Rather surprisingly, trigonometric functions of 
for 
an integer can be expressed in terms of sums, products, and
finite root extractions because 17 is a Fermat
prime. This makes the heptadecagon a constructible,
as first proved by Gauss. Although Gauss did not actually explicitly provide a construction,
he did derive the trigonometric formulas below using a series of intermediate variables
from which the final expressions were then built up.
Let
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(1)
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(2)
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(3)
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(4)
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(5)
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then
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(6)
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(7)
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(8)
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(9)
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(10)
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(11)
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 |  | ![1/(16)[delta+sqrt(2)(alpha+epsilon^*)]](/images/equations/TrigonometryAnglesPi17/Inline38.svg) |
(12)
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(13)
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 |  | ![1/(128)[sqrt(2)delta+2(alpha+epsilon^*)][4epsilon^*^2-2sqrt(2)deltaepsilon^*+8sqrt(2)epsilon-(sqrt(2)delta+2epsilon^*)alpha]^(1/2)](/images/equations/TrigonometryAnglesPi17/Inline44.svg) |
(14)
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(15)
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 |  | ![1/(16)[136-8sqrt(17)+8sqrt(2)epsilon-2(sqrt(34)-3sqrt(2))epsilon^*+2beta(delta+sqrt(2)epsilon^*)]^(1/2)](/images/equations/TrigonometryAnglesPi17/Inline50.svg) |
(16)
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(17)
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(18)
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(19)
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There are some interesting analytic formulas involving the trigonometric functions of 
.
Define
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(20)
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(21)
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(22)
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 |  | ![1/4[g_i(x)-1]](/images/equations/TrigonometryAnglesPi17/Inline72.svg) |
(23)
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(24)
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where 
or 4. Then
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(25)
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(26)
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Another interesting identity is given by
![tan(1/4tan^(-1)4)=2[cos((6pi)/(17))+cos((10pi)/(17))],](/images/equations/TrigonometryAnglesPi17/NumberedEquation1.svg) |
(27)
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where both sides are equal to
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(28)
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(Wickner 1999).
