[b]1. Numero complesso: dalla forma cartesiana alla forma trigonometrica[/b][br][br]Un numero complesso [math]z=a+ib[/math] può essere scritto in [b]forma[br]polare o trigonometrica[/b]: [math]z=r\cdot\left(cos\left(\alpha\right)+i\cdot sin\left(\alpha\right)\right)[/math][br][br]dove: [br][list][*] [math]r=\left|z\right|=\sqrt{a^2+b^2}[/math] è il [b]modulo[/b] [/*][*][math]\alpha=arctan\left(\frac{b}{a}\right)[/math] è l'[b]argomento [/b] ([b]l’angolo orientato[/b] che il vettore (a,b) forma con il semiasse reale positivo) [/*][/list]Questa forma è molto comoda per svolgere moltiplicazioni e potenze di Numeri complessi, mentre la forma cartesiana più utile per somme e sottrazioni[br][br][b]2. Numero complesso: dalla forma trigonometrica alla forma esponenziale[br][/b]Allo stesso modo sarà comoda e utile, per le operazioni tra complessi, anche la [b]forma esponenziale[/b] di un numero complesso:[br][br][math]z=r\cdot e^{i\alpha}[/math][br][br]che si basa sulla [b]formula di Eulero: [/b][b][math]e^{i\alpha}=cos\left(\alpha\right)+i\cdot sin\left(\alpha\right)[/math] [br][br][/b]Basta infatti moltiplicare entrambi i membri della formula di Eulero per[math]r[/math]:[br][br] [math]re^{i\alpha}=r\cdot\left(cos\left(\alpha\right)+i\cdot sin\left(\alpha\right)\right)[/math] al fine di ottenere la forma esponenziale di un numero complesso. [br][br]La forma esponenziale è più compatta , utile per prodotti e potenze.[br]
[b]1. Prodotto di numeri complessi in forma cartesiana[/b][br][br]Dimostra che: dati i due numeri complessi [b]in forma cartesiana[/b][br][br][math]z_1=a+bi[/math] [math]z_2=c+di[/math] [br][br] il loro [b]prodotto è[/b]:[br][br][math]z_1z_2=(ac−bd)+(ad+bc)i[/math][br][br]Scrivi sul quaderno la dimostrazione della formula del prodotto tra due numeri complessi in forma cartesiana:
[b]2. Prodotto di numeri complessi in forma trigonometrica[/b][br][br]Dimostra che : dati i due numeri complessi [math]z_1=r_1\cdot\left(cos\left(\alpha_1\right)+i\cdot sin\left(\alpha_1\right)\right)[/math] e [math]z_2=r_2\cdot\left(cos\left(\alpha_2\right)+i\cdot sin\left(\alpha_2\right)\right)[/math] , [br]il prodotto tra i due numeri è: [br][br][math]z_1\cdot z_2=r_1\cdot r_2\left(cos\left(\alpha_1+\alpha_2\right)+i\cdot sin\left(\alpha_1+\alpha_2\right)\right)[/math][br][br]Notate come[b] i moduli si moltiplicano e gli argomenti si sommano. [/b]Questo fatto è di cruciale importanza per interpretare il significato geometrico della moltiplicazione tra due numeri complessi e, di conseguenza, della potenza di un numero complesso. [br][br]Scrivi sul quaderno la dimostrazione della formula del prodotto tra due numeri complessi scritta sopra:
