L’altro giorno sono andato (… cosa volete è obbligatorio per i neo-iscritti) al corso di deontologia professionale dell’ordine degli ingegneri di Ravenna. Poiché tra un discorso interessante e l’altro c’erano anche discorsi un po’ noiosi mi è venuta la voglia di fare qualche dimostrazione matematica dei vecchi studi… e visto che ultimamente l’ho usato spesso, proprio la dimostrazione di cos (α-β).
Poi metterò anche le pagine del mio blocco appunti così vedete che è vero che gli ingegneri sono un po’ matti 🙂
La dimostrazione di cos(α-β) è la base per la facile dimostrazione delle formule gemelle
cos (α-β)
sin (α-β)
sin (α+β)
cos (α-β) = ?
![ab_aprimobprimo](https://i0.wp.com/www.versionestabile.it/blog/wp-content/uploads/2015/07/ab_aprimobprimo.png?resize=436%2C278)
Il segmento AB sulla circonferenza con c’entro nell’origine lo trasliamo di un angolo β in modo che il punto B vada a coincidere sull’asse x.
Deve ovviamente essere:Â ![\overline{AB}=\overline{A'B'}](http://s0.wp.com/latex.php?latex=%5Coverline%7BAB%7D%3D%5Coverline%7BA%27B%27%7D&bg=ffffff&fg=000&s=0&c=20201002)
![pitagora](https://i0.wp.com/www.versionestabile.it/blog/wp-content/uploads/2015/07/pitagora.png?resize=325%2C185)
Per il famoso Pitagora
![\overline{AB} = \sqrt{(x_A-x_B)^2+(y_A-y_B)^2}](http://s0.wp.com/latex.php?latex=%5Coverline%7BAB%7D+%3D+%5Csqrt%7B%28x_A-x_B%29%5E2%2B%28y_A-y_B%29%5E2%7D&bg=ffffff&fg=000&s=0&c=20201002)
Ed anche
![\overline{A^\prime B^\prime} = \sqrt{(x_A^\prime-x_B^\prime)^2+(y_A^\prime-y_B^\prime)^2}](http://s0.wp.com/latex.php?latex=%5Coverline%7BA%5E%5Cprime+B%5E%5Cprime%7D+%3D+%5Csqrt%7B%28x_A%5E%5Cprime-x_B%5E%5Cprime%29%5E2%2B%28y_A%5E%5Cprime-y_B%5E%5Cprime%29%5E2%7D&bg=ffffff&fg=000&s=0&c=20201002)
E imponiamo l’uguaglianza dei segmenti:
![\overline{AB} = \overline{A^\prime B^\prime}](http://s0.wp.com/latex.php?latex=%5Coverline%7BAB%7D+%3D+%5Coverline%7BA%5E%5Cprime+B%5E%5Cprime%7D&bg=ffffff&fg=000&s=0&c=20201002)
Poiché abbiamo tutte componenti positive possiamo elevare al quadrato e avere:
![\overline{AB}^2 = \overline{A^\prime B^\prime}^2](http://s0.wp.com/latex.php?latex=%5Coverline%7BAB%7D%5E2+%3D+%5Coverline%7BA%5E%5Cprime+B%5E%5Cprime%7D%5E2&bg=ffffff&fg=000&s=0&c=20201002)
Quindi
![(x_A-x_B)^2+(y_A-y_B)^2 = (x_A^\prime-x_B^\prime)^2+(y_A^\prime-y_B^\prime)^2](http://s0.wp.com/latex.php?latex=%28x_A-x_B%29%5E2%2B%28y_A-y_B%29%5E2+%3D+%28x_A%5E%5Cprime-x_B%5E%5Cprime%29%5E2%2B%28y_A%5E%5Cprime-y_B%5E%5Cprime%29%5E2&bg=ffffff&fg=000&s=0&c=20201002)
Sempre dalla trigonometria risulta che:
![A = \begin{cases} x_A=r \cdot cos\alpha \\ y_A = r \cdot sin\alpha \end{cases}](http://s0.wp.com/latex.php?latex=A+%3D+%5Cbegin%7Bcases%7D+x_A%3Dr+%5Ccdot+cos%5Calpha+%5C%5C+y_A+%3D+r+%5Ccdot+sin%5Calpha+%5Cend%7Bcases%7D&bg=ffffff&fg=000&s=0&c=20201002)
