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Trigonometry Values Pi/18

The exact values of $\cos(\pi/18)$ and $\sin(\pi/18)$ are given by infinite Nested Radicals.

$\displaystyle \sin\left({\pi\over 18}\right)$ $\textstyle =$ $\displaystyle {\textstyle{1\over 2}}\sqrt{2-\sqrt{2+\sqrt{2+\sqrt{2-\ldots}}}}$  
  $\textstyle \approx$ $\displaystyle 0.17365,$  

where the sequence of signs +, +, $-$ repeats with period 3, and
$\displaystyle \cos\left({\pi\over 18}\right)$ $\textstyle =$ $\displaystyle {\textstyle{1\over 6}}\sqrt{3}\left({\sqrt{8-\sqrt{8-\sqrt{8+\sqrt{8-\ldots}}}}+1}\right)$  
  $\textstyle \approx$ $\displaystyle 0.98481,$  

where the sequence of signs $-$, $-$, + repeats with period 3.




© 1996-9 Eric W. Weisstein
1999-05-26