[b]3. Prodotto di numeri complessi in forma esponenziale[/b][br][br]Dimostra che: dati i due numeri complessi in forma esponenziale, [br][math]z_1=r_1\cdot e^{i\alpha_1}[/math], [math]z_2=r_2\cdot e^{i\alpha_2}[/math][br][br]il prodotto tra i due numeri è: [br][br] [math]z_3=r_1r_2\cdot e^{i\left(\alpha_1+\alpha_2\right)}[/math][br][br]Scrivi la dimostrazione sul quaderno o nello spazio ripsosta. [br][br]
E' possibile [b]visualizzare in modo molto esplicativo il prodotto tra due numeri complessi con GeoGebra.[/b] [br][br]Nella barra di inserimento di Geogebra sono stati inseriti:[br][br][list][*]il punto [math]A=1+i[/math], fissato. [br][/*][*]lo slider[math]r[/math], numero che va da 0 a 20, che esprime il modulo di un numero complesso, scritto in forma trigonometrica[/*][*]lo slider [math]\alpha[/math], angolo, che va da 0° a 360°[/*][*]il punto[math]B[/math], espresso nel seguente modo: [math]B=r\cdot\left(cos\left(\alpha\right)+i\cdot sin\left(\alpha\right)\right)[/math][/*][/list][br][math]A[/math] rappresenta un[b] numero complesso fisso[/b][br][math]B[/math] rappresenta un [b]numero complesso variabile[/b], di cui possiamo controllare modulo e argomento tramite i due slider[math]r[/math] e [math]\alpha[/math][br][br][list][*][math]C=A\cdot B[/math], prodotto tra i due numeri complessi [br][br]Inseriamo anche i due vettori:[br][/*][/list][math]OA=vettore((0,0),A)[/math], il vettore originale[br][math]OB=vettore((0,0),B)[/math], il vettore variabile[br][br][math]OC=vettore((0,0),C)[/math], il vettore trasformato, ovvero il vettore risultato del prodotto tra A e B[br][br]
Facciamo alcune osservazioni, al fine di comprendere il significato geometrico della moltiplicazione tra numeri complessi: [br][br][b]1. Fissa[/b][math]r[/math][b] e muovi lo slider [/b][math]\alpha[/math]:[br][list][*]Cosa succede alla lunghezza del vettore [math]OC?[/math][/*][*]Cosa cambia nel vettore[math]OC[/math]?[/*][*]Descrivi il movimento del punto [math]C[/math].[/*][/list]
[b]2. Ora fissa [/b][math]\alpha[/math][b] e cambia [/b][math]r[/math][b]:[/b][br][list][*]Cosa succede a [math]OC[/math]?[/*][*]Cosa puoi dire sulla lunghezza di [math]OC[/math] e di [math]OA[/math]? [/*][*]Che movimento descrive il punto [math]C[/math]?[/*][/list][br]
