∠A:∠B:∠C=1:2:4,
A=π/7
B=2π/7
C=4π/7
1/sinB+1/sinC
=(sinB+sinC)/(sinBsinC)
=4sin[(B+C)/2]cos[(B-C)/2]/[cos(B-C)-cos(B+C)]
=4sin(3π/7)cos(π/7)/[cos(2π/7)-cos(6π/7)]
=4sin(3π/7)cos(π/7)/[cos(2π/7)+cos(π/7)]
=2sin(3π/7)cos(π/7)/[cos(3π/14)cos(π/14)]
=2sin(3π/7)cos(π/7)/[cos((3π/14)sin(3π/7)]
=2cos(π/7)/[cos((3π/14)]
=2cos(π/7)sin(π/7)/[sin(π/7)cos(3π/14)]
=sin(2π/7)/[sin(π/7)cos(3π/14)]
=cos(3π/14)/[sin(π/7)cos(3π/14)]
=1/sin(π/7)
=1/sinA
1/sinB+1/sinC=1/sinA
由正弦定理
1/AB+1/AC=1/BC