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Trigonometry Angles--Pi/16

cos(pi/(16))=1/2sqrt(2+sqrt(2+sqrt(2)))
(1)
cos((3pi)/(16))=1/2sqrt(2+sqrt(2-sqrt(2)))
(2)
cos((5pi)/(16))=1/2sqrt(2-sqrt(2-sqrt(2)))
(3)
cos((7pi)/(16))=1/2sqrt(2-sqrt(2+sqrt(2)))
(4)
cot(pi/(16))=+1+sqrt(2)+sqrt(4+2sqrt(2))
(5)
cot((3pi)/(16))=-1+sqrt(2)+sqrt(4-2sqrt(2))
(6)
cot((5pi)/(16))=+1-sqrt(2)+sqrt(4-2sqrt(2))
(7)
cot((7pi)/(16))=-1-sqrt(2)+sqrt(4+2sqrt(2))
(8)
csc(pi/(16))=sqrt(8+4sqrt(2)+2sqrt(20+14sqrt(2)))
(9)
csc((3pi)/(16))=sqrt(8-4sqrt(2)+2sqrt(20-14sqrt(2)))
(10)
csc((5pi)/(16))=sqrt(8-4sqrt(2)-2sqrt(20-14sqrt(2)))
(11)
csc((7pi)/(16))=sqrt(8+4sqrt(2)-2sqrt(20+14sqrt(2)))
(12)
sec(pi/(16))=sqrt(8+4sqrt(2)-2sqrt(20+14sqrt(2)))
(13)
sec((3pi)/(16))=sqrt(8-4sqrt(2)-2sqrt(20-14sqrt(2)))
(14)
sec((5pi)/(16))=sqrt(8-4sqrt(2)+2sqrt(20-14sqrt(2)))
(15)
sec((7pi)/(16))=sqrt(8+4sqrt(2)+2sqrt(20+14sqrt(2)))
(16)
sin(pi/(16))=1/2sqrt(2-sqrt(2+sqrt(2)))
(17)
sin((3pi)/(16))=1/2sqrt(2-sqrt(2-sqrt(2)))
(18)
sin((5pi)/(16))=1/2sqrt(2+sqrt(2-sqrt(2)))
(19)
sin((7pi)/(16))=1/2sqrt(2+sqrt(2+sqrt(2)))
(20)
tan(pi/(16))=sqrt(4+2sqrt(2))-sqrt(2)-1
(21)
tan((3pi)/(16))=sqrt(4-2sqrt(2))-sqrt(2)+1
(22)
tan((5pi)/(16))=sqrt(4-2sqrt(2))+sqrt(2)-1
(23)
tan((7pi)/(16))=sqrt(4+2sqrt(2))+sqrt(2)+1.
(24)

These can be derived from the half-angle formulas

sin(pi/(16))=sin(1/2·pi/8)
(25)
=sqrt(1/2(1-cospi/8))
(26)
=sqrt(1/2(1-1/2sqrt(2+sqrt(2))))
(27)
=sqrt(1/2-1/4sqrt(2+sqrt(2)))
(28)
=1/2sqrt(2-sqrt(2+sqrt(2)))
(29)
cos(pi/(16))=cos(1/2·pi/8)
(30)
=sqrt(1/2(1+cospi/8))
(31)
=sqrt(1/2(1+1/2sqrt(2+sqrt(2))))
(32)
=sqrt(1/2+1/4sqrt(2+sqrt(2)))
(33)
=1/2sqrt(2+sqrt(2+sqrt(2)))
(34)
tan(pi/(16))=sqrt((2-sqrt(2+sqrt(2)))/(2+sqrt(2+sqrt(2))))
(35)
=sqrt(4+2sqrt(2))-sqrt(2)-1.
(36)

See also

Hexadecagon, Trigonometry Angles, Trigonometry, Trigonometry Angles--Pi/8

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Cite this as:

Weisstein, Eric W. "Trigonometry Angles--Pi/16." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/TrigonometryAnglesPi16.html

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