De tangensregel is een stelling uit de goniometrie die stelt dat in een willekeurige driehoek in het platte vlak met zijden a , b {\displaystyle a,\ b} c {\displaystyle c} α , β {\displaystyle \alpha ,\beta } γ {\displaystyle \gamma }
a − b a + b = tan 1 2 ( α − β ) tan 1 2 ( α + β ) {\displaystyle {\frac {a-b}{a+b}}={\frac {\tan {\tfrac {1}{2}}(\alpha -\beta )}{\tan {\tfrac {1}{2}}(\alpha +\beta )}}} In de zeventiende eeuw werd de tangensregel bewezen met meetkunde[1]
Omdat:
tan 1 2 ( α + β ) = tan 1 2 ( 180 ∘ − γ ) = tan ( 90 ∘ − 1 2 γ ) = cot 1 2 γ {\displaystyle \tan {\tfrac {1}{2}}(\alpha +\beta )=\tan {\tfrac {1}{2}}(180^{\circ }-\gamma )=\tan(90^{\circ }-{\tfrac {1}{2}}\gamma )=\cot {\tfrac {1}{2}}\gamma } kan de tangensregel ook worden geschreven als:
a − b a + b = tan 1 2 ( α − β ) cot 1 2 γ {\displaystyle {\frac {a-b}{a+b}}={\frac {\tan {\tfrac {1}{2}}(\alpha -\beta )}{\cot {\tfrac {1}{2}}\gamma }}} Bewijs
Volgens de sinusregel is:
a sin α = b sin β = k ( k ≠ 0 ) {\displaystyle {\frac {a}{\sin \alpha }}={\frac {b}{\sin \beta }}=k\;(k\neq 0)} Dus: a = k ⋅ sin α {\displaystyle a=k\cdot \sin \alpha } b = k ⋅ sin β {\displaystyle b=k\cdot \sin \beta }
a − b a + b = k ⋅ sin α − k ⋅ sin β k ⋅ sin α + k ⋅ sin β = sin α − sin β sin α + sin β {\displaystyle {\frac {a-b}{a+b}}={\frac {k\cdot \sin \alpha -k\cdot \sin \beta }{k\cdot \sin \alpha +k\cdot \sin \beta }}={\frac {\sin \alpha -\sin \beta }{\sin \alpha +\sin \beta }}} De som- en verschilregel voor sinussen, dat zijn twee van de vier regels van Simson , zijn:
sin α ± sin β = 2 sin 1 2 ( α ± β ) ⋅ cos 1 2 ( α ∓ β ) {\displaystyle \sin \alpha \pm \sin \beta =2\sin {\tfrac {1}{2}}(\alpha \pm \beta )\cdot \cos {\tfrac {1}{2}}(\alpha \mp \beta )} Daarmee is dan:
a − b a + b = sin 1 2 ( α − β ) ⋅ cos 1 2 ( α + β ) sin 1 2 ( α + β ) ⋅ cos 1 2 ( α − β ) = sin 1 2 ( α − β ) cos 1 2 ( α − β ) : sin 1 2 ( α + β ) cos 1 2 ( α + β ) {\displaystyle {\frac {a-b}{a+b}}={\frac {\sin {\tfrac {1}{2}}(\alpha -\beta )\cdot \cos {\tfrac {1}{2}}(\alpha +\beta )}{\sin {\tfrac {1}{2}}(\alpha +\beta )\cdot \cos {\tfrac {1}{2}}(\alpha -\beta )}}={\frac {\sin {\tfrac {1}{2}}(\alpha -\beta )}{\cos {\tfrac {1}{2}}(\alpha -\beta )}}:{\frac {\sin {\tfrac {1}{2}}(\alpha +\beta )}{\cos {\tfrac {1}{2}}(\alpha +\beta )}}} Zodat inderdaad:
a − b a + b = tan 1 2 ( α − β ) tan 1 2 ( α + β ) {\displaystyle {\frac {a-b}{a+b}}={\frac {\tan {\tfrac {1}{2}}(\alpha -\beta )}{\tan {\tfrac {1}{2}}(\alpha +\beta )}}} Externe links https://www.pyth.eu/tangensregel Bronnen, noten en/of referenties
↑ H. Hietbrink, tangensregel, Pythagoras 60-6, juni 2021 
