Спіс дакладных трыганаметрычных пастаянных

Пры рашэнні задач часта бывае карысна ведаць дакладныя алгебраічныя выразы для значэнняў трыганаметрычных функцый, у першую чаргу для таго, каб прадставіць рашэнне праз радыкалы (карані), што адкрывае магчымасці для далейшага спрашчэння.

Усе значэнні сінусаў, косінусаў і тангенсаў вуглоў, кратных 3°, выражаюцца ў радыкалах. Гэтыя значэнні атрыманы шляхам прымянення тоеснасцей для палавіннага вугла, двайнога вугла, а таксама формул для сумы і рознасці вуглоў са значэннямі 0°, 30°, 36°, і 45°.

Заўвага: градусы і радыяны звязаны суадносінамі 1° = π/180 радыян.

Згодна з тэарэмай Нівена^[1], адзінымі рацыянальнымі значэннямі функцыі сінуса пры рацыянальным аргуменце (у градусах) з'яўляюцца лікі 0, 1/2, і 1.

Значэнні сінуса, косінуса, тангенса, катангенса, секанса і касеканса для найбольш ужывальных вострых вуглоў прыведзены ў табліцы. («∞» азначае, што функцыя ў таком пункце не вызначана, а ў яго наваколлі імкнецца да бесканечнасці).

${\displaystyle \alpha \,\!}$	0°(0 рад)	30° (π/6)	45° (π/4)	60° (π/3)	90° (π/2)	180° (π)	270° (3π/2)	360° (2π)
${\displaystyle \sin \alpha \,\!}$	${\displaystyle {0}\,\!}$	${\displaystyle {\frac {1}{2}}\,\!}$	${\displaystyle {\frac {\sqrt {2}}{2}}\,\!}$	${\displaystyle {\frac {\sqrt {3}}{2}}\,\!}$	${\displaystyle {1}\,\!}$	${\displaystyle {0}\,\!}$	${\displaystyle {-1}\,\!}$	${\displaystyle {0}\,\!}$
${\displaystyle \cos \alpha \,\!}$	${\displaystyle {1}\,\!}$	${\displaystyle {\frac {\sqrt {3}}{2}}\,\!}$	${\displaystyle {\frac {\sqrt {2}}{2}}\,\!}$	${\displaystyle {\frac {1}{2}}\,\!}$	${\displaystyle {0}\,\!}$	${\displaystyle {-1}\,\!}$	${\displaystyle {0}\,\!}$	${\displaystyle {1}\,\!}$
${\displaystyle \operatorname {tg} \alpha \,\!}$	${\displaystyle {0}\,\!}$	${\displaystyle {\frac {\sqrt {3}}{3}}\,\!}$	${\displaystyle {1}\,\!}$	${\displaystyle {\sqrt {3}}\,\!}$	${\displaystyle {\infty }\,\!}$	${\displaystyle {0}\,\!}$	${\displaystyle {\infty }\,\!}$	${\displaystyle {0}\,\!}$
${\displaystyle \operatorname {ctg} \alpha \,\!}$	${\displaystyle {\infty }\,\!}$	${\displaystyle {\sqrt {3}}\,\!}$	${\displaystyle {1}\,\!}$	${\displaystyle {\frac {\sqrt {3}}{3}}\,\!}$	${\displaystyle {0}\,\!}$	${\displaystyle {\infty }\,\!}$	${\displaystyle {0}\,\!}$	${\displaystyle {\infty }\,\!}$
${\displaystyle \sec \alpha \,\!}$	${\displaystyle {1}\,\!}$	${\displaystyle {\frac {2{\sqrt {3}}}{3}}\,\!}$	${\displaystyle {\sqrt {2}}\,\!}$	${\displaystyle {2}\,\!}$	${\displaystyle {\infty }\,\!}$	${\displaystyle {-1}\,\!}$	${\displaystyle {\infty }\,\!}$	${\displaystyle {1}\,\!}$
${\displaystyle \operatorname {cosec} \alpha \,\!}$	${\displaystyle {\infty }\,\!}$	${\displaystyle {2}\,\!}$	${\displaystyle {\sqrt {2}}\,\!}$	${\displaystyle {\frac {2{\sqrt {3}}}{3}}\,\!}$	${\displaystyle {1}\,\!}$	${\displaystyle {\infty }\,\!}$	${\displaystyle {-1}\,\!}$	${\displaystyle {\infty }\,\!}$