Rispondi alle seguenti domande:[br][list][*]cosa vuol dire [b]moltiplicare A per un numero complesso[/b] [math]B=r\cdot\left(cos\left(\alpha\right)+i\cdot sin\left(\alpha\right)\right)[/math], di modulo [math]r[/math] e argomento [math]\alpha[/math]? Rispondi osservando la posizione del punto C [/*][/list][list][*]esprimi il concetto in termini di[b] trasformazioni del piano. Ovvero, componendo quali trasformazioni otteniamo il punto C, prodotto tra A e B, partendo dal punto A?[/b][/*][*][b]cosa significa moltiplicare un numero complesso per [math]i[/math], unità immaginaria?[/b][/*][/list]
Sappiamo che la potenza di un numero reale o complesso è una [b]notazione compatta [/b]per rappresentare [b]moltiplicazioni ripetute dello stesso numero[/b], ed è fondamentale perché semplifica molto i calcoli e la scrittura di numeri grandi.[br]
[b]1. Potenza di un numero complesso scritto in forma cartesiana. [br]Assegnato [math]z=a+ib[/math], la potenza n-esima del numero z, si calcola a partire dalla formula della potenza n-esima di un binomio:[br][br][math]z=\left(a+ib\right)^n[/math] [br][/b][br]Risolvi i seguenti esercizi sul quaderno o nello spazio risposta: [br][br][math]z=(1+i)^3[/math][br][br][math]z=(2+2i)^4[/math][br][br][math]z=(1-i)^5[/math][br][br]
[b]2. Potenza di un numero complesso scritto in forma trigonometrica[/b][br][br]Per calcolare la potenza [math]z^n[/math](con [math]n\in N[/math]), scriviamo [math]z[/math] in forma trigonometrica e lo moltiplichiamo per se stesso [math]n[/math] volte. Otteniamo:[br][br] [math]z^n=r^n\cdot\left(cos\alpha+isin\alpha\right)^n[/math][br][br]Prova a calcolare:[br][list][*][math]z^2[/math], [/*][*][math]z^3[/math], per questa potenza sono utili le due seguenti formule, facilmente dimostrabili:[/*][/list][br] 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[/img][list][*] [math]z^4[/math], .........[/*][/list]con un po' di calcoli, ti puoi rendere conto che la potenza di un numero complesso scritto in forma trigonometrica è data dalla seguente formula: [br][br][math]z^n=r^n\left(cos\left(n\alpha\right)+i\cdot sin\left(n\alpha\right)\right)[/math] [b] Formula di de Moivre. [/b][br][br]Questo significa che, elevare un numero complesso a potenza diventa molto semplice perché, invece di eseguire molte moltiplicazioni, basta:[br][list][*]elevare il [b]modulo[/b] [math]r[/math]alla [math]n[/math][/*][*]moltiplicare l’[b]angolo[/b] [math]\alpha[/math] per [math]n[/math][br][br][/*][/list]P.S. Al termine dell'attività troverai una bellissima dimostrazione [b]per induzione[/b] della formula di De Moivre[br][br] [math]z^n=r^n\left(cos\left(n\alpha\right)+i\cdot sin\left(n\alpha\right)\right)[/math][br][br]ESERCIZIO:[br][img]data:image/png;base64,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[/img][br][br][br]
[b]3. Potenza di un numero complesso.[br][br][/b]La potenza [math]z^n[/math] (con [math]n\in N[/math]) di un numero complesso scritto in forma esponenziale, [math]z=r\cdot e^{i\alpha}[/math][br]è data dalla formula [br][br] [math]z^n=\left(r\cdot e^{i\alpha}\right)^n=r^n\cdot e^{in\alpha}[/math], di immediata dimostrazione.[br][br]Questo significa che, elevare un numero complesso alla potenza [math]n[/math] diventa ancora più semplice perchè, invece di eseguire molte moltiplicazioni, basta:[br][list][*]elevare il [b]modulo[/b] [math]r[/math]alla [math]n[/math][/*][*]moltiplicare l’[b]angolo[/b] [math]\alpha[/math], presente nell'esponente di e, per [math]n[/math][br][/*][/list]ESERCIZIO:[br][br][img]data:image/png;base64,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[/img][br]
Riassumendo, sulla base delle formule espresse sia in forma cartesiana, sia trigonometrica che in forma esponenziale della potenza di un numero complesso [math]z[/math], dopo aver svolto gli esercizi assegnati, fai le dovute considerazioni su quale sia la forma più adeguata di un numero complesso per determinare una sua potenza. [br]
[b] Dimostrazione della formula di de Moivre [math]z^n=r^n\left(cos\left(n\alpha\right)+i\cdot sin\left(n\alpha\right)\right)[/math][/b][br][br][b]Per induzione su [/b][img width=10,height=22]data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAoAAAAWCAMAAADO+P1vAAAAAXNSR0IArs4c6QAAAFRQTFRFAAAAAAAAAAA6AABmADpmADqQOgAAOjpmOpC2OpDbZgAAZrb/kDoAkGZmkLbbkNv/tmYAtmY6tmZmttv/tv/btv//27Zm2////7Zm/9uQ//+2///be0k4qAAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAATUlEQVQYV7WOSRKAIBADE0VUXFgURfj/P535g+aU6vQhwE9pkSZflhOQfOC4F9tnoK1cUDjcwKMgcJYDCqrrvFQFspwbqhNVfHN8evEFS9AC2E8jFygAAAAASUVORK5CYII=[/img][br][list=1][br][*][b]Base[/b][br] ([img width=39,height=22]data:image/png;base64,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[/img]): [/*][br][/list][img 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induttivo[/b]: