Trigonometric functions of
radians for
an integer not divisible by 3 (e.g., 40° and 80°) cannot
be expressed in terms of sums, products, and finite root extractions on real rational numbers because 9 is not a
product of distinct Fermat Primes. This also means that the Nonagon is not a
Constructible Polygon.
However, exact expressions involving roots of complex numbers can still be derived using the trigonometric identity
![\begin{displaymath}
\sin(3\alpha)=3\sin\alpha-4\sin^3\alpha.
\end{displaymath}](t_1982.gif) |
(1) |
Let
and
. Then the above identity gives the Cubic Equation
![\begin{displaymath}
4x^3-3x+{\textstyle{1\over 2}}\sqrt{3} = 0
\end{displaymath}](t_1984.gif) |
(2) |
![\begin{displaymath}
x^3-{\textstyle{3\over 4}} x = - {\textstyle{1\over 8}}\sqrt{3}.
\end{displaymath}](t_1985.gif) |
(3) |
This cubic is of the form
![\begin{displaymath}
x^3+p x = q,
\end{displaymath}](t_1986.gif) |
(4) |
where
The Discriminant is then
There are therefore three Real distinct roots, which are approximately
, 0.3240, and 0.6428. We want the one in the first Quadrant, which is 0.3240.
Similarly,
Because of the Newton's Relations, we have the identities
![\begin{displaymath}
\sin\left({\pi\over 9}\right)\sin\left({2\pi\over 9}\right)\sin\left({4\pi\over 9}\right)= {\textstyle{1\over 8}}\sqrt{3}
\end{displaymath}](t_2003.gif) |
(10) |
![\begin{displaymath}
\cos\left({\pi\over 9}\right)\cos\left({2\pi\over 9}\right)\cos\left({4\pi\over 9}\right)= {\textstyle{1\over 8}}
\end{displaymath}](t_2004.gif) |
(11) |
![\begin{displaymath}
\tan\left({\pi\over 9}\right)\tan\left({2\pi\over 9}\right)\tan\left({4\pi\over 9}\right)= \sqrt{3}.
\end{displaymath}](t_2005.gif) |
(12) |
See also Nonagon, Star of Goliath
© 1996-9 Eric W. Weisstein
1999-05-26