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

Then

There are some interesting analytic formulas involving the trigonometric functions of . Define

where or 4. Then

**References**

Casey, J. *Plane Trigonometry.* 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.

© 1996-9

1999-05-26