Kolmio. Tangenttilause on euklidisen geometrian perustulos.[1] kolmiossa , jonka kaksi kulmaa ovat
α
{\displaystyle \alpha }
β
{\displaystyle \beta }
a
{\displaystyle a}
b
{\displaystyle b}
a
+
b
a
−
b
=
tan
α
+
β
2
tan
α
−
β
2
{\displaystyle {\frac {a+b}{a-b}}={\frac {\tan {\frac {\alpha +\beta }{2}}}{\tan {\frac {\alpha -\beta }{2}}}}}
Sinilauseen mukaan
a
sin
α
=
b
sin
β
.
{\displaystyle {\frac {a}{\sin \alpha }}={\frac {b}{\sin \beta }}.}
Olkoon
d
=
a
sin
α
=
b
sin
β
,
{\displaystyle d={\frac {a}{\sin \alpha }}={\frac {b}{\sin \beta }},}
jolloin
a
=
d
sin
α
ja
b
=
d
sin
β
.
{\displaystyle a=d\sin \alpha {\text{ ja }}b=d\sin \beta .\,}
Tällöin
a
−
b
a
+
b
=
d
sin
α
−
d
sin
β
d
sin
α
+
d
sin
β
=
sin
α
−
sin
β
sin
α
+
sin
β
.
{\displaystyle {\frac {a-b}{a+b}}={\frac {d\sin \alpha -d\sin \beta }{d\sin \alpha +d\sin \beta }}={\frac {\sin \alpha -\sin \beta }{\sin \alpha +\sin \beta }}.}
Kun käytetään identiteettiä
sin
(
α
)
±
sin
(
β
)
=
2
sin
(
α
±
β
2
)
cos
(
α
∓
β
2
)
,
{\displaystyle \sin(\alpha )\pm \sin(\beta )=2\sin \left({\frac {\alpha \pm \beta }{2}}\right)\cos \left({\frac {\alpha \mp \beta }{2}}\right),\;}
saadaan
a
−
b
a
+
b
=
2
sin
1
2
(
α
−
β
)
cos
1
2
(
α
+
β
)
2
sin
1
2
(
α
+
β
)
cos
1
2
(
α
−
β
)
=
tan
[
1
2
(
α
−
β
)
]
tan
[
1
2
(
α
+
β
)
]
.
{\displaystyle {\frac {a-b}{a+b}}={\frac {2\sin {\tfrac {1}{2}}\left(\alpha -\beta \right)\cos {\tfrac {1}{2}}\left(\alpha +\beta \right)}{2\sin {\tfrac {1}{2}}\left(\alpha +\beta \right)\cos {\tfrac {1}{2}}\left(\alpha -\beta \right)}}={\frac {\tan[{\frac {1}{2}}(\alpha -\beta )]}{\tan[{\frac {1}{2}}(\alpha +\beta )]}}.}
↑ Thompson, Jan & Martinsson, Thomas: Matematiikan käsikirja , s. 376. Helsinki: Tammi, 1994. ISBN 951-31-0471-0 .
![{\displaystyle {\frac {a-b}{a+b}}={\frac {2\sin {\tfrac {1}{2}}\left(\alpha -\beta \right)\cos {\tfrac {1}{2}}\left(\alpha +\beta \right)}{2\sin {\tfrac {1}{2}}\left(\alpha +\beta \right)\cos {\tfrac {1}{2}}\left(\alpha -\beta \right)}}={\frac {\tan[{\frac {1}{2}}(\alpha -\beta )]}{\tan[{\frac {1}{2}}(\alpha +\beta )]}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c3de025a3d4fcf624f5401fbe4cd4bb7d285e53f)