Da Wikipedia, l'enciclopedia libera.
In matematica, il teorema di Nepero afferma le seguenti identità, utilizzando la notazione standard per gli elementi di un triangolo:
Un triangolo generico con le comuni notazioni
![{\displaystyle {\frac {b+c}{b-c}}={\frac {\tan {\displaystyle {\frac {\beta +\gamma }{2}}}}{\tan {\displaystyle {\frac {\beta -\gamma }{2}}}}};}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9EnAePotoPaNGQyte2nDKNnjmNoDsOajo1nDnEajs3nDs2age0ajJF)
![{\displaystyle {\frac {c+a}{c-a}}={\frac {\tan {\displaystyle {\frac {\gamma +\alpha }{2}}}}{\tan {\displaystyle {\frac {\gamma -\alpha }{2}}}}};}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8NnDGQyta4aqeNnjnFatm1zjdEyjs3aNG4aAvCzts2nDw3yghAaAzC)
![{\displaystyle {\frac {a+b}{a-b}}={\frac {\tan {\displaystyle {\frac {\alpha +\beta }{2}}}}{\tan {\displaystyle {\frac {\alpha -\beta }{2}}}}}.}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO80zjw2ygi2aDdFnDhDnjGOzge0aDhEagaOatlEzgrDytdEaNCNzjBA)
Siano
,
,
le lunghezze dei lati di un triangolo, e siano
,
,
le ampiezze degli angoli opposti, rispettivamente.
![{\displaystyle {\frac {b+c}{b-c}}={\frac {(b+c)^{2}}{(b+c)(b-c)}}={\frac {a^{2}(b+c)^{2}}{a^{2}(b+c)(b-c)}}={\frac {a^{2}}{b^{2}-c^{2}}}\left({\frac {b+c}{a}}\right)^{2}={\frac {(a^{2}+b^{2}-c^{2})+(a^{2}-b^{2}+c^{2})}{(a^{2}+b^{2}-c^{2})-(a^{2}-b^{2}+c^{2})}}\left({\frac {b+c}{a}}\right)^{2}.}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9DnjiPnto3zjhEyja0zNs3ztdEoNs3otCQygwNnjC1yqa3zArAaji0)
Per il teorema dei seni
,
e
. Sostituendo si ottiene:
(1)
Consideriamo il secondo membro: usando le formule di prostaferesi, la formula di duplicazione del seno e l'identità
diventa
![{\displaystyle \left({\frac {\displaystyle 2\sin {\frac {\beta +\gamma }{2}}\cos {\frac {\beta -\gamma }{2}}}{\displaystyle 2\sin {\frac {\beta +\gamma }{2}}\cos {\frac {\beta +\gamma }{2}}}}\right)^{2}=\left({\frac {\displaystyle \cos {\frac {\beta -\gamma }{2}}}{\displaystyle \cos {\frac {\beta +\gamma }{2}}}}\right)^{2}.}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO82nteNzjePoNrAzNaPzte0zDvDotdAyjhBz2hCotBCaAe2zqe1nDvE)
Consideriamo il primo addendo del numeratore del primo membro:
Usando la formula di bisezione del coseno, le formule di prostaferesi, le identità
e
, otteniamo:
![{\displaystyle \sin ^{2}\alpha +\sin ^{2}\beta -\sin ^{2}\gamma =1-\cos ^{2}\alpha +1-\cos ^{2}\beta -1+\cos ^{2}\gamma =1-{\frac {\cos 2\alpha +1}{2}}-{\frac {\cos 2\beta +1}{2}}+\cos ^{2}\gamma =\cos ^{2}\gamma -{\frac {1}{2}}(\cos 2\alpha +\cos 2\beta )=}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9AoNeQngzFnjzFo2s1zAnFata0ata1zNlEzgnEaDiNoqrDzDw4njhC)
