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:[ 1]
sin α = aurkakoa hipotenusa = a c {\displaystyle \sin \alpha ={\frac {\textrm {aurkakoa}}{\textrm {hipotenusa}}}=\color {Blue}{\frac {a}{c}}} cos α = albokoa hipotenusa = b c {\displaystyle \cos \alpha ={\frac {\textrm {albokoa}}{\textrm {hipotenusa}}}=\color {Blue}{\frac {b}{c}}} tan α = aurkakoa albokoa = a b {\displaystyle \tan \alpha ={\frac {\textrm {aurkakoa}}{\textrm {albokoa}}}=\color {Blue}{\frac {a}{b}}} cot α = albokoa aurkakoa = b a {\displaystyle \cot \alpha ={\frac {\textrm {albokoa}}{\textrm {aurkakoa}}}=\color {Blue}{\frac {b}{a}}} sec α = hipotenusa albokoa = h b {\displaystyle \sec \alpha ={\frac {\textrm {hipotenusa}}{\textrm {albokoa}}}=\color {Blue}{\frac {h}{b}}} csc α = hipotenusa aurkakoa = h a {\displaystyle \csc \alpha ={\frac {\textrm {hipotenusa}}{\textrm {aurkakoa}}}=\color {Blue}{\frac {h}{a}}}
Kontzeptu orokorrak Funtzio trigonometrikoak triangelu zuzen baten bi aldeen arteko zatidura gisa defini daitezke, haien angeluekin lotuta. Funtzio trigonometrikoak, zirkulu unitate batean (erradio unitarioa) marraztutako triangelu zuzen batean, erlazio trigonometrikoaren kontzeptuaren luzapenak diren funtzioak dira. Definizio modernoagoek serie infinitu edo ekuazio diferentzial batzuen soluzio gisa deskribatzen dituzte, balio positiboetara eta negatiboetara hedatzea ahalbidetuz, eta baita zenbaki konplexuetara ere.
Oinarrizko sei funtzio trigonometriko daude. Azken laurak lehenengo bi funtzioei dagokienez definitzen dira, nahiz eta geometrikoki edo haien erlazioen bidez defini daitezkeen. Funtzio batzuk ohikoak ziren iraganean, eta lehenengo tauletan agertzen dira, baina gaur egun ez dira erabiltzen; adibidez birsena (1 − cos θ) eta exsekantea (sec θ − 1).
Funtzioa Laburdura Baliokidetasunak (radianetan) Sinu Sin sin θ ≡ 1 csc θ ≡ cos ( π 2 − θ ) ≡ cos θ cot θ {\displaystyle \sin \;\theta \equiv {\frac {1}{\csc \theta }}\equiv \cos \left({\frac {\pi }{2}}-\theta \right)\equiv {\frac {\cos \theta }{\cot \theta }}\,} Kosinua cos cos θ ≡ 1 sec θ ≡ sin ( π 2 − θ ) ≡ sin θ tan θ {\displaystyle \cos \theta \equiv {\frac {1}{\sec \theta }}\equiv \sin \left({\frac {\pi }{2}}-\theta \right)\equiv {\frac {\sin \theta }{\tan \theta }}\,} Tangentea tan tan θ ≡ 1 cot θ ≡ cot ( π 2 − θ ) ≡ sin θ cos θ {\displaystyle \tan \theta \equiv {\frac {1}{\cot \theta }}\equiv \cot \left({\frac {\pi }{2}}-\theta \right)\equiv {\frac {\sin \theta }{\cos \theta }}\,} Kotangentea cot cot θ ≡ 1 tan θ ≡ tan ( π 2 − θ ) ≡ cos θ sin θ {\displaystyle \cot \theta \equiv {\frac {1}{\tan \theta }}\equiv \tan \left({\frac {\pi }{2}}-\theta \right)\equiv {\frac {\cos \theta }{\sin \theta }}\,} Sekantea sec sec θ ≡ 1 cos θ ≡ csc ( π 2 − θ ) ≡ tan θ sin θ {\displaystyle \sec \theta \equiv {\frac {1}{\cos \theta }}\equiv \csc \left({\frac {\pi }{2}}-\theta \right)\equiv {\frac {\tan \theta }{\sin \theta }}\,} Kosekantea csc csc θ ≡ 1 sin θ ≡ sec ( π 2 − θ ) ≡ cot θ cos θ {\displaystyle \csc \theta \equiv {\frac {1}{\sin \theta }}\equiv \sec \left({\frac {\pi }{2}}-\theta \right)\equiv {\frac {\cot \theta }{\cos \theta }}\,}
