Trigonometrinės funkcijos

Trigonometrinė funkcija – realaus arba kompleksinio kintamojo elementarioji funkcija: sinusas, kosinusas, tangentas, kotangentas, sekantas, kosekantas.^[1]

Geometrine prasme trigonometrinės funkcijos nusako ryšį tarp trikampio kraštinių ir kampų.^[2]

Viena pagrindinių šių funkcijų savybių yra jų periodiškumas, tačiau ne kiekviena periodinė funkcija, kurios argumentas yra kampas, yra trigonometrinė funkcija. Pavyzdžiui, funkcija $e^{\sin x}+\cos x$ nėra trigonometrinė funkcija.

Trigonometrinių funkcijų pagrindinių reikšmių lentelė

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

Trigonometrinių funkcijų reikšmės nestandartiniams kampams

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

$\operatorname {tg} {\frac {\pi }{120}}=\operatorname {tg} 1,5^{\circ }={\sqrt {\frac {8-{\sqrt {2(2-{\sqrt {3}})(3-{\sqrt {5}})}}-{\sqrt {2(2+{\sqrt {3}})(5+{\sqrt {5}})}}}{8+{\sqrt {2(2-{\sqrt {3}})(3-{\sqrt {5}})}}+{\sqrt {2(2+{\sqrt {3}})(5+{\sqrt {5}})}}}}}$

$\cos {\frac {\pi }{240}}={\frac {1}{16}}\left({\sqrt {2-k}}\left({\sqrt {2(5+{\sqrt {5}})}}+{\sqrt {3}}-{\sqrt {15}}\right)+{\sqrt {2+k}}\left({\sqrt {6(5+{\sqrt {5}})}}+{\sqrt {5}}-1\right)\right)$ , kur $k={\sqrt {2+{\sqrt {2}}}}$ .

$\cos {\frac {\pi }{17}}={\frac {1}{8}}{\sqrt {2\left(2{\sqrt {{\sqrt {\frac {17k}{2}}}-{\sqrt {\frac {k}{2}}}-4{\sqrt {2(17+{\sqrt {17}})}}+3{\sqrt {17}}+17}}+{\sqrt {2k}}+{\sqrt {17}}+15\right)}}$ , kur $k=17-{\sqrt {17}}$ .

Redukcijos formulės

$u$	${\frac {\pi }{2}}+\alpha$	$\pi +\alpha$	${\frac {3\pi }{2}}+\alpha$	$-\alpha$	${\frac {\pi }{2}}-\alpha$	$\pi -\alpha$	${\frac {3\pi }{2}}-\alpha$
$\sin u\,$	$\cos \alpha$	$-\sin \alpha$	$-\cos \alpha$	$-\sin \alpha$	$\cos \alpha$	$\sin \alpha$	$-\cos \alpha$
$\cos u\,$	$-\sin \alpha$	$-\cos \alpha$	$\sin \alpha$	$\cos \alpha$	$\sin \alpha$	$-\cos \alpha$	$-\sin \alpha$
$\operatorname {tg} u$	$-\operatorname {ctg} \alpha$	$\operatorname {tg} \alpha$	$-\operatorname {ctg} \alpha$	$-\operatorname {tg} \alpha$	$\operatorname {ctg} \alpha$	$-\operatorname {tg} \alpha$	$\operatorname {ctg} \alpha$
$\operatorname {ctg} u$	$-\operatorname {tg} \alpha$	$\operatorname {ctg} \alpha$	$-\operatorname {tg} \alpha$	$-\operatorname {ctg} \alpha$	$\operatorname {tg} \alpha$	$-\operatorname {ctg} \alpha$	$\operatorname {tg} \alpha$

Trigonometrinių funkcijų savybės

Pagrindinės lygybės

Kadangi sinusas ir kosinusas yra atitinkamai taško, atitinkančio kampo α apskritimą, ordinatė ir abscisė, tai pagal Pitagoro teoremą:

\sin ^{2}\alpha +\cos ^{2}\alpha =1.\qquad \qquad \,

Abi šios lygties puses padalijus iš sinuso kvadrato arba kosinuso kvadrato, gaunama:

1+\mathop {\mathrm {tg} } \,^{2}\alpha ={\frac {1}{\cos ^{2}\alpha }},\qquad \qquad \,

1+\mathop {\mathrm {ctg} } \,^{2}\alpha ={\frac {1}{\sin ^{2}\alpha }}.\qquad \qquad \,

Periodiškumas

Funkcijos $y=\sin \alpha$ , $y=\cos \alpha$ , $y=\sec \alpha$ ir $y=\csc \alpha$ yra periodinės funkcijos su periodu $2\pi$ . O funkcijos $y=\operatorname {tg} \alpha$ ir $y=\operatorname {ctg} \alpha$ yra periodinės su periodu $\pi$