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 α -->
=
aurkakoa
hipotenusa
=
a
c
{\displaystyle \sin \alpha ={\frac {\textrm {aurkakoa}}{\textrm {hipotenusa}}}=\color {Blue}{\frac {a}{c}}}
cos
-->
α α -->
=
albokoa
hipotenusa
=
b
c
{\displaystyle \cos \alpha ={\frac {\textrm {albokoa}}{\textrm {hipotenusa}}}=\color {Blue}{\frac {b}{c}}}
tan
-->
α α -->
=
aurkakoa
albokoa
=
a
b
{\displaystyle \tan \alpha ={\frac {\textrm {aurkakoa}}{\textrm {albokoa}}}=\color {Blue}{\frac {a}{b}}}
cot
-->
α α -->
=
albokoa
aurkakoa
=
b
a
{\displaystyle \cot \alpha ={\frac {\textrm {albokoa}}{\textrm {aurkakoa}}}=\color {Blue}{\frac {b}{a}}}
sec
-->
α α -->
=
hipotenusa
albokoa
=
h
b
{\displaystyle \sec \alpha ={\frac {\textrm {hipotenusa}}{\textrm {albokoa}}}=\color {Blue}{\frac {h}{b}}}
csc
-->
α 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 θ -->
≡ ≡ -->
1
csc
-->
θ θ -->
≡ ≡ -->
cos
-->
(
π π -->
2
− − -->
θ θ -->
)
≡ ≡ -->
cos
-->
θ θ -->
cot
-->
θ 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 θ -->
≡ ≡ -->
1
sec
-->
θ θ -->
≡ ≡ -->
sin
-->
(
π π -->
2
− − -->
θ θ -->
)
≡ ≡ -->
sin
-->
θ θ -->
tan
-->
θ 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 θ -->
≡ ≡ -->
1
cot
-->
θ θ -->
≡ ≡ -->
cot
-->
(
π π -->
2
− − -->
θ θ -->
)
≡ ≡ -->
sin
-->
θ θ -->
cos
-->
θ 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 θ -->
≡ ≡ -->
1
tan
-->
θ θ -->
≡ ≡ -->
tan
-->
(
π π -->
2
− − -->
θ θ -->
)
≡ ≡ -->
cos
-->
θ θ -->
sin
-->
θ 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 θ -->
≡ ≡ -->
1
cos
-->
θ θ -->
≡ ≡ -->
csc
-->
(
π π -->
2
− − -->
θ θ -->
)
≡ ≡ -->
tan
-->
θ θ -->
sin
-->
θ 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 θ -->
≡ ≡ -->
1
sin
-->
θ θ -->
≡ ≡ -->
sec
-->
(
π π -->
2
− − -->
θ θ -->
)
≡ ≡ -->
cot
-->
θ θ -->
cos
-->
θ 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}
α α -->
,
β β -->
,
γ γ -->
{\displaystyle \alpha ,\beta ,\gamma }
a
,
b
,
c
{\displaystyle a,b,c}
a
sin
-->
(
α α -->
)
=
b
sin
-->
(
β β -->
)
=
c
sin
-->
(
γ γ -->
)
{\displaystyle {\frac {a}{\sin(\alpha )}}={\frac {b}{\sin(\beta )}}={\frac {c}{\sin(\gamma )}}}
Kosinuaren teorema.
A
B
C
{\displaystyle ABC}
α α -->
,
β β -->
,
γ γ -->
{\displaystyle \alpha ,\beta ,\gamma }
a
,
b
,
c
{\displaystyle a,b,c}
a
2
=
b
2
+
c
2
− − -->
2
b
c
cos
-->
(
α α -->
)
,
b
2
=
a
2
+
c
2
− − -->
2
a
c
cos
-->
(
β β -->
)
,
c
2
=
a
2
+
b
2
− − -->
2
a
b
cos
-->
(
γ 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}
α α -->
,
β β -->
,
γ γ -->
{\displaystyle \alpha ,\beta ,\gamma }
a
,
b
,
c
{\displaystyle a,b,c}
a
− − -->
b
a
+
b
=
tan
-->
(
α α -->
− − -->
β β -->
2
)
tan
-->
(
α α -->
+
β β -->
2
)
,
b
− − -->
c
b
+
c
=
tan
-->
(
β β -->
− − -->
γ γ -->
2
)
tan
-->
(
β β -->
+
γ γ -->
2
)
,
a
− − -->
c
a
+
c
=
tan
-->
(
α α -->
− − -->
γ γ -->
2
)
tan
-->
(
α α -->
+
γ 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
Kanpo estekak