${\displaystyle \alpha \,}$	${\displaystyle {\frac {2\pi }{3}}=120^{\circ }}$	${\displaystyle {\frac {3\pi }{4}}=135^{\circ }}$	${\displaystyle {\frac {5\pi }{6}}=150^{\circ }}$	${\displaystyle {\frac {7\pi }{6}}=210^{\circ }}$	${\displaystyle {\frac {5\pi }{4}}=225^{\circ }}$	${\displaystyle {\frac {4\pi }{3}}=240^{\circ }}$	${\displaystyle {\frac {5\pi }{3}}=300^{\circ }}$	${\displaystyle {\frac {7\pi }{4}}=315^{\circ }}$	${\displaystyle {\frac {11\pi }{6}}=330^{\circ }}$
${\displaystyle \sin \alpha \,}$	${\displaystyle {\frac {\sqrt {3}}{2}}}$	${\displaystyle {\frac {\sqrt {2}}{2}}}$	${\displaystyle {\frac {1}{2}}}$	${\displaystyle -{\frac {1}{2}}}$	${\displaystyle -{\frac {\sqrt {2}}{2}}}$	${\displaystyle -{\frac {\sqrt {3}}{2}}}$	${\displaystyle -{\frac {\sqrt {3}}{2}}}$	${\displaystyle -{\frac {\sqrt {2}}{2}}}$	${\displaystyle -{\frac {1}{2}}}$
${\displaystyle \cos \alpha \,}$	${\displaystyle -{\frac {1}{2}}}$	${\displaystyle -{\frac {\sqrt {2}}{2}}}$	${\displaystyle -{\frac {\sqrt {3}}{2}}}$	${\displaystyle -{\frac {\sqrt {3}}{2}}}$	${\displaystyle -{\frac {\sqrt {2}}{2}}}$	${\displaystyle -{\frac {1}{2}}}$	${\displaystyle {\frac {1}{2}}}$	${\displaystyle {\frac {\sqrt {2}}{2}}}$	${\displaystyle {\frac {\sqrt {3}}{2}}}$
${\displaystyle \operatorname {tg} \,\alpha }$	${\displaystyle -{\sqrt {3}}}$	${\displaystyle {-1}\,\!}$	${\displaystyle -{\frac {\sqrt {3}}{3}}}$	${\displaystyle {\frac {\sqrt {3}}{3}}}$	${\displaystyle {1}\,\!}$	${\displaystyle {\sqrt {3}}}$	${\displaystyle -{\sqrt {3}}}$	${\displaystyle {-1}\,\!}$	${\displaystyle -{\frac {\sqrt {3}}{3}}}$
${\displaystyle \operatorname {ctg} \,\alpha }$	${\displaystyle -{\frac {\sqrt {3}}{3}}}$	${\displaystyle {-1}\,\!}$	${\displaystyle -{\sqrt {3}}}$	${\displaystyle {\sqrt {3}}}$	${\displaystyle {1}\,\!}$	${\displaystyle {\frac {\sqrt {3}}{3}}}$	${\displaystyle -{\frac {\sqrt {3}}{3}}}$	${\displaystyle {-1}\,\!}$	${\displaystyle -{\sqrt {3}}}$

