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Trigonometry Angles--Pi/17


Rather surprisingly, trigonometric functions of npi/17 for n 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

epsilon=sqrt(17+sqrt(17))
(1)
epsilon^*=sqrt(17-sqrt(17))
(2)
delta=sqrt(17)-1
(3)
alpha=sqrt(34+6sqrt(17)+sqrt(2)(sqrt(17)-1)epsilon^*-8sqrt(2)epsilon)
(4)
beta=2sqrt(17+3sqrt(17)-2sqrt(2)epsilon-sqrt(2)epsilon^*),
(5)

then

sin(pi/(17))=1/8sqrt(2)sqrt(epsilon^*^2-sqrt(2)(alpha+epsilon^*))
(6)
 approx 0.18375
(7)
cos(pi/(17))=1/8sqrt(2)sqrt(15+sqrt(17)+sqrt(2)(alpha+epsilon^*))
(8)
 approx 0.98297
(9)
sin((2pi)/(17))=1/(16)sqrt(2)sqrt(4epsilon^*^2-2sqrt(2)deltaepsilon^*+8sqrt(2)epsilon-(sqrt(2)delta+2epsilon^*)alpha)
(10)
 approx 0.36124
(11)
cos((2pi)/(17))=1/(16)[delta+sqrt(2)(alpha+epsilon^*)]
(12)
 approx 0.93247
(13)
sin((4pi)/(17))=1/(128)[sqrt(2)delta+2(alpha+epsilon^*)][4epsilon^*^2-2sqrt(2)deltaepsilon^*+8sqrt(2)epsilon-(sqrt(2)delta+2epsilon^*)alpha]^(1/2)
(14)
 approx 0.67370
(15)
sin((8pi)/(17))=1/(16)[136-8sqrt(17)+8sqrt(2)epsilon-2(sqrt(34)-3sqrt(2))epsilon^*+2beta(delta+sqrt(2)epsilon^*)]^(1/2)
(16)
 approx 0.99573
(17)
cos((8pi)/(17))=1/(16)(delta+sqrt(2)epsilon^*-2sqrt(17+3sqrt(17)-sqrt(2)epsilon^*-2sqrt(2)epsilon))
(18)
 approx 0.09227.
(19)

There are some interesting analytic formulas involving the trigonometric functions of npi/17. Define

P(x)=(x-1)(x-2)(x^2+1)
(20)
g_1(x)=(2+sqrt(P(x)))/(1-x)
(21)
g_4(x)=(2-sqrt(P(x)))/(1-x)
(22)
f_i(x)=1/4[g_i(x)-1]
(23)
a=1/4tan^(-1)4,
(24)

where i=1 or 4. Then

f_1(tana)=cos((2pi)/(17))
(25)
f_4(tana)=cos((8pi)/(17)).
(26)

Another interesting identity is given by

 tan(1/4tan^(-1)4)=2[cos((6pi)/(17))+cos((10pi)/(17))],
(27)

where both sides are equal to

 C=(sqrt(2(17+sqrt(17)))-sqrt(17)-1)/4
(28)

(Wickner 1999).


See also

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

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