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)

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

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