Trigonometrinės funkcijos

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

Geometrine prasme trigonometrinės funkcijos nusako ryšį tarp trikampio kraštinių ir kampų.

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

Trigonometrinių funkcijų pagrindinių reikšmių lentelė

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

Trigonometrinių funkcijų reikšmės nestandartiniams kampams

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

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

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

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

Redukcijos formulės

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

Trigonometrinių funkcijų savybės

Pagrindinės lygybės

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

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

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

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

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

Periodiškumas

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

Lyginės ir nelyginės funkcijos

Kosinusas yra lyginė funkcija, nes

${\displaystyle \cos(-\alpha )=\cos \alpha .}$

Sinusas yra nelyginė funkcija, nes

${\displaystyle \sin(-\alpha )=-\sin \alpha .}$

Tangentas ir kotangentas yra nelyginės funkcijos, t. y.

${\displaystyle {\text{tg}}(-\alpha )=-{\text{tg}}\;\alpha ;}$

${\displaystyle {\text{ctg}}(-\alpha )=-{\text{ctg}}\;\alpha .}$

Kai kurios lygybės

${\displaystyle \cos x=\sin {\Big (}x+{\frac {\pi }{2}}{\Big )}.}$

Į formulę

${\displaystyle \cos(\alpha -\beta )=\cos \alpha \cos \beta +\sin \alpha \sin \beta \quad (1)}$

įstačius ${\displaystyle {\frac {\pi }{2}}}$ vietoje ${\displaystyle \alpha }$ ir įstačius ${\displaystyle \alpha }$ vietoje ${\displaystyle \beta }$ gausime

${\displaystyle \cos({\frac {\pi }{2}}-\alpha )=\cos {\frac {\pi }{2}}\cos \alpha +\sin {\frac {\pi }{2}}\sin \alpha =\sin \alpha .}$

Gautoje formulėje

${\displaystyle \sin \alpha =\cos({\frac {\pi }{2}}-\alpha )\quad (2)}$

įstačius ${\displaystyle \alpha +\beta }$ vietoje ${\displaystyle \alpha ,}$ gausime

${\displaystyle \sin(\alpha +\beta )=\cos({\frac {\pi }{2}}-\alpha -\beta )=\cos(({\frac {\pi }{2}}-\alpha )-\beta ).}$

Toliau į (1) formulę įstačius ${\displaystyle {\frac {\pi }{2}}-\alpha }$ vietoje ${\displaystyle \alpha ,}$ gausime

${\displaystyle \sin(\alpha +\beta )=\cos(({\frac {\pi }{2}}-\alpha )-\beta )=}$

${\displaystyle =\cos({\frac {\pi }{2}}-\alpha )\cos \beta +\sin({\frac {\pi }{2}}-\alpha )\sin \beta =}$

${\displaystyle =\sin \alpha \cos \beta +\cos \alpha \sin \beta .}$

Pasinaudojome formule

${\displaystyle \sin({\frac {\pi }{2}}-\alpha )=\cos \alpha ,\quad (3)}$

kuri išplaukia iš formulės (2) įstačius į ją ${\displaystyle {\frac {\pi }{2}}-\alpha }$ vietoje ${\displaystyle \alpha ;}$ tada

[${\displaystyle \sin \alpha =\cos({\frac {\pi }{2}}-\alpha )\quad (2)}$]

${\displaystyle \sin({\frac {\pi }{2}}-\alpha )=\cos({\frac {\pi }{2}}-({\frac {\pi }{2}}-\alpha ))=\cos \alpha .}$

Taigi, gavome formulę (3).

Šaltiniai

1. trigonometrinės funkcijos. Visuotinė lietuvių enciklopedija (tikrinta 2023-11-05).
2. Udo Quak. Kaip suprasti matematiką. Teminis žinynas. – Kaunas: Šviesa, 2003. – 91 p. ISBN 5-430-03555-6