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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.

Рацыянальныя значэнні трыганаметрычных функцый

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

Стандартныя «школьныя» вуглы

Асноўныя вострыя вуглы

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)
$\{\displaystyle \sin \alpha \,\!\}$	$\{\displaystyle \{0\}\,\!\}$	$\{\displaystyle \{\frac \{1\}\{2\}\}\,\!\}$	$\{\displaystyle \{\frac \{\sqrt \{2\}\}\{2\}\}\,\!\}$	$\{\displaystyle \{\frac \{\sqrt \{3\}\}\{2\}\}\,\!\}$	$\{\displaystyle \{1\}\,\!\}$	$\{\displaystyle \{0\}\,\!\}$	$\{\displaystyle \{-1\}\,\!\}$	$\{\displaystyle \{0\}\,\!\}$
$\{\displaystyle \cos \alpha \,\!\}$	$\{\displaystyle \{1\}\,\!\}$	$\{\displaystyle \{\frac \{\sqrt \{3\}\}\{2\}\}\,\!\}$	$\{\displaystyle \{\frac \{\sqrt \{2\}\}\{2\}\}\,\!\}$	$\{\displaystyle \{\frac \{1\}\{2\}\}\,\!\}$	$\{\displaystyle \{0\}\,\!\}$	$\{\displaystyle \{-1\}\,\!\}$	$\{\displaystyle \{0\}\,\!\}$	$\{\displaystyle \{1\}\,\!\}$
$\{\displaystyle \operatorname \{tg\} \alpha \,\!\}$	$\{\displaystyle \{0\}\,\!\}$	$\{\displaystyle \{\frac \{\sqrt \{3\}\}\{3\}\}\,\!\}$	$\{\displaystyle \{1\}\,\!\}$	$\{\displaystyle \{\sqrt \{3\}\}\,\!\}$	$\{\displaystyle \{\infty \}\,\!\}$	$\{\displaystyle \{0\}\,\!\}$	$\{\displaystyle \{\infty \}\,\!\}$	$\{\displaystyle \{0\}\,\!\}$
$\{\displaystyle \operatorname \{ctg\} \alpha \,\!\}$	$\{\displaystyle \{\infty \}\,\!\}$	$\{\displaystyle \{\sqrt \{3\}\}\,\!\}$	$\{\displaystyle \{1\}\,\!\}$	$\{\displaystyle \{\frac \{\sqrt \{3\}\}\{3\}\}\,\!\}$	$\{\displaystyle \{0\}\,\!\}$	$\{\displaystyle \{\infty \}\,\!\}$	$\{\displaystyle \{0\}\,\!\}$	$\{\displaystyle \{\infty \}\,\!\}$
$\{\displaystyle \sec \alpha \,\!\}$	$\{\displaystyle \{1\}\,\!\}$	$\{\displaystyle \{\frac \{2\{\sqrt \{3\}\}\}\{3\}\}\,\!\}$	$\{\displaystyle \{\sqrt \{2\}\}\,\!\}$	$\{\displaystyle \{2\}\,\!\}$	$\{\displaystyle \{\infty \}\,\!\}$	$\{\displaystyle \{-1\}\,\!\}$	$\{\displaystyle \{\infty \}\,\!\}$	$\{\displaystyle \{1\}\,\!\}$
$\{\displaystyle \operatorname \{cosec\} \alpha \,\!\}$	$\{\displaystyle \{\infty \}\,\!\}$	$\{\displaystyle \{2\}\,\!\}$	$\{\displaystyle \{\sqrt \{2\}\}\,\!\}$	$\{\displaystyle \{\frac \{2\{\sqrt \{3\}\}\}\{3\}\}\,\!\}$	$\{\displaystyle \{1\}\,\!\}$	$\{\displaystyle \{\infty \}\,\!\}$	$\{\displaystyle \{-1\}\,\!\}$	$\{\displaystyle \{\infty \}\,\!\}$

Значэнні трыганаметрычных функцый вуглоў, кратных 30° ці 45°

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

Значэнні для іншых распаўсюджаных вуглоў

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

Пашыраны спіс значэнняў трыганаметрычных функцый

sin ⁡

π 60

= cos ⁡

= sin ⁡

∘

= cos ⁡

∘

(

1 ) (

− 1 ) − 2 (

− 1 )

5 +

{\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}},}

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

cos ⁡

π 60

= sin ⁡

= cos ⁡

∘

= sin ⁡

∘

(

− 1 ) (

− 1 ) + 2 (

1 )

