Funtzio trigonometriko

Triangelu angeluzuzen bateko angeluen eta aldeen notazioa

Triangelu angeluzuzen batean, funtzio trigonometrikoa aldeen neurrien arteko erlazioak adierazten dituzten funtzioetako edozein da. Funtzio nagusiak sei dira: sinua, kosinua, tangentea, kosekantea, sekantea eta kotangentea. (Ikusi irudia) ABC triangelu angeluzuzen bat izanik, C angelu zuzena dela eta a, b eta c, hurrenez hurren, A, B eta C angeluen aurrez aurreko aldeak direla, funtzio trigonometrikoak hauek dira:

$\sin \alpha ={\frac {\textrm {aurkakoa}}{\textrm {hipotenusa}}}=\color {Blue}{\frac {a}{c}}$

$\cos \alpha ={\frac {\textrm {albokoa}}{\textrm {hipotenusa}}}=\color {Blue}{\frac {b}{c}}$

$\tan \alpha ={\frac {\textrm {aurkakoa}}{\textrm {albokoa}}}=\color {Blue}{\frac {a}{b}}$

$\cot \alpha ={\frac {\textrm {albokoa}}{\textrm {aurkakoa}}}=\color {Blue}{\frac {b}{a}}$

$\sec \alpha ={\frac {\textrm {hipotenusa}}{\textrm {albokoa}}}=\color {Blue}{\frac {h}{b}}$

$\csc \alpha ={\frac {\textrm {hipotenusa}}{\textrm {aurkakoa}}}=\color {Blue}{\frac {h}{a}}$

Angelu nabarien funtzio trigonometrikoak

	0°	30°	45°	60°	90°
sin	$0$	${\frac {1}{2}}$	${\frac {\sqrt {2}}{2}}$	${\frac {\sqrt {3}}{2}}$	$1$
cos	$1$	${\frac {\sqrt {3}}{2}}$	${\frac {\sqrt {2}}{2}}$	${\frac {1}{2}}$	$0$
tan	$0$	${\frac {\sqrt {3}}{3}}$	$1$	${\sqrt {3}}$	$\infty$
cot	$\infty$	${\sqrt {3}}$	$1$	${\frac {\sqrt {3}}{3}}$	$0$
sec	$1$	${\frac {2{\sqrt {3}}}{3}}$	${\sqrt {2}}$	$2$	$\infty$
csc	$\infty$	$2$	${\sqrt {2}}$	${\frac {2{\sqrt {3}}}{3}}$	$1$

Adierazpen grafikoak

Identitateak

Identitate pitagorikoak

$\sin ^{2}(x)+\cos ^{2}(x)=1,\qquad \sec ^{2}(x)-\tan ^{2}(x)=1,\qquad \csc ^{2}(x)-\cot ^{2}(x)=1$

Angelu batuketa eta kenketa

$\sin(x\pm y)=\sin(x)\cos(y)\pm \cos(x)\sin(y)$	,	$\csc(x\pm y)={\frac {1}{\sin(x\pm y)}}$
$\cos(x\pm y)=\cos(x)\cos(y)\mp \sin(x)\sin(y)$	,	$\sec(x\pm y)={\frac {1}{\cos(x\pm y)}}$
$\tan(x\pm y)={\frac {\tan(x)\pm \tan(y)}{1\mp \tan(x)\tan(y)}}$	,	$\cot(x\pm y)={\frac {\cot(x)\cot(y)\mp 1}{\cot(y)\pm \cot(x)}}$

Angelu bikoitza eta erdia

$\sin(2x)={\frac {2\tan(x)}{1+\tan ^{2}(x)}}=2\sin(x)\cos(x)$	,	$\csc(2x)={\frac {1}{\sin(2x)}}$
$\cos(2x)={\frac {1-\tan ^{2}(x)}{1+\tan ^{2}(x)}}=\cos ^{2}(x)-\sin ^{2}(x)=2\cos ^{2}(x)-1$	,	$\sec(2x)={\frac {1}{\cos(2x)}}$
$\tan(2x)={\frac {2\tan(x)}{1-\tan ^{2}(x)}}$	,	$\cot(2x)={\frac {\cot ^{2}(x)-1}{2\cot(x)}}$

$\sin(x/2)=\pm {\sqrt {\frac {1-\cos(x)}{2}}}$	,	$\csc(x/2)={\frac {1}{\sin(x/2)}}$
$\cos(x/2)=\pm {\sqrt {\frac {1+\cos(x)}{2}}}$	,	$\sec(x/2)={\frac {1}{\cos(x/2)}}$
$\tan(x/2)=\csc(x)-\cot(x)=\pm {\sqrt {\frac {1-\cos(x)}{1+\cos(x)}}}={\frac {\sin(x)}{1+\cos(x)}}$	,	$\cot(x/2)=\csc(x)+\cot(x)$

