解方程:sin(x/2)+cosx=1

题目

解方程:sin(x/2)+cosx=1

答案

cosx=1-2sin²(x/2)
sin(x/2)+1-2sin²(x/2)=1
sin(x/2)[1-2sin(x/2)]=0
sin(x/2)=0 1-2sin(x/2)=0
x/2=kπ x=2kπ
1-2sin(x/2)=0
sin(x/2)=1/2
x/2=π/6+2kπ 或
x/2=5π/6+2kπ
解得x=π/3+4kπ x=5π/3+4kπ
综上所述
x=2kπ x=π/3+4kπ x=5π/3+4kπ
k∈Z