supponiamo che valga per [img width=9,height=22]data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAkAAAAWCAMAAAAlz0ZsAAAAAXNSR0IArs4c6QAAAFdQTFRFAAAAAAAAAABmADpmADqQAGa2OgAAOgA6OgBmOpCQOpDbZjoAZrb/kDoAkGY6kGaQkJC2kNv/tmYAtpBmttv/tv//25A62//b2////7Zm/9uQ/9u2///bjf122wAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAASUlEQVQYV62LDQ6AIBSCsbQy7c/SfOb9zxneQTY24BtAZ9Vd+c+pFbifYM+cdAbqZoBAo8wXm2eSIaIskakB0e9BwJtMYxs76Acw7QLctpahQgAAAABJRU5ErkJggg==[/img]: [/*][/list][br][img 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[/img]=[br][b][br]Ricordando il passo induttivo[/b] per il primo fattore del secondo membro[b]:[br][br][/b][img 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7In9NaN4jgtI9kuaQ8PUqRWIgSgco1k1JmzOfDmKMRg0L2KI0N49cyxnBi3n7MK4ZKhxcp7hykeXzJLOTUS63mZZTzP2fZ65Vrn2Ytub0Sxi5ZOjCHa4G5EaKS/dQp+iqLac4m+HKqSnyApQ67KKZddNhlucguSYr7kE72B4x05Au61zNI10FGvLkvvPQjuge+KaJ6pl+A/E17FZRw5f0ivBIi9/IY0rJzqOsaFan10v3kKfr67zw0F8Mo8KPwylVy10g3C1nikOUzfPnmZcn6SE6WRtpCjNW9eV6HCkrPlSwmOTKZDtNivIcGP7NRLIdLHg8Qu0sTSkzoln1OpH7L7wOPUBaP3L7A5yyFk/9N4nEN9xjOEeABvKrgIew3O34G9ovR2SXYz7FM758x8SyW1XyuBoBZETKpR3iuChEyTXsvEx0fVi9hrJzII9D3uPWBZWDgjW1xzoNzhlRH1HNeRz3nA9Rzrunw1wx0VQu6mrL/qKDes0Kad2PJlOl/wqdE0FLyKthR+FSzNuCVWR4/o06X6LgOHR3QMb0tzgkd9ZyOS9DRY5c6aG+CCH40oa7k7TlCS3dD5KmSnhbzwT6qeL7O3t6hUPu3pn3iOr396TpNYF5RD5sOmz79Z/h0Houk+bmXf76y88fi8A1xsAfzC//49n2ksudI18/SUmbX/vopuv3Lk/AHnYy3zeuryStw35CyDEhZ8oQyiWe9SJacfQKBz9veO/32zsStiAWp+p3vVdRncjSzWe57aMZlknnApg4EQMj6/eTvryVI5WhXdMxO7ocgoUaMbJQmsT+N4az4Kz6HaJky1L1p2N12QQ6NkAvkEPWSXYIY9oOQIvI4Zl+tJQhFg5fJMRmjCL7UppNBCiYdNIkP68lgL9kXwzQf0wwhR7D3PPIxcdSp+cSY8vMVRwQTlVlQEcl83rKbpL62XDVBqPOCXO1nOYwDCatiREYNmEBuhf2HNB0+BRxB4NNB6nX4aDp4DqltoFYlj708m7OLCrvMltjFsRihpOHV7TQEu1wgA3ZxmHaRtWHFdoFV9DGPjzpBSmJxu2qkZjNjjkeKpDCukKqHKaN6sowmv/9QuRQ5RddXQI4gcx9kvjR7jjyQ2V9BZkmK+y4oUhZ1tyxX4CPNSSu6QhEaveYhAyG2kt92ktx20Ye7Iqi5+sdOEInYCvBIBzPjjtUsHleBh1GEh9jXsf9IuQQbXr8LPDZmTTyCkyYeFXwnGvw92c/dpTBZ6FIKz1Wqnc7h/nN/6CMHnqukF/bThmi27wLBMMBsYpf9gBkgeyCfKxt5Hl6h9cxdHNug6LiDLuHcNZyLFJ17AOL4RnRs2+5pV2lpJxPS7hDqKqnzk3XyekFI3/g/2+P2VVU+vRXMIQgtZFKFTO2mTDLNavrUFQK/KZCtOp+bei4Tnx1Cx8c7d4HFBi2Pt9N0Vscwzp3EuTV5TrGkI2Pb7Seh4zfQ8Xdv0Xibhzo+eUwfeEFI/3Nwe7x9Va3lbRF87tl/LezTRpfSeL7S7fTruzuwz3+hdjxf/+k7e2ZAhX2O+fF83aN2yy4seoRPi205M37yV2XnV965asncNrYtJwlhR0Ce33+oyufrux26uvEHOtbuo4/wfP1UpQ4BdtmBLBbxfGVl2cnbRcgCD0a2vGR7zdFFyr01fJdUMXg9xuFrGIEsAkwyX3BXMEkMYmGRosYRq19EExElfPqczTVJM8458n0VpZDLVH5uNUWpkGagGF2x+q0iilkVlcSWVdFsaSP4EBwSXx/9VrPesnsTP8u7pmQDi5h7dVdkMxe1XRz05K953rIjvEK15NqtfCOizc+GcRdNedE4gcilYfNkBnsmlMU6/PRff7CqsIsOuyjCLvE5t7qFesu3NuO60TWlOBBFM4xoZvWeoThnCpFNSfZ6HcriIJoCVMxpSJKgCJmThh0k9APVeiFgJJAeFz8MtO5ywnTRP35gykyQecBZj8wpWpM+sksWpKwXBzwgwU9Hbk1i6kAXOiJAWTEkmRE0xke6+X9sIDfJFV/5xiRZjzlCH8L/izP06SQcGhGoSmWgc2tZPGxe9VWJx8XqeAgbQgeyuTODTh8qXqsbMbH1N7Dfhj4r7Gc4KBZyqU8QZe3+d4SuOj807UepDPy3JAorI6ACs9PA7MsvLcbAf9ge/5Oijtz+gB58qVhurC3o9OoFOrF1g1a8aM75ReGciLwNaL/BZ8I18n29TFddppDXYPtz3oHMLYUs0FW/XzV1m6RVhMZktNVqNu3IqF9fu7p42k3tkGF3I0/uvE9E5XbbESS12I7rd59ugMHnCxprTD2+/4tiuSl1PA8db0JHNN38QlXPfoNzIJg3ReRSm6Qrzj+R75soXXVL/iZ1/LXXniHFa3n1vKnj/Somur9ltWj2jczsI1Ljhp1if+uyPEG95ZubKxnq/Ae1HXIYA8lt2EftwQ8SXzayKXU9Bix+7payOgb0bUyjVppfe7Py8wUfsMCftvF8qbU/E5K0JuwCWazx79u6utbQdCSIp8MiZJn6xZfyWMlsjg5UKt8iUa+JjDlvjMC+EWCSuW8I9zfAvgmM/MJ00iC+Gbd2FzmWOScip91d5ud6IJBQQinku5GWFREQpDiNyDxS5Qt1LC8kU4uot5w36y1zEcLRKXxzJ3KCQC6PDeezGGXvGUT+XB47KNkFXtXk2mXS/USb94XxXlwrHaHFez00c92st9QIpLP3gjI69Q7NzQmzLwtD0yCi2fFcfjp7zyDy5+IYZFZDybBOwx4QG5n61sPwC7dt/53kT6uE9DQGLsgSplsVUtamzy6odOICXThRGKl8d/mTyjWjImLp7Kg7UirwuCrwOAM8QPMkHteRGm/G0kbZZ+TTz8x6S3TKgUBeUEbAiAUhNjfTf/Mp/ew9p5E/9z7AB8FGkB6ive50LqxZDPoGPi92nUOa1NLdaQiyoVq+/L8WIxnehu3VduiKNH8mfB2p8ge1lkYyZRLBdlmDKGSCH55G/txbUpdoCgM7WujE+d12LNNdXqE2EZ8dj4DDz0XUOZ+Tzn3OpHCuh35e9GtBzPdmh57Bxxx0dFmM9dvbdOak+roKkgsdl6Cjx15HJzf0Ev756Z/NessBEgTSr579rYv+NpuRAK2TPp2XK3vP6T+bxwKB+5ar67czcn6LTH1nlvB81TX/M38mmLLEcrnyDVN+IUu8tGD0mUfmGxiBehBgklkPSk28Zj8Epj6x4oQf7DJ1Lr+zsuliZ5aVBhYSihXpeiNm1nW67XUuL5RNLU7lGnjkF90o5TmmJJBEHUVCRi+j9hOd5v31kFh5XxNkb4hc9SGfu2rPGD/bNObVsrYSkdBcylcSyBGaekchyTHL6L982UtxdJrHHWh28fvNHyALCdXaj7R6bEiHXyge+MUSljfC8isqcrF7kezpe9Ip6SNHd/sIOs1jkGUL9aONmah0lOhXc+QcSJaNlFaaT+DxKsitsQI8UDfpcYwDj6uNxUNMLmsrOwspe/lcmVFX077mc/V6MWa/x3MlMAPp9d8AZm8cqQFbnNZSBj4TzKhlLt3tPI3oHJLtgc8TFqGrvmLWdXraoWu1pZEkcd0lk3jW0Wke+/5Li//m2gGs2yl0LHS1m9FEv9oD5inJ+cJ96HgeOpp1nULHb6BjreWFSNZWdtI/oAbAI+0hSNtZ+i2eL8kx4dOPdtknIrFYopXvVWBhLl1UbX7UMzTG7TfwfMEu7SW+gmddyDL1lzb/7ihmY2blURiBEgSYZL7oDmHzEFJx5LsclCnvXN3lcHKSNBANeW46SJF5zRARynT0MpqMRAQSUY/UpqFMd8iGIpEqH+roqvtLXXxRI0dfaBDKR4fSlNw8YqRtDlS3CZKYppBo7gmCwJ4XDS4u3JONdB6E7PXI1UCf0FADu1eM9yLG8j+j6QR2catZwvjWJpp8OlAXl6aZsbCeDmXtgi9Dw67o0SN3yCXsorvIne00h8y6Ot0h18HEpto7u2T6fC/yVL3HZoeP3JM+okOW5SPfkufCKBoehCxFTe8NnDhXB9r5M6LPouP6J91/RNmI2UluQz0a3VtFQUXpykkSj0vA4+4HAgbVfrQTKdYm4IHRo/+MqkkQxlx3u9b974i6diLqukHB8bDu+aNpvzVgJuzX/UoRZoK8CKxeexfP/QT5Pg7S+lUvZNVk3eVw8pz6u/+omOcuzdLPrnszWBPRsoEfDT4RnUMk9G76k0zbt0fpMXS9RajXPNopU7tVTfAaZEKBhpDJa1MydyCTW9hxx0U/otKx0GneQEPis2PA+SvyfTJLj6HjANaeTEPHM9Bx6jXFPPfRLP1Xh1cSR9Rk0kT8bfoEEby7G+NSx7XvpD0ttnp0zIoe+RqfeyBp7yLtLcsC3txEYw2eL9Sgz45HMy7YpxM+LbGwqxIL4dPf7LjRpGP6NO7JWD7qwDqYzz7/MyH4Gp4vxZTl5ml159s//Ispy7aLfrKo5e0CEn1PkGh0nv+0O0L8TJPzxYyAiQCTzBfcE0QkVItFjNVekfL2QRunsTkzLzu85QdaDCELGaEU58g4j67fmVQM6WqkxOWalaJuLXsu23lerbNcDLI7GpqD0BnHWL2rsjtdkN0Qmlp6kZaVmXg0+0RS3pKu9YOQXaava8jVUJewdwDV8hibSymZ1WfuXpD9+dL9ZMyLKKIDBADcP+4QtYv6eZ8mO8TLyWjWVl5QnIOioUVcYtbW9sAujqxdZJ1nZDTT60J1V5Fd3MX1j/ZOyCzq4KRfIPIi3sAzD+KDe9OzmT7PnCL6Ajx9HZScJ/2M7DbP7q949fT7DpI1k7HqMss0dHhE7/MgbCmy0PCR8LqXUBa47wXeK9lQpNuRGtYnPGcUunKd5nxFSxVJWy3SU+tzyuMTu/C4LomgaALq81wDHgbwOErJ64r+d54/AY/sfhYPYUOJhwEbIhpcbnkks7YS9pMRxcLmjGPuvkeyw9uGOa+Gz2LOdlFgC8yu0BIwK+5aF+lvbSWcWe0TKe8J8TagzAh09aKz/OJFdI+vpDKECOW77Tgn6m/R+X5lfQXrayoW5Y2j2z00oWZtjx8XZuf5j3/5Unael8MVdrRcXRoRMqnFMoFgNm1ReJH+1u4uZVaPeZDyLtbRbpkW61LeXYeOZ9STxTo+ho5IiSvq0W3YU223SP82dUTn+U81Fj4vjoaKn19rWTCc9yboddjnNuyDZhrL1dtnt4+dfF21oK39bTTY3Bb2KR4bdY/Cn15v85n4I70+sjQv56d0EPdeUw3c+y78af26kvnlyWuU31/xZjb+vh3Pl44obDvpV9YzE0ex+H6Zxethl7Y/3j67c+zd1y1tOVke460+/1T5rT4bqEs20I1erimooZ+LPNihQYBJ5ktgakGcRMo736GeL4KUtU9Yu6IfHb6F9nUQTKk11gpU+nFff3FneyK/2FBFZHIpftFclNtEZ3Q/2ojFULnGoeJjqBgVtZ5Pjfm8ZZd6F8laSa5GuUU1jOW5LGZCjgXAWbx/Q/xKsF4odGejcQPmqmYXlZDi7i+6LvCvVrXYLuL2ABWO5fQvHjaw8IN6AhOdKPaL+wvkR1cEJML6kjgnpcBamUJm7JuXZtfORJMJDplbLZkT1pLxEjfM5qJmbgmrX5XyPblB/mKXlJH1Cfoikiaft9AFLtfURPr4RFEdKN3HvXk8cE4KjDrQG4SS0aJ9NAeJphscMjfM2V+hry5Xb3oC15zIXhNI/A3wMce7j7HMxpTCMTHnfaz1ubthRRA/kfLO165izLUnpgiChNKrJyBTQaEnGEOcDqyRpZyuOUJVyS54pizFeguZmr1JHY9Dx+MFbAs6ouazko5EluPA5nixPf8KfP5aXWJZR3oczUVPbhawvG+1HD+PsXDrX9E4ZA5ROFbOPvCnivPD7PnxxL03bxKmNMeX+6IBCw1OOJT1p4Is5aSHXdqK779/s7Jdct3oPac/o6m/oClobVdTULMNyuO/lAgwyXwpzcpKMQKHBwGxGPoJKhCzvWouSJ72jyO6z/OxEv6gvlda7nUuvu9pBGBHi7TjFqNzIAhEf0+/Qn3p1//LhnS6ozllDgeiGE96kAgwyTxI9HluRoARaCkEEPlRk1fwLvK+WfEuctnYg3rlumuVW0oZFoYRqBMB881A31o+XvuOfvonld9dXidufFltBJhk1saIr2AEGIFDhICIjCanSP/4WJ9y9LMVJpqHyPaHUdX3Xrmzc+z3A5bfPr4rlqpignkYnaCJOjPJbCK4PDQjwAi8mAgsJLawZFHpu81fTE1YakagOgKf/9Dfdtxq1vbyxgg0GgEmmY1GlMdjBBgBRoARYAQYAUaAEaD/D9giHaahJOLMAAAAAElFTkSuQmCC[/img][br]Ma per le [b]formule di addizione del coseno e seno[/b]:[br][br][br][img 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[/img][br][br]Induzione completata.[br][br][br]