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Trigonometry Angles--Pi/8

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cos(pi/8)=1/2sqrt(2+sqrt(2))
(1)
cos((3pi)/8)=1/2sqrt(2-sqrt(2))
(2)
cot(pi/8)=1+sqrt(2)
(3)
cot((3pi)/8)=sqrt(2)-1
(4)
csc(pi/8)=sqrt(4+2sqrt(2))
(5)
csc((3pi)/8)=sqrt(4-2sqrt(2))
(6)
sec(pi/8)=sqrt(4-2sqrt(2))
(7)
sec((3pi)/8)=sqrt(4+2sqrt(2))
(8)
sin(pi/8)=1/2sqrt(2-sqrt(2))
(9)
sin((3pi)/8)=1/2sqrt(2+sqrt(2))
(10)
tan(pi/8)=sqrt(2)-1
(11)
tan((3pi)/8)=1+sqrt(2).
(12)

To derive these formulas, use the half-angle formulas

sin(pi/8)=sin(1/2·pi/4)
(13)
=sqrt(1/2(1-cospi/4))
(14)
=sqrt(1/2(1-1/2sqrt(2)))
(15)
=1/2sqrt(2-sqrt(2))
(16)
cos(pi/8)=cos(1/2·pi/4)
(17)
=sqrt(1/2(1+cospi/4))
(18)
=sqrt(1/2(1+(sqrt(2))/2))
(19)
=1/2sqrt(2+sqrt(2))
(20)
tan(pi/8)=sqrt((2-sqrt(2))/(2+sqrt(2)))
(21)
=sqrt(((2-sqrt(2))^2)/(4-2))
(22)
=sqrt((4+2-4sqrt(2))/2)
(23)
=sqrt(3-2sqrt(2))
(24)
=sqrt(2)-1
(25)
cot(pi/8)=1/(sqrt(2)-1)
(26)
=(sqrt(2)+1)/(2-1)
(27)
=sqrt(2)+1.
(28)

See also

Octagon, Trigonometry Angles, Trigonometry, Trigonometry Angles--Pi/4

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Cite this as:

Weisstein, Eric W. "Trigonometry Angles--Pi/8." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/TrigonometryAnglesPi8.html

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