Trigonometry Angles--Pi/8
[画像:cos(pi/8)] = 1/2sqrt(2+sqrt(2))
(1)
[画像:cos((3pi)/8)] = 1/2sqrt(2-sqrt(2))
(2)
[画像:cot(pi/8)] = 1+sqrt(2)
(3)
[画像:cot((3pi)/8)] = sqrt(2)-1
(4)
[画像:csc(pi/8)] = sqrt(4+2sqrt(2))
(5)
[画像:csc((3pi)/8)] = sqrt(4-2sqrt(2))
(6)
[画像:sec(pi/8)] = sqrt(4-2sqrt(2))
(7)
[画像:sec((3pi)/8)] = sqrt(4+2sqrt(2))
(8)
[画像:sin(pi/8)] = 1/2sqrt(2-sqrt(2))
(9)
[画像:sin((3pi)/8)] = 1/2sqrt(2+sqrt(2))
(10)
[画像:tan(pi/8)] = sqrt(2)-1
(11)
[画像:tan((3pi)/8)] = 1+sqrt(2).
(12)
To derive these formulas, use the half-angle formulas
= [画像:sqrt(1/2(1-cospi/4))]
(14)
= sqrt(1/2(1-1/2sqrt(2)))
(15)
= 1/2sqrt(2-sqrt(2))
(16)
= [画像:sqrt(1/2(1+cospi/4))]
(18)
= [画像:sqrt(1/2(1+(sqrt(2))/2))]
(19)
= 1/2sqrt(2+sqrt(2))
(20)
= [画像:sqrt(((2-sqrt(2))^2)/(4-2))]
(22)
= [画像:sqrt((4+2-4sqrt(2))/2)]
(23)
= sqrt(3-2sqrt(2))
(24)
= sqrt(2)-1
(25)
= [画像:(sqrt(2)+1)/(2-1)]
(27)
= sqrt(2)+1.
(28)
See also
Octagon, Trigonometry Angles, Trigonometry, Trigonometry Angles--Pi/4Explore with Wolfram|Alpha
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Cite this as:
Weisstein, Eric W. "Trigonometry Angles--Pi/8." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/TrigonometryAnglesPi8.html