${\displaystyle \alpha \,}$	${\displaystyle {\frac {\pi }{12}}=15^{\circ }}$	${\displaystyle {\frac {\pi }{10}}=18^{\circ }}$	${\displaystyle {\frac {\pi }{8}}=22{,}5^{\circ }}$	${\displaystyle {\frac {\pi }{5}}=36^{\circ }}$	${\displaystyle {\frac {3\,\pi }{10}}=54^{\circ }}$	${\displaystyle {\frac {3\,\pi }{8}}=67{,}5^{\circ }}$	${\displaystyle {\frac {2\,\pi }{5}}=72^{\circ }}$	${\displaystyle {\frac {5\,\pi }{12}}=75^{\circ }}$
${\displaystyle \sin \alpha \,}$	${\displaystyle {\frac {{\sqrt {3}}-1}{2\,{\sqrt {2}}}}}$	${\displaystyle {\frac {{\sqrt {5}}-1}{4}}}$	${\displaystyle {\frac {\sqrt {2-{\sqrt {2}}}}{2}}}$	${\displaystyle {\frac {\sqrt {5-{\sqrt {5}}}}{2\,{\sqrt {2}}}}}$	${\displaystyle {\frac {{\sqrt {5}}+1}{4}}}$	${\displaystyle {\frac {\sqrt {2+{\sqrt {2}}}}{2}}}$	${\displaystyle {\frac {\sqrt {5+{\sqrt {5}}}}{2\,{\sqrt {2}}}}}$	${\displaystyle {\frac {{\sqrt {3}}+1}{2\,{\sqrt {2}}}}}$
${\displaystyle \cos \alpha \,}$	${\displaystyle {\frac {{\sqrt {3}}+1}{2\,{\sqrt {2}}}}}$	${\displaystyle {\frac {\sqrt {5+{\sqrt {5}}}}{2\,{\sqrt {2}}}}}$	${\displaystyle {\frac {\sqrt {2+{\sqrt {2}}}}{2}}}$	${\displaystyle {\frac {{\sqrt {5}}+1}{4}}}$	${\displaystyle {\frac {\sqrt {5-{\sqrt {5}}}}{2\,{\sqrt {2}}}}}$	${\displaystyle {\frac {\sqrt {2-{\sqrt {2}}}}{2}}}$	${\displaystyle {\frac {{\sqrt {5}}-1}{4}}}$	${\displaystyle {\frac {{\sqrt {3}}-1}{2\,{\sqrt {2}}}}}$
${\displaystyle \operatorname {tg} \,\alpha }$	${\displaystyle 2-{\sqrt {3}}}$	${\displaystyle {\sqrt {1-{\frac {2}{\sqrt {5}}}}}}$	${\displaystyle {\sqrt {2}}-1}$	${\displaystyle {\sqrt {5-2\,{\sqrt {5}}}}}$	${\displaystyle {\sqrt {1+{\frac {2}{\sqrt {5}}}}}}$	${\displaystyle {\sqrt {2}}+1}$	${\displaystyle {\sqrt {5+2\,{\sqrt {5}}}}}$	${\displaystyle 2+{\sqrt {3}}}$
${\displaystyle \operatorname {ctg} \,\alpha }$	${\displaystyle 2+{\sqrt {3}}}$	${\displaystyle {\sqrt {5+2\,{\sqrt {5}}}}}$	${\displaystyle {\sqrt {2}}+1}$	${\displaystyle {\sqrt {1+{\frac {2}{\sqrt {5}}}}}}$	${\displaystyle {\sqrt {5-2\,{\sqrt {5}}}}}$	${\displaystyle {\sqrt {2}}-1}$	${\displaystyle {\sqrt {1-{\frac {2}{\sqrt {5}}}}}}$	${\displaystyle 2-{\sqrt {3}}}$

${\displaystyle \sin {\frac {\pi }{60}}=\cos {\frac {29\,\pi }{60}}=\sin 3^{\circ }=\cos 87^{\circ }={\frac {{\sqrt {2}}({\sqrt {3}}+1)({\sqrt {5}}-1)-2({\sqrt {3}}-1){\sqrt {5+{\sqrt {5}}}}}{16}},}$

${\displaystyle \cos {\frac {\pi }{60}}=\sin {\frac {29\,\pi }{60}}=\cos 3^{\circ }=\sin 87^{\circ }={\frac {{\sqrt {2}}({\sqrt {3}}-1)({\sqrt {5}}-1)+2({\sqrt {3}}+1){\sqrt {5+{\sqrt {5}}}}}{16}},}$

