cosx)=(1/2)*√2*(1/√2)(sinx+cosx)=(√2/2)(√2/2*sinx+√2/2*cosx)而sin(π/4)=cos(π/4)=√2/2所以就变成了上式=(√2/2)(cos(π/4)*sinx+sin(π/4)*cosx)根据和角公...