Trigonometry Values Pi/9

Trigonometric functions of $n\pi/9$ 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

$\begin{displaymath} \sin(3\alpha)=3\sin\alpha-4\sin^3\alpha. \end{displaymath}$

(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

, 0.3240, 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