${\displaystyle \operatorname {tg} {\frac {\pi }{60}}=\operatorname {ctg} {\frac {29\,\pi }{60}}=\operatorname {tg} 3^{\circ }=\operatorname {ctg} 87^{\circ }={\frac {2({\sqrt {5}}+2)-{\sqrt {3}}({\sqrt {5}}+3)+(2-{\sqrt {3}})({\sqrt {3}}({\sqrt {5}}+1)-2){\sqrt {5-2{\sqrt {5}}}}}{2}},}$

${\displaystyle \operatorname {ctg} {\frac {\pi }{60}}=\operatorname {tg} {\frac {29\,\pi }{60}}=\operatorname {ctg} 3^{\circ }=\operatorname {tg} 87^{\circ }={\frac {2(2({\sqrt {5}}+2)+{\sqrt {3}}({\sqrt {5}}+3))+({\sqrt {3}}({\sqrt {5}}-1)+2){\sqrt {2(25+11{\sqrt {5}})}}}{4}},}$

${\displaystyle \sin {\frac {\pi }{30}}=\cos {\frac {7\,\pi }{15}}=\sin 6^{\circ }=\cos 84^{\circ }={\frac {{\sqrt {6(5-{\sqrt {5}})}}-{\sqrt {5}}-1}{8}},}$

${\displaystyle \cos {\frac {\pi }{30}}=\sin {\frac {7\,\pi }{15}}=\cos 6^{\circ }=\sin 84^{\circ }={\frac {{\sqrt {2(5-{\sqrt {5}})}}+{\sqrt {3}}({\sqrt {5}}+1)}{8}},}$

${\displaystyle \operatorname {tg} {\frac {\pi }{30}}=\operatorname {ctg} {\frac {7\,\pi }{15}}=\operatorname {tg} 6^{\circ }=\operatorname {ctg} 84^{\circ }={\frac {{\sqrt {2(5-{\sqrt {5}})}}-{\sqrt {3}}({\sqrt {5}}-1)}{2}},}$

${\displaystyle \operatorname {ctg} {\frac {\pi }{30}}=\operatorname {tg} {\frac {7\,\pi }{15}}=\operatorname {ctg} 6^{\circ }=\operatorname {tg} 84^{\circ }={\frac {{\sqrt {2(25+11{\sqrt {5}})}}+{\sqrt {3}}({\sqrt {5}}+3)}{2}},}$

${\displaystyle \sin {\frac {\pi }{20}}=\cos {\frac {9\,\pi }{20}}=\sin 9^{\circ }=\cos 81^{\circ }={\frac {{\sqrt {2}}({\sqrt {5}}+1)-2{\sqrt {5-{\sqrt {5}}}}}{8}},}$

${\displaystyle \cos {\frac {\pi }{20}}=\sin {\frac {9\,\pi }{20}}=\cos 9^{\circ }=\sin 81^{\circ }={\frac {{\sqrt {2}}({\sqrt {5}}+1)+2{\sqrt {5-{\sqrt {5}}}}}{8}},}$

${\displaystyle \operatorname {tg} {\frac {\pi }{20}}=\operatorname {ctg} {\frac {9\,\pi }{20}}=\operatorname {tg} 9^{\circ }=\operatorname {ctg} 81^{\circ }={{\sqrt {5}}+1-{\sqrt {5+2{\sqrt {5}}}}},}$

${\displaystyle \operatorname {ctg} {\frac {\pi }{20}}=\operatorname {tg} {\frac {9\,\pi }{20}}=\operatorname {ctg} 9^{\circ }=\operatorname {tg} 81^{\circ }={{\sqrt {5}}+1+{\sqrt {5+2{\sqrt {5}}}}},}$

${\displaystyle \sin {\frac {\pi }{15}}=\cos {\frac {13\,\pi }{30}}=\sin 12^{\circ }=\cos 78^{\circ }={\frac {{\sqrt {2(5+{\sqrt {5}})}}-{\sqrt {3}}({\sqrt {5}}-1)}{8}},}$

