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).