Angelu nabarien funtzio trigonometrikoak 0° 30° 45° 60° 90° sin 0 {\displaystyle 0} 1 2 {\displaystyle {\frac {1}{2}}} 2 2 {\displaystyle {\frac {\sqrt {2}}{2}}} 3 2 {\displaystyle {\frac {\sqrt {3}}{2}}} 1 {\displaystyle 1} cos 1 {\displaystyle 1} 3 2 {\displaystyle {\frac {\sqrt {3}}{2}}} 2 2 {\displaystyle {\frac {\sqrt {2}}{2}}} 1 2 {\displaystyle {\frac {1}{2}}} 0 {\displaystyle 0} tan 0 {\displaystyle 0} 3 3 {\displaystyle {\frac {\sqrt {3}}{3}}} 1 {\displaystyle 1} 3 {\displaystyle {\sqrt {3}}} ∞ {\displaystyle \infty } cot ∞ {\displaystyle \infty } 3 {\displaystyle {\sqrt {3}}} 1 {\displaystyle 1} 3 3 {\displaystyle {\frac {\sqrt {3}}{3}}} 0 {\displaystyle 0} sec 1 {\displaystyle 1} 2 3 3 {\displaystyle {\frac {2{\sqrt {3}}}{3}}} 2 {\displaystyle {\sqrt {2}}} 2 {\displaystyle 2} ∞ {\displaystyle \infty } csc ∞ {\displaystyle \infty } 2 {\displaystyle 2} 2 {\displaystyle {\sqrt {2}}} 2 3 3 {\displaystyle {\frac {2{\sqrt {3}}}{3}}} 1 {\displaystyle 1}
Adierazpen grafikoak
Identitateak
Identitate pitagorikoak sin 2 ( x ) + cos 2 ( x ) = 1 , sec 2 ( x ) − tan 2 ( x ) = 1 , csc 2 ( x ) − cot 2 ( x ) = 1 {\displaystyle \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 ± y ) = sin ( x ) cos ( y ) ± cos ( x ) sin ( y ) {\displaystyle \sin(x\pm y)=\sin(x)\cos(y)\pm \cos(x)\sin(y)} , csc ( x ± y ) = 1 sin ( x ± y ) {\displaystyle \csc(x\pm y)={\frac {1}{\sin(x\pm y)}}} cos ( x ± y ) = cos ( x ) cos ( y ) ∓ sin ( x ) sin ( y ) {\displaystyle \cos(x\pm y)=\cos(x)\cos(y)\mp \sin(x)\sin(y)} , sec ( x ± y ) = 1 cos ( x ± y ) {\displaystyle \sec(x\pm y)={\frac {1}{\cos(x\pm y)}}} tan ( x ± y ) = tan ( x ) ± tan ( y ) 1 ∓ tan ( x ) tan ( y ) {\displaystyle \tan(x\pm y)={\frac {\tan(x)\pm \tan(y)}{1\mp \tan(x)\tan(y)}}} , cot ( x ± y ) = cot ( x ) cot ( y ) ∓ 1 cot ( y ) ± cot ( x ) {\displaystyle \cot(x\pm y)={\frac {\cot(x)\cot(y)\mp 1}{\cot(y)\pm \cot(x)}}}
Angelu bikoitza eta erdia sin ( 2 x ) = 2 tan ( x ) 1 + tan 2 ( x ) = 2 sin ( x ) cos ( x ) {\displaystyle \sin(2x)={\frac {2\tan(x)}{1+\tan ^{2}(x)}}=2\sin(x)\cos(x)} , csc ( 2 x ) = 1 sin ( 2 x ) {\displaystyle \csc(2x)={\frac {1}{\sin(2x)}}} cos ( 2 x ) = 1 − tan 2 ( x ) 1 + tan 2 ( x ) = cos 2 ( x ) − sin 2 ( x ) = 2 cos 2 ( x ) − 1 {\displaystyle \cos(2x)={\frac {1-\tan ^{2}(x)}{1+\tan ^{2}(x)}}=\cos ^{2}(x)-\sin ^{2}(x)=2\cos ^{2}(x)-1} , sec ( 2 x ) = 1 cos ( 2 x ) {\displaystyle \sec(2x)={\frac {1}{\cos(2x)}}} tan ( 2 x ) = 2 tan ( x ) 1 − tan 2 ( x ) {\displaystyle \tan(2x)={\frac {2\tan(x)}{1-\tan ^{2}(x)}}} , cot ( 2 x ) = cot 2 ( x ) − 1 2 cot ( x ) {\displaystyle \cot(2x)={\frac {\cot ^{2}(x)-1}{2\cot(x)}}}
