 TOPICS # Trigonometry Angles--Pi/17

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)   (2)   (3)   (4)   (5)

then   (6)   (7)   (8)   (9)   (10)   (11)   (12)   (13)   (14)   (15)   (16)   (17)   (18)   (19)

There are some interesting analytic formulas involving the trigonometric functions of . Define   (20)   (21)   (22)   (23)   (24)

where or 4. Then   (25)   (26)

Another interesting identity is given by (27)

where both sides are equal to (28)

(Wickner 1999).

Constructible Polygon, Fermat Prime, Heptadecagon, Trigonometry Angles, Trigonometry

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## References

Casey, J. A Treatise on Plane Trigonometry, Containing an Account of Hyperbolic Functions, with Numerous Examples. Dublin: Hodges, Figgis, & Co., p. 220, 1888.Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, pp. 192-194 and 229-230, 1996.Dörrie, H. "The Regular Heptadecagon." §37 in 100 Great Problems of Elementary Mathematics: Their History and Solutions. New York: Dover, pp. 177-184, 1965.Ore, Ø. Number Theory and Its History. New York: Dover, 1988.Smith, D. E. A Source Book in Mathematics. New York: Dover, p. 348, 1994.Wickner, J. "Solution to Problem 1562: A Tangent and Cosine Identity." Math. Mag. 72, pp. 412-413, 1999.

## Cite this as:

Weisstein, Eric W. "Trigonometry Angles--Pi/17." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/TrigonometryAnglesPi17.html