5 +

{\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}},}

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

tg ⁡

π 60

= ctg ⁡

= tg ⁡

∘

= ctg ⁡

∘

2 (

2 ) −

(

3 ) + ( 2 −

) (

(

1 ) − 2 )

5 − 2

{\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}},}

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

ctg ⁡

π 60

= tg ⁡

= ctg ⁡

∘

= tg ⁡

∘

2 ( 2 (

2 ) +

(

3 ) ) + (

(

− 1 ) + 2 )

2 ( 25 + 11

)

{\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}},}

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

sin ⁡

π 30

= cos ⁡

= sin ⁡

∘

= cos ⁡

∘

6 ( 5 −

)

−

− 1

{\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}},}

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

cos ⁡

π 30

= sin ⁡

= cos ⁡

∘

= sin ⁡

∘

2 ( 5 −

)

(

1 )

{\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}},}

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

tg ⁡

π 30

= ctg ⁡

= tg ⁡

∘

= ctg ⁡

∘

2 ( 5 −

)

−

(

− 1 )

{\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}},}

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

ctg ⁡

π 30

= tg ⁡

= ctg ⁡

∘

= tg ⁡

∘

2 ( 25 + 11

)

(

3 )

{\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}},}

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

sin ⁡

π 20

= cos ⁡

= sin ⁡

∘

= cos ⁡

∘

(

1 ) − 2

5 −

{\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}},}

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

cos ⁡

π 20

= sin ⁡

= cos ⁡

∘

= sin ⁡

∘

(

1 ) + 2

5 −

{\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}},}

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

tg ⁡

π 20

= ctg ⁡

= tg ⁡

∘

= ctg ⁡

∘

1 −

5 + 2

{\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}}}}},}

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

ctg ⁡

π 20

= tg ⁡

= ctg ⁡

∘

= tg ⁡

∘

1 +

5 + 2

{\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}}}}},}

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

sin ⁡

π 15

= cos ⁡

= sin ⁡

∘

= cos ⁡

∘

2 ( 5 +

)

−

(

− 1 )

{\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}},}

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

cos ⁡

π 15

= sin ⁡

= cos ⁡

∘

= sin ⁡

∘

6 ( 5 +

)

− 1

{\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}},}

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

tg ⁡

π 15

= ctg ⁡

= tg ⁡

∘

= ctg ⁡

∘

( 3 −

) −

2 ( 25 − 11

)

{\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}},}

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

ctg ⁡

π 15

= tg ⁡

= ctg ⁡

∘

= tg ⁡

∘

(

1 ) +

2 ( 5 +

)

{\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}},}

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

sin ⁡

= cos ⁡

= sin ⁡

∘

= cos ⁡

∘

−

(

− 1 ) (

1 ) + 2 (

1 )

5 −

{\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}},}

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

cos ⁡

= sin ⁡

= cos ⁡

∘

= sin ⁡

∘

(

1 ) (

1 ) + 2 (

− 1 )

5 −

{\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}},}

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

tg ⁡

= ctg ⁡

= tg ⁡

∘

= ctg ⁡

∘

2 ( 2 (

− 2 ) −

( 3 −

) ) + (

(

1 ) − 2 )

2 ( 25 − 11

)

{\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}},}

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

ctg ⁡

= tg ⁡

= ctg ⁡

∘

= tg ⁡

∘

2 ( 2 (

− 2 ) +

( 3 −

) ) + (

(

1 ) + 2 )

2 ( 25 − 11

)

{\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}},}

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

sin ⁡

= cos ⁡

= sin ⁡

∘

= cos ⁡

∘

(

1 ) −

2 ( 5 −

)

{\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}},}

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

cos ⁡

= sin ⁡

= cos ⁡

∘

= sin ⁡

∘

1 +

6 ( 5 −

)

{\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}},}

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

tg ⁡

= ctg ⁡

= tg ⁡

∘

= ctg ⁡

∘

−

( 3 +

) +

2 ( 25 + 11

)

{\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}},}

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

ctg ⁡

= tg ⁡

= ctg ⁡

∘

= tg ⁡

∘

(

− 1 ) +

2 ( 5 −

)

