Re: [中學] 四邊形角度問題

[JianMing (小明)]

※ 引述《WRD (旅行是我的神)》之銘言：
: http://imgur.com/IhR3kIB
: 我只有想到用斜率tan去算
: 但這是國中題目
: 不知道有什麼更直觀的解法嗎

先證明 tan(27°)tan(87°)tan(21°)tan(15°)=1

proof:

(1):

sin(27°)sin(87°)sin(21°)sin(15°)

=1/2(cos(60°)-cos(114°)) 1/2(cos(6°)-cos(36°))


(2):

cos(27°)cos(87°)cos(21°)cos(15°)

=1/2(cos(60°)+cos(114°)) 1/2(cos(6°)+cos(36°))


(1)-(2):

sin(27°)sin(87°)sin(21°)sin(15°)

-cos(27°)cos(87°)cos(21°)cos(15°)

=-1/2(cos(60°)cos(36°)+cos(114°)cos(6°))

=-1/2(1/2cos(36°)+1/2(cos(120°)+cos(108°)))

=-1/4(cos(36°)+cos(120°)+cos(108°))

=-1/4((√5+1)/4-1/2-(√5-1)/4)=0


所以tan(27°)tan(87°)tan(21°)tan(15°)=1 得證




若tan(27°)tan(87°)tan(21°)tan(x)=1 (0°<x<90°)

可得tan(x)=tan(15°)

x=15°

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