不等式スレッド
-log(cos(x)) (|x|≦π/2)の凸不等式をつかう。
(x+y)/2≧0のとき
0≦4sin((x+y)/2)cos(x/2)cos(y/2)=4cos(π/2-(x+y)/2)cos(x/2)cos(y/2)≦4(cos(π/6))＾3=3(√3)/2
(x+y)/2<0のとき
0≧4sin((x+y)/2)cos(x/2)cos(y/2)=-4cos(-π/2-(x+y)/2)cos(x/2)cos(y/2)≧-4(cos(-π/6))＾3=-3(√3)/2
∴-3(√3)/2≦sin(x)+sin(y)+sin(x+y)≦3(√3)/2
(x+y)/2≧0のとき
0≦4sin((x+y)/2)cos(x/2)cos(y/2)=4cos(π/2-(x+y)/2)cos(x/2)cos(y/2)≦4(cos(π/6))＾3=3(√3)/2
(x+y)/2<0のとき
0≧4sin((x+y)/2)cos(x/2)cos(y/2)=-4cos(-π/2-(x+y)/2)cos(x/2)cos(y/2)≧-4(cos(-π/6))＾3=-3(√3)/2
∴-3(√3)/2≦sin(x)+sin(y)+sin(x+y)≦3(√3)/2
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