{\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}},}

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

sin ⁡

= cos ⁡

= sin ⁡

∘

= cos ⁡

∘

−

(

− 1 ) + 2

5 +

{\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}},}

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

cos ⁡

= sin ⁡

= cos ⁡

∘

= sin ⁡

∘

(

− 1 ) + 2

5 +

{\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}},}

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

tg ⁡

= ctg ⁡

= tg ⁡

∘

= ctg ⁡

∘

− 1 −

5 − 2

{\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}}}}},}

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

ctg ⁡

= tg ⁡

= ctg ⁡

∘

= tg ⁡

∘

− 1 +

5 − 2

{\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}}}}},}

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

sin ⁡

= cos ⁡

= sin ⁡

∘

= cos ⁡

∘

(

1 ) (

− 1 ) + 2 (

− 1 )

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}},}

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

cos ⁡

= sin ⁡

= cos ⁡

∘

= sin ⁡

∘

−

(

− 1 ) (

− 1 ) + 2 (

1 )

5 +

{\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}},}

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

tg ⁡

= ctg ⁡

= tg ⁡

∘

= ctg ⁡

∘

− 2 (

2 ) +

( 3 +

) + ( 2 −

) (

(

1 ) − 2 )

5 − 2

{\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}},}

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

ctg ⁡

= tg ⁡

= ctg ⁡

∘

= tg ⁡

∘

− 2 ( 2 (

2 ) +

( 3 +

) ) + (

(

− 1 ) + 2 )

2 ( 25 + 11

)

{\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}},}

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

sin ⁡

= cos ⁡

= sin ⁡

∘

= cos ⁡

∘

(

1 ) (

1 ) − 2 (

− 1 )

5 −

{\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}},}

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

cos ⁡

= sin ⁡

= cos ⁡

∘

= sin ⁡

∘

(

− 1 ) (

1 ) + 2 (

1 )

5 −

{\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}},}

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

tg ⁡

= ctg ⁡

= tg ⁡

∘

= ctg ⁡

∘

− 2 ( 2 (

− 2 ) +

( 3 −

) ) + (

(

1 ) + 2 )

2 ( 25 − 11

)

{\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}},}

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

ctg ⁡

= tg ⁡

= ctg ⁡

∘

= tg ⁡

∘

− 2 ( 2 (

− 2 ) −

( 3 −

) ) + (

(

1 ) − 2 )

2 ( 25 − 11

)

{\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}},}

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

sin ⁡

= cos ⁡

= sin ⁡

∘

= cos ⁡

∘

− (

− 1 ) +

6 ( 5 +

)

{\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}},}

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

cos ⁡

= sin ⁡

= cos ⁡

∘

= sin ⁡

∘

(

− 1 ) +

2 ( 5 +

)

{\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}},}

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

tg ⁡

= ctg ⁡

= tg ⁡

∘

= ctg ⁡

∘

(

1 ) −

2 ( 5 +

)

{\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}},}

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

ctg ⁡

= tg ⁡

= ctg ⁡

∘

= tg ⁡

∘

( 3 −

) +

2 ( 25 − 11

)

{\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}},}

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

tg ⁡

π 120

= ctg ⁡

120

= tg ⁡

1.5

∘

= ctg ⁡

88.5

∘

8 −

2 ( 2 −

) ( 3 −

)

−

2 ( 2 +

) ( 5 +

)

8 +

2 ( 2 −

) ( 3 −

)

2 ( 2 +

) ( 5 +

)

{\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}})}}}}},}

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

cos ⁡

π 240

= sin ⁡

119

240

= cos ⁡

0.75

∘

= sin ⁡

89.25

∘

1 16

(

2 −

2 +

(

2 ( 5 +

)

( 1 −

)

{\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.}

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

2 +

(

6 ( 5 +

)

− 1

)

{\displaystyle \left.+{\sqrt {2+{\sqrt {2+{\sqrt {2}}}}}}\left({\sqrt {6(5+{\sqrt {5}})}}+{\sqrt {5}}-1\right)\right),}

$\{\displaystyle \left.+\{\sqrt \{2+\{\sqrt \{2+\{\sqrt \{2\}\}\}\}\}\}\left(\{\sqrt \{6(5+\{\sqrt \{5\}\})\}\}+\{\sqrt \{5\}\}-1\right)\right),\}$

Асобыя вуглы

cos ⁡

π 17

= sin ⁡

1 8

(

−

2 ( 85 + 19

)

2 ( 17 −

)

{\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)}}.}

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

Зноскі

↑ Гл. вынік 3.12 у кнізе Ivan Niven. Irrational Numbers.. — Wiley, 1956. — С. 41.

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