sin ( x / 2 ) = ± 1 − cos ( x ) 2 {\displaystyle \sin(x/2)=\pm {\sqrt {\frac {1-\cos(x)}{2}}}} , csc ( x / 2 ) = 1 sin ( x / 2 ) {\displaystyle \csc(x/2)={\frac {1}{\sin(x/2)}}} cos ( x / 2 ) = ± 1 + cos ( x ) 2 {\displaystyle \cos(x/2)=\pm {\sqrt {\frac {1+\cos(x)}{2}}}} , sec ( x / 2 ) = 1 cos ( x / 2 ) {\displaystyle \sec(x/2)={\frac {1}{\cos(x/2)}}} tan ( x / 2 ) = csc ( x ) − cot ( x ) = ± 1 − cos ( x ) 1 + cos ( x ) = sin ( x ) 1 + cos ( x ) {\displaystyle \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 ) {\displaystyle \cot(x/2)=\csc(x)+\cot(x)}
Biderketatik batuketara sin ( x ) sin ( y ) = cos ( x − y ) − cos ( x + y ) 2 {\displaystyle \sin(x)\sin(y)={\frac {\cos(x-y)-\cos(x+y)}{2}}} , sin ( x ) cos ( y ) = sin ( x + y ) + sin ( x − y ) 2 {\displaystyle \sin(x)\cos(y)={\frac {\sin(x+y)+\sin(x-y)}{2}}} cos ( x ) cos ( y ) = cos ( x + y ) + cos ( x − y ) 2 {\displaystyle \cos(x)\cos(y)={\frac {\cos(x+y)+\cos(x-y)}{2}}} , cos ( x ) sin ( y ) = sin ( x + y ) − sin ( x − y ) 2 {\displaystyle \cos(x)\sin(y)={\frac {\sin(x+y)-\sin(x-y)}{2}}}
sin 2 ( x ) − sin 2 ( y ) = sin ( x + y ) sin ( x − y ) {\displaystyle \sin ^{2}(x)-\sin ^{2}(y)=\sin(x+y)\sin(x-y)}
cos 2 ( x ) − sin 2 ( y ) = cos ( x + y ) cos ( x − y ) {\displaystyle \cos ^{2}(x)-\sin ^{2}(y)=\cos(x+y)\cos(x-y)}
sin 2 ( x ) cos 2 ( x ) = 1 − cos ( 4 x ) 8 {\displaystyle \sin ^{2}(x)\cos ^{2}(x)={\frac {1-\cos(4x)}{8}}}
Batuketatik biderketara sin ( x ) + sin ( y ) = 2 sin ( x + y 2 ) cos ( x − y 2 ) {\displaystyle \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 ( x − y 2 ) cos ( x + y 2 ) {\displaystyle \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 ( x + y 2 ) cos ( x − y 2 ) {\displaystyle \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 ( x + y 2 ) sin ( x − y 2 ) {\displaystyle \cos(x)-\cos(y)=-2\sin \left({\frac {x+y}{2}}\right)\sin \left({\frac {x-y}{2}}\right)} tan ( x ) + tan ( y ) = sin ( x + y ) cos ( x ) cos ( y ) {\displaystyle \tan(x)+\tan(y)={\frac {\sin(x+y)}{\cos(x)\cos(y)}}} , tan ( x ) − tan ( y ) = sin ( x − y ) cos ( x ) cos ( y ) {\displaystyle \tan(x)-\tan(y)={\frac {\sin(x-y)}{\cos(x)\cos(y)}}}
Berreketak sin 2 ( x ) = 1 − cos ( 2 x ) 2 {\displaystyle \sin ^{2}(x)={\frac {1-\cos(2x)}{2}}} {\displaystyle \quad } cos 2 ( x ) = 1 + cos ( 2 x ) 2 {\displaystyle \cos ^{2}(x)={\frac {1+\cos(2x)}{2}}} {\displaystyle \quad } tan 2 ( x ) = 1 − cos ( 2 x ) 1 + cos ( 2 x ) {\displaystyle \tan ^{2}(x)={\frac {1-\cos(2x)}{1+\cos(2x)}}}
sin 2 ( x ) − sin 2 ( y ) = sin ( x + y ) sin ( x − y ) {\displaystyle \sin ^{2}(x)-\sin ^{2}(y)=\sin(x+y)\sin(x-y)}