e
![B = \begin{cases} x_A=r \cdot cos\beta \\ y_A = r \cdot sin\beta \end{cases}](http://s0.wp.com/latex.php?latex=B+%3D+%5Cbegin%7Bcases%7D+x_A%3Dr+%5Ccdot+cos%5Cbeta+%5C%5C+y_A+%3D+r+%5Ccdot+sin%5Cbeta+%5Cend%7Bcases%7D&bg=ffffff&fg=000&s=0&c=20201002)
e
![A^\prime = \begin{cases} x_A=r \cdot cos(\alpha-\beta) \\ y_A = r \cdot sin(\alpha-\beta)) \end{cases}](http://s0.wp.com/latex.php?latex=A%5E%5Cprime+%3D+%5Cbegin%7Bcases%7D+x_A%3Dr+%5Ccdot+cos%28%5Calpha-%5Cbeta%29+%5C%5C+y_A+%3D+r+%5Ccdot+sin%28%5Calpha-%5Cbeta%29%29+%5Cend%7Bcases%7D&bg=ffffff&fg=000&s=0&c=20201002)
e
![B^\prime = \begin{cases} x_A=r \cdot cos(\beta-\beta) \\ y_A = r \cdot sin(\beta-\beta)) \end{cases}](http://s0.wp.com/latex.php?latex=B%5E%5Cprime+%3D+%5Cbegin%7Bcases%7D+x_A%3Dr+%5Ccdot+cos%28%5Cbeta-%5Cbeta%29+%5C%5C+y_A+%3D+r+%5Ccdot+sin%28%5Cbeta-%5Cbeta%29%29+%5Cend%7Bcases%7D&bg=ffffff&fg=000&s=0&c=20201002)
Quindi
![\overline{AB}^2 = (r\cdot cos\alpha - r\cdot cos\beta)^2+(r\cdot sin\alpha-r\cdot sin\beta)^2](http://s0.wp.com/latex.php?latex=%5Coverline%7BAB%7D%5E2+%3D+%28r%5Ccdot+cos%5Calpha+-+r%5Ccdot+cos%5Cbeta%29%5E2%2B%28r%5Ccdot+sin%5Calpha-r%5Ccdot+sin%5Cbeta%29%5E2&bg=ffffff&fg=000&s=0&c=20201002)
e
![\overline{A^\prime B^\prime}^2 = (r\cdot cos(\alpha-\beta) - r\cdot cos0)^2+(r\cdot sin(\alpha-\beta) - r\cdot sin0)^2 = (r\cdot cos(\alpha-\beta) - r)^2+(r\cdot sin(\alpha-\beta))^2](http://s0.wp.com/latex.php?latex=%5Coverline%7BA%5E%5Cprime+B%5E%5Cprime%7D%5E2+%3D+%28r%5Ccdot+cos%28%5Calpha-%5Cbeta%29+-+r%5Ccdot+cos0%29%5E2%2B%28r%5Ccdot+sin%28%5Calpha-%5Cbeta%29+-+r%5Ccdot+sin0%29%5E2+%3D+%28r%5Ccdot+cos%28%5Calpha-%5Cbeta%29+-+r%29%5E2%2B%28r%5Ccdot+sin%28%5Calpha-%5Cbeta%29%29%5E2&bg=ffffff&fg=000&s=0&c=20201002)
Imponiamo di avere un raggio pari ad 1: ![r=1](http://s0.wp.com/latex.php?latex=r%3D1&bg=ffffff&fg=000&s=0&c=20201002)
Abbiamo:
![\overline{AB}^2 = \cos^2\alpha + \cos^2\beta - 2\cos\alpha\cos\beta + \sin^2\alpha + \sin^2\beta-2\sin\alpha\sin\beta](http://s0.wp.com/latex.php?latex=%5Coverline%7BAB%7D%5E2+%3D+%5Ccos%5E2%5Calpha+%2B+%5Ccos%5E2%5Cbeta+-+2%5Ccos%5Calpha%5Ccos%5Cbeta+%2B+%5Csin%5E2%5Calpha+%2B+%5Csin%5E2%5Cbeta-2%5Csin%5Calpha%5Csin%5Cbeta&bg=ffffff&fg=000&s=0&c=20201002)
e
![\overline{A^\prime B^\prime}^2 = \cos^2(\alpha-\beta) + 1 - 2\cos(\alpha-\beta) + \sin^2(\alpha-\beta)](http://s0.wp.com/latex.php?latex=%5Coverline%7BA%5E%5Cprime+B%5E%5Cprime%7D%5E2+%3D+%5Ccos%5E2%28%5Calpha-%5Cbeta%29+%2B+1+-+2%5Ccos%28%5Calpha-%5Cbeta%29+%2B+%5Csin%5E2%28%5Calpha-%5Cbeta%29&bg=ffffff&fg=000&s=0&c=20201002)
Poiché equivale l’uguaglianza: ![\cos^2\theta + \sin^2\theta = 1](http://s0.wp.com/latex.php?latex=%5Ccos%5E2%5Ctheta+%2B+%5Csin%5E2%5Ctheta+%3D+1&bg=ffffff&fg=000&s=0&c=20201002)
Abbiamo