${\displaystyle \cos {\frac {\pi }{15}}=\sin {\frac {13\,\pi }{30}}=\cos 12^{\circ }=\sin 78^{\circ }={\frac {{\sqrt {6(5+{\sqrt {5}})}}+{\sqrt {5}}-1}{8}},}$

${\displaystyle \operatorname {tg} {\frac {\pi }{15}}=\operatorname {ctg} {\frac {13\,\pi }{30}}=\operatorname {tg} 12^{\circ }=\operatorname {ctg} 78^{\circ }={\frac {{\sqrt {3}}(3-{\sqrt {5}})-{\sqrt {2(25-11{\sqrt {5}})}}}{2}},}$

${\displaystyle \operatorname {ctg} {\frac {\pi }{15}}=\operatorname {tg} {\frac {13\,\pi }{30}}=\operatorname {ctg} 12^{\circ }=\operatorname {tg} 78^{\circ }={\frac {{\sqrt {3}}({\sqrt {5}}+1)+{\sqrt {2(5+{\sqrt {5}})}}}{2}},}$

${\displaystyle \sin {\frac {7\,\pi }{60}}=\cos {\frac {23\,\pi }{60}}=\sin 21^{\circ }=\cos 69^{\circ }={\frac {-{\sqrt {2}}({\sqrt {3}}-1)({\sqrt {5}}+1)+2({\sqrt {3}}+1){\sqrt {5-{\sqrt {5}}}}}{16}},}$

${\displaystyle \cos {\frac {7\,\pi }{60}}=\sin {\frac {23\,\pi }{60}}=\cos 21^{\circ }=\sin 69^{\circ }={\frac {{\sqrt {2}}({\sqrt {3}}+1)({\sqrt {5}}+1)+2({\sqrt {3}}-1){\sqrt {5-{\sqrt {5}}}}}{16}},}$

${\displaystyle \operatorname {tg} {\frac {7\,\pi }{60}}=\operatorname {ctg} {\frac {23\,\pi }{60}}=\operatorname {tg} 21^{\circ }=\operatorname {ctg} 69^{\circ }={\frac {2(2({\sqrt {5}}-2)-{\sqrt {3}}(3-{\sqrt {5}}))+({\sqrt {3}}({\sqrt {5}}+1)-2){\sqrt {2(25-11{\sqrt {5}})}}}{4}},}$

${\displaystyle \operatorname {ctg} {\frac {7\,\pi }{60}}=\operatorname {tg} {\frac {23\,\pi }{60}}=\operatorname {ctg} 21^{\circ }=\operatorname {tg} 69^{\circ }={\frac {2(2({\sqrt {5}}-2)+{\sqrt {3}}(3-{\sqrt {5}}))+({\sqrt {3}}({\sqrt {5}}+1)+2){\sqrt {2(25-11{\sqrt {5}})}}}{4}},}$

${\displaystyle \sin {\frac {2\,\pi }{15}}=\cos {\frac {11\,\pi }{30}}=\sin 24^{\circ }=\cos 66^{\circ }={\frac {{\sqrt {3}}({\sqrt {5}}+1)-{\sqrt {2(5-{\sqrt {5}})}}}{8}},}$

${\displaystyle \cos {\frac {2\,\pi }{15}}=\sin {\frac {11\,\pi }{30}}=\cos 24^{\circ }=\sin 66^{\circ }={\frac {{\sqrt {5}}+1+{\sqrt {6(5-{\sqrt {5}})}}}{8}},}$

${\displaystyle \operatorname {tg} {\frac {2\,\pi }{15}}=\operatorname {ctg} {\frac {11\,\pi }{30}}=\operatorname {tg} 24^{\circ }=\operatorname {ctg} 66^{\circ }={\frac {-{\sqrt {3}}(3+{\sqrt {5}})+{\sqrt {2(25+11{\sqrt {5}})}}}{2}},}$

${\displaystyle \operatorname {ctg} {\frac {2\,\pi }{15}}=\operatorname {tg} {\frac {11\,\pi }{30}}=\operatorname {ctg} 24^{\circ }=\operatorname {tg} 66^{\circ }={\frac {{\sqrt {3}}({\sqrt {5}}-1)+{\sqrt {2(5-{\sqrt {5}})}}}{2}},}$

