Trigonometry Values Pi/20

$$\sin\left({\pi\over 20}\right) = \sin \left({{1\over 2}{\pi\over 10}}\right)= \sqrt{{1\over 2} \left({1-\cos{\pi\over 10}}\right)}$$

$$= {\textstyle{1\over 4}}\sqrt{8-2\sqrt{10+2\sqrt{5}}}$$

$$\approx 0.15643$$ (1)

$$\cos\left({\pi\over 20}\right) = \cos \left({{1\over 2}{\pi\over 10}}\right)= \sqrt{{1\over 2} \left({1+\cos{\pi\over 10}}\right)}$$

$$= {\textstyle{1\over 4}}\sqrt{8+2\sqrt{10+2\sqrt{5}}}$$

$$\approx 0.98768$$ (2)

$$\tan\left({\pi\over 20}\right) = 1+\sqrt{5}-\sqrt{5+2\sqrt{5}}$$

$$\approx 0.15838.$$ (3)

An interesting near-identity is given by

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

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