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 ${\displaystyle e^{\sin x}+\cos x}$ nėra trigonometrinė funkcija.

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

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

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

${\displaystyle \cos {\frac {\pi }{240}={\frac {1}{16}\left({\sqrt {2-k}\left({\sqrt {2(5+{\sqrt {5})}+{\sqrt {3}-{\sqrt {15}\right)+{\sqrt {2+k}\left({\sqrt {6(5+{\sqrt {5})}+{\sqrt {5}-1\right)\right)}$, 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 ${\displaystyle k=17-{\sqrt {17}$ .

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

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:

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

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

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 ${\displaystyle \pi }$

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.

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

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

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

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).

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