![\overline{AB}^2 = (\cos^2\alpha + \sin^2\alpha) + (\cos^2\beta + \sin^2\beta) - 2\cos\alpha\cos\beta -2\sin\alpha\sin\beta = 1 + 1 - 2\cos\alpha\cos\beta -2\sin\alpha\sin\beta = 2 - 2\cos\alpha\cos\beta -2\sin\alpha\sin\beta](http://s0.wp.com/latex.php?latex=%5Coverline%7BAB%7D%5E2+%3D+%28%5Ccos%5E2%5Calpha+%2B+%5Csin%5E2%5Calpha%29+%2B+%28%5Ccos%5E2%5Cbeta+%2B+%5Csin%5E2%5Cbeta%29+-+2%5Ccos%5Calpha%5Ccos%5Cbeta+-2%5Csin%5Calpha%5Csin%5Cbeta+%3D+1+%2B+1+-+2%5Ccos%5Calpha%5Ccos%5Cbeta+-2%5Csin%5Calpha%5Csin%5Cbeta+%3D+2+-+2%5Ccos%5Calpha%5Ccos%5Cbeta+-2%5Csin%5Calpha%5Csin%5Cbeta&bg=ffffff&fg=000&s=0&c=20201002)
e
![\overline{A^\prime B^\prime}^2 = (\cos^2(\alpha-\beta) + \sin^2(\alpha-\beta))+ 1 - 2\cos(\alpha-\beta) = 1 + 1 - 2\cos(\alpha-\beta) = 2 - 2\cos(\alpha-\beta)](http://s0.wp.com/latex.php?latex=%5Coverline%7BA%5E%5Cprime+B%5E%5Cprime%7D%5E2+%3D+%28%5Ccos%5E2%28%5Calpha-%5Cbeta%29+%2B+%5Csin%5E2%28%5Calpha-%5Cbeta%29%29%2B+1+-+2%5Ccos%28%5Calpha-%5Cbeta%29+%3D+1+%2B+1+-+2%5Ccos%28%5Calpha-%5Cbeta%29+%3D+2+-+2%5Ccos%28%5Calpha-%5Cbeta%29&bg=ffffff&fg=000&s=0&c=20201002)
Quindi se ![\overline{AB}^2 = \overline{A^\prime B^\prime}^2](http://s0.wp.com/latex.php?latex=%5Coverline%7BAB%7D%5E2+%3D+%5Coverline%7BA%5E%5Cprime+B%5E%5Cprime%7D%5E2&bg=ffffff&fg=000&s=0&c=20201002)
si ha:
![2 - 2\cos\alpha\cos\beta - 2\sin\alpha\sin\beta = 2 - 2\cos(\alpha-\beta)](http://s0.wp.com/latex.php?latex=2+-+2%5Ccos%5Calpha%5Ccos%5Cbeta+-+2%5Csin%5Calpha%5Csin%5Cbeta+%3D+2+-+2%5Ccos%28%5Calpha-%5Cbeta%29&bg=ffffff&fg=000&s=0&c=20201002)
→
![(2 - 2) - 2\cos\alpha\cos\beta - 2\sin\alpha\sin\beta = - \cos(\alpha-\beta)](http://s0.wp.com/latex.php?latex=%282+-+2%29+-+2%5Ccos%5Calpha%5Ccos%5Cbeta+-+2%5Csin%5Calpha%5Csin%5Cbeta+%3D+-+%5Ccos%28%5Calpha-%5Cbeta%29&bg=ffffff&fg=000&s=0&c=20201002)
→
![2\cos(\alpha-\beta) = 2\cos\alpha\cos\beta + 2\sin\alpha\sin\beta](http://s0.wp.com/latex.php?latex=2%5Ccos%28%5Calpha-%5Cbeta%29+%3D+2%5Ccos%5Calpha%5Ccos%5Cbeta+%2B+2%5Csin%5Calpha%5Csin%5Cbeta&bg=ffffff&fg=000&s=0&c=20201002)
→
![\cos(\alpha-\beta) = \cos\alpha\cos\beta + \sin\alpha\sin\beta](http://s0.wp.com/latex.php?latex=%5Ccos%28%5Calpha-%5Cbeta%29+%3D+%5Ccos%5Calpha%5Ccos%5Cbeta+%2B+%5Csin%5Calpha%5Csin%5Cbeta&bg=ffffff&fg=000&s=0&c=20201002)
CVD.
Lo so… gli ingegneri tendono alla follia a volte… 😀
![cos_alpha_men_beta_1](https://i0.wp.com/www.versionestabile.it/blog/wp-content/uploads/2015/07/cos_alpha_men_beta_1.png?resize=640%2C522)
![cos_alpha_men_beta_2](https://i0.wp.com/www.versionestabile.it/blog/wp-content/uploads/2015/07/cos_alpha_men_beta_2.png?resize=640%2C547)
![alpha_men_beta_3](https://i0.wp.com/www.versionestabile.it/blog/wp-content/uploads/2015/07/alpha_men_beta_3.png?resize=640%2C523)
Mi piace:
Mi piace Caricamento...