角OAC的度数 x,角OAB=80-x,
△AOB,OA/sin40=OB/sin(80-x) (正弦定理)
△AOC,OA/sin30=OC/sinx
sin40/sin30=(OB/OC)*sinx/sin(80-x) (正弦定理)
(OB/OC)*sinx/sin(80-x)=sin40/(1/2)
(OB/OC)*sinx/sin(80-x)=2sin40.(1)
△OBC,OB/sin20=OC/sin10,(正弦定理)
OB/OC=sin20/sin10=2sin10cos10/sin10=2cos10代入(1)
2cos10sinx/sin(80-x)=2sin40
sin80sinx=sin40sin(80-x)
2sin40cos40=sin40sin(80-x)
2cos40sinx=sin(80-x)=sin80cosx-cos80sinx
(2cos40+cos80)sinx=sin80cosx
tanx=sin80/(2cos40+cos80)
x=arctan[sin80/(2cos40+cos80)]