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
(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
(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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(12)
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(13)
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(14)
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(15)
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(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
(20)
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(21)
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(22)
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(23)
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(24)
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where or 4. Then
(25)
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(26)
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Another interesting identity is given by
(27)
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where both sides are equal to
(28)
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(Wickner 1999).