2cosx=sinx+1求x=?

题目

2cosx=sinx+1求x=?

答案

记t=tan(x/2),由万能公式,做代换
得sinx=2t/(1+t²),cost=(1-t²)/(1+t²)
∴2(1-t²)/(1+t²)=2t/(1+t²)+1
=>2(1-t²)=2t+1+t² => 3t²+2t-1=0
=>(3t-1)(t+1)=0 => t=1/3或-1
即tan(x/2)=1/3或-1
∴x/2=arctan(1/3)+kπ 或 -π/4+kπ,k为任意整数
即x=2arctan(1/3)+2kπ 或 -π/2+2kπ,k为任意整数