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Trigonometry Values Pi/16


$\displaystyle \sin\left({\pi\over 16}\right)$ $\textstyle =$ $\displaystyle \sin\left({{1\over 2}\cdot{\pi\over 8}}\right)$  
  $\textstyle =$ $\displaystyle \sqrt{{\textstyle{1\over 2}}\left({1-\cos{\pi\over 8}}\right)}=\s...
...{\textstyle{1\over 2}}\left({1-{\textstyle{1\over 2}}\sqrt{2+\sqrt{2}}}\right)}$  
  $\textstyle =$ $\displaystyle \sqrt{{\textstyle{1\over 2}}-{\textstyle{1\over 4}}\sqrt{2+\sqrt{2}}} ={\textstyle{1\over 2}}\sqrt{2-\sqrt{2+\sqrt{2}}}$ (1)
$\displaystyle \cos\left({\pi\over 16}\right)$ $\textstyle =$ $\displaystyle \cos\left({{1\over 2}\cdot {\pi\over 8}}\right)$  
  $\textstyle =$ $\displaystyle \sqrt{{1\over 2}\left({1+\cos{\pi\over 8}}\right)}=\sqrt{{\textstyle{1\over 2}}\left({1+{\textstyle{1\over 2}}\sqrt{2+\sqrt{2}}}\right)}$  
  $\textstyle =$ $\displaystyle \sqrt{{\textstyle{1\over 2}}+{\textstyle{1\over 4}}\sqrt{2+\sqrt{2}}} ={\textstyle{1\over 2}}\sqrt{2+\sqrt{2+\sqrt{2}}}$ (2)
$\displaystyle \tan\left({\pi\over 16}\right)$ $\textstyle =$ $\displaystyle \sqrt{{2-\sqrt{2+\sqrt{2}}}\over {2+\sqrt{2+\sqrt{2}}}}$  
  $\textstyle =$ $\displaystyle \sqrt{4+2\sqrt{2}}-\sqrt{2}-1.$ (3)

Summarizing,
$\displaystyle \sin\left({\pi\over 16}\right)$ $\textstyle =$ $\displaystyle {\textstyle{1\over 2}}\sqrt{2-\sqrt{2+\sqrt{2}}} \approx 0.19509$ (4)
$\displaystyle \sin\left({3\pi\over 16}\right)$ $\textstyle =$ $\displaystyle {\textstyle{1\over 2}}\sqrt{2-\sqrt{2-\sqrt{2}}} \approx 0.55557$ (5)
$\displaystyle \cos\left({\pi\over 16}\right)$ $\textstyle =$ $\displaystyle {\textstyle{1\over 2}}\sqrt{2+\sqrt{2+\sqrt{2}}} \approx 0.98079$ (6)
$\displaystyle \cos\left({3\pi\over 16}\right)$ $\textstyle =$ $\displaystyle {\textstyle{1\over 2}}\sqrt{2+\sqrt{2-\sqrt{2}}} \approx 0.83147$ (7)
$\displaystyle \tan\left({\pi\over 16}\right)$ $\textstyle =$ $\displaystyle \sqrt{4+2\sqrt{2}}-\sqrt{2}-1 \approx 0.19891.$ (8)




© 1996-9 Eric W. Weisstein
1999-05-26