${\displaystyle \sin {\frac {3\,\pi }{20}}=\cos {\frac {7\,\pi }{20}}=\sin 27^{\circ }=\cos 63^{\circ }={\frac {-{\sqrt {2}}({\sqrt {5}}-1)+2{\sqrt {5+{\sqrt {5}}}}}{8}},}$

${\displaystyle \cos {\frac {3\,\pi }{20}}=\sin {\frac {7\,\pi }{20}}=\cos 27^{\circ }=\sin 63^{\circ }={\frac {{\sqrt {2}}({\sqrt {5}}-1)+2{\sqrt {5+{\sqrt {5}}}}}{8}},}$

${\displaystyle \operatorname {tg} {\frac {3\,\pi }{20}}=\operatorname {ctg} {\frac {7\,\pi }{20}}=\operatorname {tg} 27^{\circ }=\operatorname {ctg} 63^{\circ }={{\sqrt {5}}-1-{\sqrt {5-2{\sqrt {5}}}}},}$

${\displaystyle \operatorname {ctg} {\frac {3\,\pi }{20}}=\operatorname {tg} {\frac {7\,\pi }{20}}=\operatorname {ctg} 27^{\circ }=\operatorname {tg} 63^{\circ }={{\sqrt {5}}-1+{\sqrt {5-2{\sqrt {5}}}}},}$

${\displaystyle \sin {\frac {11\,\pi }{60}}=\cos {\frac {19\,\pi }{60}}=\sin 33^{\circ }=\cos 57^{\circ }={\frac {{\sqrt {2}}({\sqrt {3}}+1)({\sqrt {5}}-1)+2({\sqrt {3}}-1){\sqrt {5+{\sqrt {5}}}}}{16}},}$

${\displaystyle \cos {\frac {11\,\pi }{60}}=\sin {\frac {19\,\pi }{60}}=\cos 33^{\circ }=\sin 57^{\circ }={\frac {-{\sqrt {2}}({\sqrt {3}}-1)({\sqrt {5}}-1)+2({\sqrt {3}}+1){\sqrt {5+{\sqrt {5}}}}}{16}},}$

${\displaystyle \operatorname {tg} {\frac {11\,\pi }{60}}=\operatorname {ctg} {\frac {19\,\pi }{60}}=\operatorname {tg} 33^{\circ }=\operatorname {ctg} 57^{\circ }={\frac {-2({\sqrt {5}}+2)+{\sqrt {3}}(3+{\sqrt {5}})+(2-{\sqrt {3}})({\sqrt {3}}({\sqrt {5}}+1)-2){\sqrt {5-2{\sqrt {5}}}}}{2}},}$

${\displaystyle \operatorname {ctg} {\frac {11\,\pi }{60}}=\operatorname {tg} {\frac {19\,\pi }{60}}=\operatorname {ctg} 33^{\circ }=\operatorname {tg} 57^{\circ }={\frac {-2(2({\sqrt {5}}+2)+{\sqrt {3}}(3+{\sqrt {5}}))+({\sqrt {3}}({\sqrt {5}}-1)+2){\sqrt {2(25+11{\sqrt {5}})}}}{4}},}$

${\displaystyle \sin {\frac {13\,\pi }{60}}=\cos {\frac {17\,\pi }{60}}=\sin 39^{\circ }=\cos 51^{\circ }={\frac {{\sqrt {2}}({\sqrt {3}}+1)({\sqrt {5}}+1)-2({\sqrt {3}}-1){\sqrt {5-{\sqrt {5}}}}}{16}},}$

${\displaystyle \cos {\frac {13\,\pi }{60}}=\sin {\frac {17\,\pi }{60}}=\cos 39^{\circ }=\sin 51^{\circ }={\frac {{\sqrt {2}}({\sqrt {3}}-1)({\sqrt {5}}+1)+2({\sqrt {3}}+1){\sqrt {5-{\sqrt {5}}}}}{16}},}$

