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)

$\displaystyle \sin\left({\pi\over 16}\right)$	$\textstyle =$	$\displaystyle {\textstyle{1\over 2}}\sqrt{2-\sqrt{2+\sqrt{2}}} \approx 0.19509$	(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)