![{\displaystyle =\cos ^{2}\gamma +\cos \gamma \cos(\alpha -\beta )=\cos \gamma (\cos \gamma +\cos(\alpha -\beta ))=2\cos \gamma \cos {\frac {\alpha -\beta +\gamma }{2}}\cos {\frac {-\alpha +\beta +\gamma }{2}}=2\sin \alpha \sin \beta \cos \gamma .}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9FzqdCatoOago4nDhBnjK3aqe1zjsNzqrCotJByjwPntlDnjG5a2zD)
Allo stesso modo si ottiene che
![{\displaystyle \displaystyle \sin ^{2}\alpha -\sin ^{2}\beta +\sin ^{2}\gamma =2\sin \alpha \cos \beta \sin \gamma .}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8QaqoOatoNoAe0aqzCzAe3zAsNyjnCyjFDnjo1o2hFoNFDaNFFo2zE)
Sostituendo le espressioni trovate per il primo e il secondo membro nella (1) e usando la formula di somma del seno, otteniamo
![{\displaystyle {\frac {b+c}{b-c}}={\frac {2\sin \alpha \sin \beta \cos \gamma +2\sin \alpha \cos \beta \sin \gamma }{2\sin \alpha \sin \beta \cos \gamma -2\sin \alpha \cos \beta \sin \gamma }}\left({\frac {\displaystyle \cos {\frac {\beta -\gamma }{2}}}{\displaystyle \cos {\frac {\beta +\gamma }{2}}}}\right)^{2}={\frac {\sin(\beta +\gamma )}{\sin(\beta -\gamma )}}\left({\frac {\displaystyle \cos {\frac {\beta -\gamma }{2}}}{\displaystyle \cos {\frac {\beta +\gamma }{2}}}}\right)^{2}.}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9FnjJFnjoPoNGNaja5yta3ajm0oqsQotnBaqzAnAaPaAzFytGOaqdD)
Usando la formula di duplicazione del seno otteniamo
![{\displaystyle {\frac {b+c}{b-c}}={\frac {\sin(\beta +\gamma )}{\sin(\beta -\gamma )}}\left({\frac {\displaystyle \cos {\frac {\beta -\gamma }{2}}}{\displaystyle \cos {\frac {\beta +\gamma }{2}}}}\right)^{2}={\frac {\displaystyle 2\sin {\frac {\beta +\gamma }{2}}\cos {\frac {\beta +\gamma }{2}}}{\displaystyle 2\sin {\frac {\beta -\gamma }{2}}\cos {\frac {\beta -\gamma }{2}}}}\left({\frac {\displaystyle \cos {\frac {\beta -\gamma }{2}}}{\displaystyle \cos {\frac {\beta +\gamma }{2}}}}\right)^{2}={\frac {\displaystyle \tan {\frac {\beta +\gamma }{2}}}{\displaystyle \tan {\frac {\beta -\gamma }{2}}}}.}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9DngrEzDdEa2dFa2sOaDFCyjvDaNzDygvAaji5zqaOzNs4aNeQoqs0)
- Nepero, teorema di, in Enciclopedia della Matematica, Istituto dell'Enciclopedia Italiana, 2013.
![Modifica su Wikidata](https://amansaja.com/index.php?q=Mfv0Kfa6bO91KgPRoqwSJ2BVMq1BngBFbA9OnO93MqTXKgrCMqiRo29TLq9SKO90MfrToE83bNKNb0dUJqrHKgrSo2BUbZz2nO8Pafl4bsdUJqrHKgrSo2BUbZz2nO5QLAK%3D)
- (EN) law of tangents, su Enciclopedia Britannica, Encyclopædia Britannica, Inc.
![Modifica su Wikidata](https://amansaja.com/index.php?q=Mfv0Kfa6bO91KgPRoqwSJ2BVMq1BngBFbA9OnO93MqTXKgrCMqiRo29TLq9SKO90MfrToE83bNKNb0dUJqrHKgrSo2BUbZz2nO8Pafl4bsdUJqrHKgrSo2BUbZz2nO5QLAK%3D)
- (EN) Eric W. Weisstein, Law of Tangents, su MathWorld, Wolfram Research.
![Modifica su Wikidata](https://amansaja.com/index.php?q=Mfv0Kfa6bO91KgPRoqwSJ2BVMq1BngBFbA9OnO93MqTXKgrCMqiRo29TLq9SKO90MfrToE83bNKNb0dUJqrHKgrSo2BUbZz2nO8Pafl4bsdUJqrHKgrSo2BUbZz2nO5QLAK%3D)