${\displaystyle \operatorname {tg} {\frac {13\,\pi }{60}}=\operatorname {ctg} {\frac {17\,\pi }{60}}=\operatorname {tg} 39^{\circ }=\operatorname {ctg} 51^{\circ }={\frac {-2(2({\sqrt {5}}-2)+{\sqrt {3}}(3-{\sqrt {5}}))+({\sqrt {3}}({\sqrt {5}}+1)+2){\sqrt {2(25-11{\sqrt {5}})}}}{4}},}$

${\displaystyle \operatorname {ctg} {\frac {13\,\pi }{60}}=\operatorname {tg} {\frac {17\,\pi }{60}}=\operatorname {ctg} 39^{\circ }=\operatorname {tg} 51^{\circ }={\frac {-2(2({\sqrt {5}}-2)-{\sqrt {3}}(3-{\sqrt {5}}))+({\sqrt {3}}({\sqrt {5}}+1)-2){\sqrt {2(25-11{\sqrt {5}})}}}{4}},}$

${\displaystyle \sin {\frac {7\,\pi }{30}}=\cos {\frac {8\,\pi }{30}}=\sin 42^{\circ }=\cos 48^{\circ }={\frac {-({\sqrt {5}}-1)+{\sqrt {6(5+{\sqrt {5}})}}}{8}},}$

${\displaystyle \cos {\frac {7\,\pi }{30}}=\sin {\frac {8\,\pi }{30}}=\cos 42^{\circ }=\sin 48^{\circ }={\frac {{\sqrt {3}}({\sqrt {5}}-1)+{\sqrt {2(5+{\sqrt {5}})}}}{8}},}$

${\displaystyle \operatorname {tg} {\frac {7\,\pi }{30}}=\operatorname {ctg} {\frac {8\,\pi }{30}}=\operatorname {tg} 42^{\circ }=\operatorname {ctg} 48^{\circ }={\frac {{\sqrt {3}}({\sqrt {5}}+1)-{\sqrt {2(5+{\sqrt {5}})}}}{2}},}$

${\displaystyle \operatorname {ctg} {\frac {7\,\pi }{30}}=\operatorname {tg} {\frac {8\,\pi }{30}}=\operatorname {ctg} 42^{\circ }=\operatorname {tg} 48^{\circ }={\frac {{\sqrt {3}}(3-{\sqrt {5}})+{\sqrt {2(25-11{\sqrt {5}})}}}{2}},}$

${\displaystyle \operatorname {tg} {\frac {\pi }{120}}=\operatorname {ctg} {\frac {59\,\pi }{120}}=\operatorname {tg} 1.5^{\circ }=\operatorname {ctg} 88.5^{\circ }={\sqrt {\frac {8-{\sqrt {2(2-{\sqrt {3}})(3-{\sqrt {5}})}}-{\sqrt {2(2+{\sqrt {3}})(5+{\sqrt {5}})}}}{8+{\sqrt {2(2-{\sqrt {3}})(3-{\sqrt {5}})}}+{\sqrt {2(2+{\sqrt {3}})(5+{\sqrt {5}})}}}}},}$

${\displaystyle \cos {\frac {\pi }{240}}=\sin {\frac {119\,\pi }{240}}=\cos 0.75^{\circ }=\sin 89.25^{\circ }={\frac {1}{16}}\left({\sqrt {2-{\sqrt {2+{\sqrt {2}}}}}}\left({\sqrt {2(5+{\sqrt {5}})}}+{\sqrt {3}}(1-{\sqrt {5}})\right)+\right.}$ ${\displaystyle \left.+{\sqrt {2+{\sqrt {2+{\sqrt {2}}}}}}\left({\sqrt {6(5+{\sqrt {5}})}}+{\sqrt {5}}-1\right)\right),}$

${\displaystyle \cos {\frac {\pi }{17}}=\sin {\frac {15\,\pi }{34}}={\frac {1}{8}}{\sqrt {2\left(2{\sqrt {3{\sqrt {17}}-{\sqrt {2(85+19{\sqrt {17}})}}+17}}+{\sqrt {2(17-{\sqrt {17}})}}+{\sqrt {17}}+15\right)}}.}$