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三角函数角--Pi/20

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三角函数的值可以精确地表示为 pi/20 的整数倍。对于 cosx，

cos(pi/(20))=1/4sqrt(8+2sqrt(10+2sqrt(5)))
(1)
cos((3pi)/(20))=1/4sqrt(8+2sqrt(10-2sqrt(5)))
(2)
cos((7pi)/(20))=1/4sqrt(8-2sqrt(10-2sqrt(5)))
(3)
cos((9pi)/(20))=1/4sqrt(8-2sqrt(10+2sqrt(5)))
(4)

对于 cotx，

cot(pi/(20))=1+sqrt(5)+sqrt(5+2sqrt(5))
(5)
cot((3pi)/(20))=-1+sqrt(5)+sqrt(5-2sqrt(5))
(6)
cot((7pi)/(20))=-1+sqrt(5)-sqrt(5-2sqrt(5))
(7)
cot((9pi)/(20))=1+sqrt(5)-sqrt(5+2sqrt(5))
(8)

对于 cscx，

csc(pi/(20))=sqrt(12+4sqrt(5)+2sqrt(50+22sqrt(5)))
(9)
csc((3pi)/(20))=sqrt(12-4sqrt(5)+2sqrt(50-22sqrt(5)))
(10)
csc((7pi)/(20))=sqrt(12-4sqrt(5)-2sqrt(50-22sqrt(5)))
(11)
csc((9pi)/(20))=sqrt(12+4sqrt(5)-2sqrt(50+22sqrt(5)))
(12)

对于 secx，

sec(pi/(20))=sqrt(12+4sqrt(5)-2sqrt(50+22sqrt(5)))
(13)
sec((3pi)/(20))=sqrt(12-4sqrt(5)-2sqrt(50-22sqrt(5)))
(14)
sec((7pi)/(20))=sqrt(12-4sqrt(5)+2sqrt(50-22sqrt(5)))
(15)
sec((9pi)/(20))=sqrt(12+4sqrt(5)+2sqrt(50+22sqrt(5)))
(16)

对于 sinx，

sin(pi/(20))=1/4sqrt(8-2sqrt(10+2sqrt(5)))
(17)
sin((3pi)/(20))=1/4sqrt(8-2sqrt(10-2sqrt(5)))
(18)
sin((7pi)/(20))=1/4sqrt(8+2sqrt(10-2sqrt(5)))
(19)
sin((9pi)/(20))=1/4sqrt(8+2sqrt(10+2sqrt(5)))
(20)

以及对于 tanx，

tan(pi/(20))=1+sqrt(5)-sqrt(5+2sqrt(5))
(21)
tan((3pi)/(20))=-1+sqrt(5)-sqrt(5-2sqrt(5))
(22)
tan((7pi)/(20))=-1+sqrt(5)+sqrt(5-2sqrt(5))
(23)
tan((9pi)/(20))=1+sqrt(5)+sqrt(5+2sqrt(5)).
(24)

这些可以从半角公式推导出来

sin(pi/(20))=sin(1/2·pi/(10))
(25)
=sqrt(1/2(1-cospi/(10)))
(26)
=1/4sqrt(8-2sqrt(10+2sqrt(5)))
(27)
cos(pi/(20))=cos(1/2·pi/(10))
(28)
=sqrt(1/2(1+cospi/(10)))
(29)
=1/4sqrt(8+2sqrt(10+2sqrt(5)))
(30)
tan(pi/(20))=1+sqrt(5)-sqrt(5+2sqrt(5)).
(31)

参见

三角函数角, 三角学三角函数角--Pi/10

使用 Wolfram|Alpha 探索

引用为

Weisstein, Eric W. "三角函数角--Pi/20。" 来自 MathWorld -- Wolfram Web 资源。 https://mathworld.net.cn/TrigonometryAnglesPi20.html

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