求证sinπ/3+sin2π/3+...+sinnπ/3=2sinnπ/6sin(n+1)π/6
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解决时间 2021-02-13 03:04
- 提问者网友:痞子房西
- 2021-02-12 07:02
求证sinπ/3+sin2π/3+...+sinnπ/3=2sinnπ/6sin(n+1)π/6
最佳答案
- 二级知识专家网友:哥在撩妹请勿打扰
- 2021-02-12 08:18
证明:(数学归纳法)
(1)当n=1时,左边=sin(π/3)=√3/2
右边=2sin(π/6)sin(2π/6)=2(1/2)(√3/2)=√3/2
则左边=右边,原命题成立。
(2)假定当n=k时原命题成立,即
sin(π/3)+sin(2π/3)+...+sin(kπ/3)=2sin(kπ/6)sin((k+1)π/6)
那么,当n=k+1时,有
sin(π/3)+sin(2π/3)+...+sin(kπ/3)+sin((k+1)π/3)
=2sin(kπ/6)sin((k+1)π/6)+sin((k+1)π/3)
=2sin(kπ/6)sin((k+1)π/6)+2sin((k+1)π/6)cos((k+1)π/6) (应用倍角公式)
=2sin((k+1)π/6)[sin(kπ/6)+cos((k+1)π/6)]
=2sin((k+1)π/6)[cos(π/2-kπ/6)+cos((k+1)π/6)] (应用诱导公式)
=2sin((k+1)π/6)[cos((3-k)π/6)+cos((k+1)π/6)]
=2sin((k+1)π/6)[2cos(π/3)cos((k-1)π/6)] (应用和差化积公式)
=2sin((k+1)π/6)[2(1/2)cos((k-1)π/6)]
=2sin((k+1)π/6)cos((1-k)π/6)
=2sin((k+1)π/6)sin(π/2-(1-k)π/6) (应用诱导公式)
=2sin((k+1)π/6)sin((k+2)π/6)
则左边=右边,原命题成立。
于是,有数学归纳法知
sin(π/3)+sin(2π/3)+...+sin(nπ/3)=2sin(nπ/6)sin((n+1)π/6)成立,证毕。
(1)当n=1时,左边=sin(π/3)=√3/2
右边=2sin(π/6)sin(2π/6)=2(1/2)(√3/2)=√3/2
则左边=右边,原命题成立。
(2)假定当n=k时原命题成立,即
sin(π/3)+sin(2π/3)+...+sin(kπ/3)=2sin(kπ/6)sin((k+1)π/6)
那么,当n=k+1时,有
sin(π/3)+sin(2π/3)+...+sin(kπ/3)+sin((k+1)π/3)
=2sin(kπ/6)sin((k+1)π/6)+sin((k+1)π/3)
=2sin(kπ/6)sin((k+1)π/6)+2sin((k+1)π/6)cos((k+1)π/6) (应用倍角公式)
=2sin((k+1)π/6)[sin(kπ/6)+cos((k+1)π/6)]
=2sin((k+1)π/6)[cos(π/2-kπ/6)+cos((k+1)π/6)] (应用诱导公式)
=2sin((k+1)π/6)[cos((3-k)π/6)+cos((k+1)π/6)]
=2sin((k+1)π/6)[2cos(π/3)cos((k-1)π/6)] (应用和差化积公式)
=2sin((k+1)π/6)[2(1/2)cos((k-1)π/6)]
=2sin((k+1)π/6)cos((1-k)π/6)
=2sin((k+1)π/6)sin(π/2-(1-k)π/6) (应用诱导公式)
=2sin((k+1)π/6)sin((k+2)π/6)
则左边=右边,原命题成立。
于是,有数学归纳法知
sin(π/3)+sin(2π/3)+...+sin(nπ/3)=2sin(nπ/6)sin((n+1)π/6)成立,证毕。
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