请问,已知△ABC中,a=10,B=60°,C=45°,则c=?

题目

请问,已知△ABC中,a=10,B=60°,C=45°,则c=?

答案

A=180°-B-C=180°-60°-45°=75°
1-2sin²75°=cos150°
sin²75°=(1-cos150°)/2=(1+√3/2)/2=(2+√3)/4
sin75°=√[(2+√3)/4]=√[(4+2√3)/8]=√{(√3+1)²/8}=(√3+1)/(2√2)
a/sinA=c/sinC
c=asinC/sinA
=10×sin45°/sin75°
=10×(√2/2)/[(√3+1)/(2√2)]
=20/(√3+1)
=10(√3-1)
=10√3 -10