info prev up next book cdrom email home

Trigonometry Values Pi/15


$\displaystyle \sin\left({\pi\over 15}\right)$ $\textstyle =$ $\displaystyle \sin\left({{\pi\over 6}-{\pi\over 10}}\right)$  
  $\textstyle =$ $\displaystyle \sin\left({\pi\over 6}\right)\cos\left({\pi\over 10}\right)-\sin\left({\pi\over 10}\right)\cos\left({\pi\over 6}\right)$  
  $\textstyle =$ $\displaystyle {1\over 2} \sqrt{{1\over 8}\left({5+\sqrt{5}}\right)} - {\sqrt{3}\over 2}{1\over 4}(\sqrt{5}-1)$  
  $\textstyle =$ $\displaystyle {\textstyle{1\over 16}}(2\sqrt{3}-2\sqrt{15}+\sqrt{40+8\sqrt{5}}\,)$ (1)

and
$\displaystyle \cos\left({ \pi\over 15}\right)$ $\textstyle =$ $\displaystyle \cos\left({{\pi\over 6}-{\pi\over 10}}\right)$  
  $\textstyle =$ $\displaystyle \cos\left({\pi\over 6}\right)\cos\left({\pi\over 10}\right)+\sin\left({\pi\over 6}\right)\sin\left({\pi\over 10}\right)$  
  $\textstyle =$ $\displaystyle {\sqrt{3}\over 2}\sqrt{{1\over 8}\left({5+\sqrt{5}}\right)} + {1\over 2} {1\over 4}(\sqrt{5}-1)$  
  $\textstyle =$ $\displaystyle {\textstyle{1\over 8}}(\sqrt{30+6\sqrt{5}}+\sqrt{5}-1).$ (2)

Summarizing,
$\displaystyle \sin\left({\pi\over 15}\right)$ $\textstyle =$ $\displaystyle {\textstyle{1\over 16}}(2\sqrt{3}-2\sqrt{15}+\sqrt{40+8\sqrt{5}}\,)$ (3)
  $\textstyle \approx$ $\displaystyle 0.20791$ (4)
$\displaystyle \sin\left({2\pi\over 15}\right)$ $\textstyle =$ $\displaystyle {\textstyle{1\over 8}}(\sqrt{3}+\sqrt{15}-\sqrt{10-2\sqrt{5}}\,)$ (5)
  $\textstyle \approx$ $\displaystyle 0.40674$ (6)
$\displaystyle \cos\left({ \pi\over 15}\right)$ $\textstyle =$ $\displaystyle {\textstyle{1\over 8}}(\sqrt{30+6\sqrt{5}}+\sqrt{5}-1) \approx 0.97815$ (7)
$\displaystyle \cos\left({2\pi\over 15}\right)$ $\textstyle =$ $\displaystyle {\textstyle{1\over 8}}(\sqrt{30-6\sqrt{5}}+1) \approx 0.91355$ (8)
$\displaystyle \tan\left({ \pi\over 15}\right)$ $\textstyle =$ $\displaystyle {\textstyle{1\over 2}}(3\sqrt{3}-\sqrt{15}-\sqrt{50-22\sqrt{5}}\,)$ (9)
  $\textstyle \approx$ $\displaystyle 0.21256.$ (10)




© 1996-9 Eric W. Weisstein
1999-05-26