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Trigonometry Values Pi/12


$\displaystyle \sin\left({\pi\over 12}\right)$ $\textstyle =$ $\displaystyle \sin\left({\pi\over 3}-{\pi\over 4}\right)$  
  $\textstyle =$ $\displaystyle -\sin\left({\pi\over 4}\right)\cos\left({\pi\over 3}\right)+ \sin\left({\pi\over 3}\right)\cos\left({\pi\over 4}\right)$  
  $\textstyle =$ $\displaystyle -{\textstyle{1\over 2}}\sqrt{2}({\textstyle{1\over 2}})+{\textstyle{1\over 2}}\sqrt{3}({\textstyle{1\over 2}}\sqrt{2}\,)$  
  $\textstyle =$ $\displaystyle {\textstyle{1\over 4}}(\sqrt{6}-\sqrt{2}\,).$ (1)

Similarly,
$\displaystyle \cos\left({\pi\over 12}\right)$ $\textstyle =$ $\displaystyle \cos\left({\pi\over 3}-{\pi \over 4}\right)$  
  $\textstyle =$ $\displaystyle \cos\left({\pi\over 4}\right)\cos\left({\pi\over 3}\right)-\sin\left({\pi\over 3}\right)\sin\left({\pi\over 4}\right)$  
  $\textstyle =$ $\displaystyle {\textstyle{1\over 2}}({\textstyle{1\over 2}}\sqrt{2}\,)+{\textstyle{1\over 2}}\sqrt{3}(-{\textstyle{1\over 2}}\sqrt{2}\,)$  
  $\textstyle =$ $\displaystyle {\textstyle{1\over 4}}(\sqrt{6}+\sqrt{2}).$ (2)

Summarizing,
$\displaystyle \sin\left({\pi\over 12}\right)$ $\textstyle =$ $\displaystyle {\textstyle{1\over 4}}(\sqrt{6}-\sqrt{2}) \approx 0.25882$ (3)
$\displaystyle \cos\left({\pi\over 12}\right)$ $\textstyle =$ $\displaystyle {\textstyle{1\over 4}}(\sqrt{6}+\sqrt{2}) \approx 0.96593$ (4)
$\displaystyle \tan\left({\pi\over 12}\right)$ $\textstyle =$ $\displaystyle 2-\sqrt{3} \approx 0.26795$ (5)
$\displaystyle \csc\left({\pi\over 12}\right)$ $\textstyle =$ $\displaystyle \sqrt{6}+\sqrt{2} \approx 3.86370$ (6)
$\displaystyle \sec\left({\pi\over 12}\right)$ $\textstyle =$ $\displaystyle \sqrt{6}-\sqrt{2} \approx 1.03528$ (7)
$\displaystyle \cot\left({\pi\over 12}\right)$ $\textstyle =$ $\displaystyle 2+\sqrt{3} \approx 3.73205.$ (8)




© 1996-9 Eric W. Weisstein
1999-05-26