## Trigonometry Values 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

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.