|sinx|+|cosx|=π/4 ( ) A.无解 B.有唯一解 C.有不止一个解但解的个数有限 D.有无穷解

分四种情况

1）2kπ<=x<2kπ+π/2

sinx+cosx=π/4

sin(x+π/4)=根号2π/8

2kπ+π/4=<x+π/4<2kπ+3π/4

根号2/2=<sin(x+π/4)<=1

根号2π/8<根号2/2

方程无解

2）2kπ+π/2<=x<2kπ+π

sinx-cosx=π/4

sin(x-π/4)=根号2π/8

2kπ+π/4=<x-π/4<2kπ+3π/4

也无解

3）2kπ+π<=x<2kπ+3π/2

-sinx-cosx=π/4

sinx+cosx=-π/4

sin(x+π/4)=-根号2π/8

2kπ+π5/4=<x+π/4<2kπ+7π/4

-1=<sin(x+π/4)<=-根号2/2

-根号2π/8>-根号2/2

也无解

4）2kπ+3π/2<=x<2kπ+2π

同样无解，

所以方程无解