解: f(A)=m.n
=1*(-cosA+1*sinA.
=sinA-cosA.
=√2sin(A+π/4).
1. 当sin(A+π/4)=-1, 则 f(A)=-√2;√√√
当sin((A+π/4)=1, 则 f(A)=√2.
∴ f(x)∈[-√2,√2].
2. |m|=√2, |n|=1.
cos=m.n/|m|.|n|=cosπ/3.
√2sin(A+π/4)/(√2*1)=(1/2).
√2sin(A+π/4)=√2/2.
sin(A+π/4)=1/2.
A+π/4=π/6; A=-π/12 (舍去).
或,A+π/4=5π/6.
∴ ∠A=7π/12.(=105°)
已知∠C=60°.
∴∠B=180-105-60=15° .
由正弦定理,得: c/sinC=b/sinB.
b=csinB/sinC.
=√6*[(√2)/4](√3-1)/(1/2).
=√3(√3-1).
∴b=3-√3.