cos 2 ( x ) − sin 2 ( y ) = cos ( x + y ) cos ( x − y ) {\displaystyle \cos ^{2}(x)-\sin ^{2}(y)=\cos(x+y)\cos(x-y)}
sin 2 ( x ) cos 2 ( x ) = 1 − cos ( 4 x ) 8 {\displaystyle \sin ^{2}(x)\cos ^{2}(x)={\frac {1-\cos(4x)}{8}}}
Deribatuak d d x sin ( x ) = cos ( x ) {\displaystyle {\frac {d}{dx}}\sin(x)=\cos(x)}
d d x cos ( x ) = − sin ( x ) {\displaystyle {\frac {d}{dx}}\cos(x)=-\sin(x)}
d d x tan ( x ) = sec 2 ( x ) = 1 + tan 2 ( x ) {\displaystyle {\frac {d}{dx}}\tan(x)=\sec ^{2}(x)=1+\tan ^{2}(x)}
d d x csc ( x ) = − csc ( x ) cot ( x ) {\displaystyle {\frac {d}{dx}}\csc(x)=-\csc(x)\cot(x)}
d d x sec ( x ) = sec ( x ) tan ( x ) {\displaystyle {\frac {d}{dx}}\sec(x)=\sec(x)\tan(x)}
d d x cot ( x ) = − csc 2 ( x ) = − ( 1 + cot 2 ( x ) ) {\displaystyle {\frac {d}{dx}}\cot(x)=-\csc ^{2}(x)=-(1+\cot ^{2}(x))}
Integralak Funtzio trigonometrikoen integralen zerrenda
∫ sin ( x ) d x = − cos ( x ) + C {\displaystyle \int \sin(x)dx=-\cos(x)+C}
∫ cos ( x ) d x = sin ( x ) + C {\displaystyle \int \cos(x)dx=\sin(x)+C}
∫ tan ( x ) d x = − ln | cos ( x ) | + C {\displaystyle \int \tan(x)dx=-\ln |\cos(x)|+C}
∫ csc ( x ) d x = − ln | csc ( x ) + cot ( x ) | + C {\displaystyle \int \csc(x)dx=-\ln |\csc(x)+\cot(x)|+C}
∫ sec ( x ) d x = ln | sec ( x ) + tan ( x ) | + C {\displaystyle \int \sec(x)dx=\ln |\sec(x)+\tan(x)|+C}
∫ cot ( x ) d x = ln | sin ( x ) | + C {\displaystyle \int \cot(x)dx=\ln |\sin(x)|+C}
Teoremak Sinuaren teorema. A B C {\displaystyle ABC} triangelu batean α , β , γ {\displaystyle \alpha ,\beta ,\gamma } hurrenez hurren a , b , c {\displaystyle a,b,c} aldeen aurkako angeluak baldin badira, orduan betetzen da:
a sin ( α ) = b sin ( β ) = c sin ( γ ) {\displaystyle {\frac {a}{\sin(\alpha )}}={\frac {b}{\sin(\beta )}}={\frac {c}{\sin(\gamma )}}}
Kosinuaren teorema. A B C {\displaystyle ABC} triangelu batean α , β , γ {\displaystyle \alpha ,\beta ,\gamma } hurrenez hurren a , b , c {\displaystyle a,b,c} aldeen aurkako angeluak baldin badira, orduan betetzen da:
a 2 = b 2 + c 2 − 2 b c cos ( α ) , b 2 = a 2 + c 2 − 2 a c cos ( β ) , c 2 = a 2 + b 2 − 2 a b cos ( γ ) {\displaystyle 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. A B C {\displaystyle ABC} triangelu batean α , β , γ {\displaystyle \alpha ,\beta ,\gamma } hurrenez hurren a , b , c {\displaystyle a,b,c} aldeen aurkako angeluak baldin badira, orduan betetzen da:
a − b a + b = tan ( α − β 2 ) tan ( α + β 2 ) , b − c b + c = tan ( β − γ 2 ) tan ( β + γ 2 ) , a − c a + c = tan ( α − γ 2 ) tan ( α + γ 2 ) {\displaystyle {\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)}}}
Erreferentziak eta oharrak ↑ Elhuyar Zientzia eta Teknologiaren Hiztegi Entziklopedikoa. Funtzio trigonometriko. .
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