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Trigonometry Values Pi/20


$\displaystyle \sin\left({\pi\over 20}\right)$ $\textstyle =$ $\displaystyle \sin \left({{1\over 2}{\pi\over 10}}\right)= \sqrt{{1\over 2} \left({1-\cos{\pi\over 10}}\right)}$  
  $\textstyle =$ $\displaystyle {\textstyle{1\over 4}}\sqrt{8-2\sqrt{10+2\sqrt{5}}}$  
  $\textstyle \approx$ $\displaystyle 0.15643$ (1)
$\displaystyle \cos\left({\pi\over 20}\right)$ $\textstyle =$ $\displaystyle \cos \left({{1\over 2}{\pi\over 10}}\right)= \sqrt{{1\over 2} \left({1+\cos{\pi\over 10}}\right)}$  
  $\textstyle =$ $\displaystyle {\textstyle{1\over 4}}\sqrt{8+2\sqrt{10+2\sqrt{5}}}$  
  $\textstyle \approx$ $\displaystyle 0.98768$ (2)
$\displaystyle \tan\left({\pi\over 20}\right)$ $\textstyle =$ $\displaystyle 1+\sqrt{5}-\sqrt{5+2\sqrt{5}}$  
  $\textstyle \approx$ $\displaystyle 0.15838.$ (3)


An interesting near-identity is given by

\begin{displaymath}
{1\over 4}\left[{\cos({\textstyle{1\over 10}})+\cosh({\texts...
...}\,)\cosh({\textstyle{1\over 20}}\sqrt{2}\,)}\right]\approx 1.
\end{displaymath} (4)

In fact, the left-hand side is approximately equal to $1+2.480\times 10^{-13}$.




© 1996-9 Eric W. Weisstein
1999-05-26