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Trigonometry Values Pi/9

Trigonometric functions of $n\pi/9$ radians for $n$ 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} (1)

Let $\alpha\equiv{\pi/9}$ and $x\equiv\sin\alpha$. Then the above identity gives the Cubic Equation
\begin{displaymath}
4x^3-3x+{\textstyle{1\over 2}}\sqrt{3} = 0
\end{displaymath} (2)


\begin{displaymath}
x^3-{\textstyle{3\over 4}} x = - {\textstyle{1\over 8}}\sqrt{3}.
\end{displaymath} (3)

This cubic is of the form
\begin{displaymath}
x^3+p x = q,
\end{displaymath} (4)

where
$\displaystyle p$ $\textstyle =$ $\displaystyle - {\textstyle{3\over 4}}$ (5)
$\displaystyle q$ $\textstyle =$ $\displaystyle - {\textstyle{1\over 8}}\sqrt{3}.$ (6)

The Discriminant is then
$\displaystyle D$ $\textstyle \equiv$ $\displaystyle {\left({p\over 3}\right)}^3+{\left({q\over 2}\right)}^2$  
  $\textstyle =$ $\displaystyle {\left(-{1\over 4}\right)}^3+{\left({\sqrt{3}\over 16}\right)}^2 = - {1\over{16\cdot 4}}+{3\over{16\cdot 16}} = {-4+3 \over 256}$  
  $\textstyle =$ $\displaystyle -{1\over 256} < 0.$ (7)

There are therefore three Real distinct roots, which are approximately $-0.9848$, 0.3240, and 0.6428. We want the one in the first Quadrant, which is 0.3240.
$\displaystyle \sin\left({\pi \over 9}\right)$ $\textstyle =$ $\displaystyle {\root 3 \of {-{\sqrt{3} \over 16}+\sqrt{-{1\over 256}}}} + {\root 3 \of {-{\sqrt{3} \over 16}-\sqrt{-{1\over 256}}}}$  
  $\textstyle =$ $\displaystyle {\root 3 \of {-{\sqrt{3} \over 16} + {1\over 16} i}} - {\root 3 \of {{\sqrt{3}\over 16} + {1\over 16} i}}$  
  $\textstyle =$ $\displaystyle 2^{-4/3}(\root 3 \of {i-\sqrt{3}}-\root 3 \of{i+\sqrt{3}})$  
  $\textstyle \approx$ $\displaystyle 0.34202.$ (8)

Similarly,
$\displaystyle \cos\left({\pi\over 9}\right)$ $\textstyle =$ $\displaystyle 2^{-4/3}(\root 3 \of {1+i\sqrt{3}}+\root 3 \of{1-i\sqrt{3}}\,)$  
  $\textstyle \approx$ $\displaystyle 0.93969.$ (9)

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} (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} (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} (12)

See also Nonagon, Star of Goliath



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© 1996-9 Eric W. Weisstein
1999-05-26