Biderketatik batuketara

$\sin(x)\sin(y)={\frac {\cos(x-y)-\cos(x+y)}{2}}$	,	$\sin(x)\cos(y)={\frac {\sin(x+y)+\sin(x-y)}{2}}$
$\cos(x)\cos(y)={\frac {\cos(x+y)+\cos(x-y)}{2}}$	,	$\cos(x)\sin(y)={\frac {\sin(x+y)-\sin(x-y)}{2}}$

$\sin ^{2}(x)-\sin ^{2}(y)=\sin(x+y)\sin(x-y)$

$\cos ^{2}(x)-\sin ^{2}(y)=\cos(x+y)\cos(x-y)$

$\sin ^{2}(x)\cos ^{2}(x)={\frac {1-\cos(4x)}{8}}$

Batuketatik biderketara

$\sin(x)+\sin(y)=2\sin \left({\frac {x+y}{2}}\right)\cos \left({\frac {x-y}{2}}\right)$	,	$\sin(x)-\sin(y)=2\sin \left({\frac {x-y}{2}}\right)\cos \left({\frac {x+y}{2}}\right)$
$\cos(x)+\cos(y)=2\cos \left({\frac {x+y}{2}}\right)\cos \left({\frac {x-y}{2}}\right)$	,	$\cos(x)-\cos(y)=-2\sin \left({\frac {x+y}{2}}\right)\sin \left({\frac {x-y}{2}}\right)$
$\tan(x)+\tan(y)={\frac {\sin(x+y)}{\cos(x)\cos(y)}}$	,	$\tan(x)-\tan(y)={\frac {\sin(x-y)}{\cos(x)\cos(y)}}$

Berreketak

$\sin ^{2}(x)={\frac {1-\cos(2x)}{2}}$

$\quad$

$\cos ^{2}(x)={\frac {1+\cos(2x)}{2}}$

$\quad$

$\tan ^{2}(x)={\frac {1-\cos(2x)}{1+\cos(2x)}}$

$\sin ^{2}(x)-\sin ^{2}(y)=\sin(x+y)\sin(x-y)$

$\cos ^{2}(x)-\sin ^{2}(y)=\cos(x+y)\cos(x-y)$

$\sin ^{2}(x)\cos ^{2}(x)={\frac {1-\cos(4x)}{8}}$

Deribatuak

${\frac {d}{dx}}\sin(x)=\cos(x)$

${\frac {d}{dx}}\cos(x)=-\sin(x)$

${\frac {d}{dx}}\tan(x)=\sec ^{2}(x)=1+\tan ^{2}(x)$

${\frac {d}{dx}}\csc(x)=-\csc(x)\cot(x)$

${\frac {d}{dx}}\sec(x)=\sec(x)\tan(x)$

${\frac {d}{dx}}\cot(x)=-\csc ^{2}(x)=-(1+\cot ^{2}(x))$

Integralak

Funtzio trigonometrikoen integralen zerrenda

$\int \sin(x)dx=-\cos(x)+C$

$\int \cos(x)dx=\sin(x)+C$

$\int \tan(x)dx=-\ln |\cos(x)|+C$

$\int \csc(x)dx=-\ln |\csc(x)+\cot(x)|+C$

$\int \sec(x)dx=\ln |\sec(x)+\tan(x)|+C$

$\int \cot(x)dx=\ln |\sin(x)|+C$

Teoremak

Sinuaren teorema. $ABC$ triangelu batean $\alpha ,\beta ,\gamma$ hurrenez hurren $a,b,c$ aldeen aurkako angeluak baldin badira, orduan betetzen da:

${\frac {a}{\sin(\alpha )}}={\frac {b}{\sin(\beta )}}={\frac {c}{\sin(\gamma )}}$

Kosinuaren teorema. $ABC$ triangelu batean $\alpha ,\beta ,\gamma$ hurrenez hurren $a,b,c$ aldeen aurkako angeluak baldin badira, orduan betetzen da:

$a^{2}=b^{2}+c^{2}-2bc\cos(\alpha ),\quad b^{2}=a^{2}+c^{2}-2ac\cos(\beta ),\quad c^{2}=a^{2}+b^{2}-2ab\cos(\gamma )$

Tangentearen teorema. $ABC$ triangelu batean $\alpha ,\beta ,\gamma$ hurrenez hurren $a,b,c$ aldeen aurkako angeluak baldin badira, orduan betetzen da:

${\frac {a-b}{a+b}}={\frac {\tan \left({\frac {\alpha -\beta }{2}}\right)}{\tan \left({\frac {\alpha +\beta }{2}}\right)}},\quad {\frac {b-c}{b+c}}={\frac {\tan \left({\frac {\beta -\gamma }{2}}\right)}{\tan \left({\frac {\beta +\gamma }{2}}\right)}},\quad {\frac {a-c}{a+c}}={\frac {\tan \left({\frac {\alpha -\gamma }{2}}\right)}{\tan \left({\frac {\alpha +\gamma }{2}}\right)}}$

Kanpo estekak

• Datuak: Q93344
• Multimedia: Trigonometric functions