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Saturday, January 24, 2015

Trigonometry of regular polgons

In school we learned that the exterior angle of a regular n-sided polygon is 360/n degrees or 2π/n radians.  Similarly the interior angle of a regular n-side polygon is 180(n2)/n degrees or (n2)π/n radians.  It can be shown that the trigonometry functions (cos, sin, tan, ...) of a rational multiple of π is an algebraic number, i.e, a root of a polynomial with integer coefficients.  Gauss showed that n being a power of 2 multiplied by distinct Fermat primes (i.e. primes of the form 22m+1) is sufficient to ensure the polygon can be constructed with straightedge and compass and Wantzel showed that this is also necessary (http://mathworld.wolfram.com/TrigonometryAngles.html).  This implies that the expression of sin(kπ/n), cos(kπ/n), etc. can be written explicitly using radicals. Some expressions for these angles are given in (http://en.wikipedia.org/wiki/Exact_trigonometric_constants) for angles where the number of minimal nesting of radicals is 2 or less.

Other values of n for which the minimal nesting of radicals is 2 or less include n=40 and n=120.  For instance for n=40, we have:

sin(π40)=(158)2+2+28225+5
cos(π40)=(518)22+282+25+5
sin(3π40)=(518)22+282+255
cos(3π40)=(5+18)2+2+282255
sin(7π40)=(5+18)2+2282255
cos(7π40)=(5+18)22+282+255
sin(9π40)=(518)2+2+28225+5
cos(9π40)=(158)22+282+25+5
sin(11π40)=(158)22+282+25+5
cos(11π40)=(518)2+2+28225+5
sin(13π40)=(1+58)22+282+255
cos(13π40)=(5+18)2+2282255
sin(17π40)=(1+58)2+2+282+255
cos(17π40)=(518)22+282+255
sin(19π40)=(518)22+282+25+5
cos(19π40)=(158)2+2+28225+5






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