[sin(x/2)]^2>1/2
sinx=2sin(x/2)cos(x/2)=2/3
sin(x/2)cos(x/2)=1/3
cos(x/2)=1/[3sin(x/2)]
令a=sin(x/2)
[sin(x/2)]^2+[cos(x/2)]^2=1
所以a^2+(1/3a)^2=1
兩邊乘a^2
a^4+1/9=a^2
a^4-a^2+1/9=0
a^2=[1±√5/3]/2=(3±√5)/6
因?yàn)閇sin(x/2)]^2>1/2
所以a^2=(3+√5)/6=(18+6√5)/36
分子=18+2√(9*5)=(√15)^2+2√15*√3+(√3)^2=(√15+√3)^2
所以a^2=(√15+√3)^2/6^2
sin(x/2)>sinpi/4=√2/2>0
所以sin(x/2)=(√15+√3)/6