Trigonometry Angles--Pi/20
Values of the trigonometric functions can be expressed exactly for integer multiples of pi/20. For cosx,
[画像:cos(pi/(20))] = 1/4sqrt(8+2sqrt(10+2sqrt(5)))
(1)
[画像:cos((3pi)/(20))] = 1/4sqrt(8+2sqrt(10-2sqrt(5)))
(2)
[画像:cos((7pi)/(20))] = 1/4sqrt(8-2sqrt(10-2sqrt(5)))
(3)
[画像:cos((9pi)/(20))] = 1/4sqrt(8-2sqrt(10+2sqrt(5)))
(4)
for cotx,
[画像:cot(pi/(20))] = 1+sqrt(5)+sqrt(5+2sqrt(5))
(5)
[画像:cot((3pi)/(20))] = -1+sqrt(5)+sqrt(5-2sqrt(5))
(6)
[画像:cot((7pi)/(20))] = -1+sqrt(5)-sqrt(5-2sqrt(5))
(7)
[画像:cot((9pi)/(20))] = 1+sqrt(5)-sqrt(5+2sqrt(5))
(8)
for cscx,
[画像:csc(pi/(20))] = sqrt(12+4sqrt(5)+2sqrt(50+22sqrt(5)))
(9)
[画像:csc((3pi)/(20))] = sqrt(12-4sqrt(5)+2sqrt(50-22sqrt(5)))
(10)
[画像:csc((7pi)/(20))] = sqrt(12-4sqrt(5)-2sqrt(50-22sqrt(5)))
(11)
[画像:csc((9pi)/(20))] = sqrt(12+4sqrt(5)-2sqrt(50+22sqrt(5)))
(12)
for secx,
[画像:sec(pi/(20))] = sqrt(12+4sqrt(5)-2sqrt(50+22sqrt(5)))
(13)
[画像:sec((3pi)/(20))] = sqrt(12-4sqrt(5)-2sqrt(50-22sqrt(5)))
(14)
[画像:sec((7pi)/(20))] = sqrt(12-4sqrt(5)+2sqrt(50-22sqrt(5)))
(15)
[画像:sec((9pi)/(20))] = sqrt(12+4sqrt(5)+2sqrt(50+22sqrt(5)))
(16)
for sinx,
[画像:sin(pi/(20))] = 1/4sqrt(8-2sqrt(10+2sqrt(5)))
(17)
[画像:sin((3pi)/(20))] = 1/4sqrt(8-2sqrt(10-2sqrt(5)))
(18)
[画像:sin((7pi)/(20))] = 1/4sqrt(8+2sqrt(10-2sqrt(5)))
(19)
[画像:sin((9pi)/(20))] = 1/4sqrt(8+2sqrt(10+2sqrt(5)))
(20)
and for tanx,
[画像:tan(pi/(20))] = 1+sqrt(5)-sqrt(5+2sqrt(5))
(21)
[画像:tan((3pi)/(20))] = -1+sqrt(5)-sqrt(5-2sqrt(5))
(22)
[画像:tan((7pi)/(20))] = -1+sqrt(5)+sqrt(5-2sqrt(5))
(23)
[画像:tan((9pi)/(20))] = 1+sqrt(5)+sqrt(5+2sqrt(5)).
(24)
These can be derived from the half-angle formulas
= [画像:sqrt(1/2(1-cospi/(10)))]
(26)
= 1/4sqrt(8-2sqrt(10+2sqrt(5)))
(27)
= [画像:sqrt(1/2(1+cospi/(10)))]
(29)
= 1/4sqrt(8+2sqrt(10+2sqrt(5)))
(30)
[画像:tan(pi/(20))] = 1+sqrt(5)-sqrt(5+2sqrt(5)).
(31)
See also
Trigonometry Angles, Trigonometry, Trigonometry Angles--Pi/10Explore with Wolfram|Alpha
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Cite this as:
Weisstein, Eric W. "Trigonometry Angles--Pi/20." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/TrigonometryAnglesPi20.html