{"id":7546,"date":"2019-03-06T14:20:29","date_gmt":"2019-03-06T12:20:29","guid":{"rendered":"https:\/\/www.ripetizioni.it\/blog\/?p=7546"},"modified":"2019-03-06T17:19:43","modified_gmt":"2019-03-06T15:19:43","slug":"formule-trigonometriche-trucchi-per-impararle","status":"publish","type":"post","link":"https:\/\/www.ripetizioni.it\/blog\/formule-trigonometriche-trucchi-per-impararle\/","title":{"rendered":"Formule trigonometriche: trucchi per impararle"},"content":{"rendered":"<p><span style=\"font-weight: 400;\">Se stai leggendo questo articolo sicuramente sarai alle prese con gli <\/span><b>odiosissimi<\/b><span style=\"font-weight: 400;\"> esercizi di trigonometria. Bene, allora posso consolarti nel dirti che tutti i tuoi dubbi a breve cesseranno di esistere. E\u2019 noto che la risoluzione degli esercizi di trigonometria, nella maggior parte dei casi, si risolvono riducendo le equazioni utilizzando una serie di formule dall&#8217;aspetto non invitante e di difficile memorizzazione. <\/span><\/p>\n<p>&nbsp;<\/p>\n<p><a href=\"https:\/\/www.ripetizioni.it\/lezioni-private\/matematica?utm_source=blog-ripetizioni&amp;utm_medium=refarral&amp;utm_campaign=formule-trigonometriche-trucchi-per-impararle\"><b><i>Sei interessato ad approfondire questi argomenti? Prendi ripetizioni di trigonometria tramite il portale di Skuola.net | Ripetizioni e inizia subito a migliorare i tuoi voti!<\/i><\/b><\/a><\/p>\n<p>&nbsp;<\/p>\n<p><span style=\"font-weight: 400;\">Ecco, in questo articolo cercher\u00f2 di farti vedere come sia possibile <\/span><b>memorizzarle<\/b><span style=\"font-weight: 400;\"> nella maggior parte dei casi. \u00a0<\/span><\/p>\n<p>&nbsp;<\/p>\n<p><span style=\"font-weight: 400;\">Questo perch\u00e9, non tutte le formule presentano una forma che si presta ad essere memorizzata con facilit\u00e0. Nei casi in cui la memorizzazione fallisce il ragionamento ci salva. Infatti, in queste situazioni la strada migliore da perseguire \u00e8 quella della dimostrazione. Dimostrazioni che in questi casi richiedono solo \u00a0la conoscenza di alcuni teoremi di geometria elementare (<\/span><b>Pitagora, Euclide<\/b><span style=\"font-weight: 400;\">).<\/span><\/p>\n<p>&nbsp;<\/p>\n<p><img decoding=\"async\" class=\" wp-image-6740 aligncenter\" src=\"https:\/\/www.ripetizioni.it\/blog\/wp-content\/uploads\/2018\/10\/matite-1.jpg\" alt=\"matematica\" width=\"1090\" height=\"624\" \/><\/p>\n<p>&nbsp;<\/p>\n<h5><strong>Le identit\u00e0 fondamenti delle trigonometria da memorizzare sono le formule di:<\/strong><\/h5>\n<p>&nbsp;<\/p>\n<ul>\n<li><b>Addizione e sottrazione.<\/b><\/li>\n<\/ul>\n<p>&nbsp;<\/p>\n<ul>\n<li>Duplicazione.<\/li>\n<\/ul>\n<p>&nbsp;<\/p>\n<ul>\n<li>Bisezione.<\/li>\n<\/ul>\n<p>&nbsp;<\/p>\n<ul>\n<li>Prostaferesi.<\/li>\n<\/ul>\n<p>&nbsp;<\/p>\n<ul>\n<li>Werner.<\/li>\n<\/ul>\n<p>&nbsp;<\/p>\n<p><span style=\"font-weight: 400;\">La buone notizia \u00e8 che le formule di <\/span><b>bisezione<\/b><span style=\"font-weight: 400;\">, <\/span><b>duplicazione ,Werner<\/b><span style=\"font-weight: 400;\"> e di <\/span><b>prostaferesi<\/b> <b>\u00a0<\/b><span style=\"font-weight: 400;\">si ricavano facilmente e rapidamente se troviamo un modo di memorizzare le formule di <\/span><b>addizione <\/b><span style=\"font-weight: 400;\">e <\/span><b>sottrazione<\/b><span style=\"font-weight: 400;\">. Se non si volessero memorizzare queste formule, la soluzione migliore \u00e8 utilizzare il ragionamento matematico, ossia, fare la <\/span><b>dimostrazione.<\/b><span style=\"font-weight: 400;\"> Naturalmente questo \u00e8 vero per chi non vuole o non riesce a memorizzarle tutte. <\/span><\/p>\n<p>&nbsp;<\/p>\n<h2><span style=\"color: #007dc3;\"><b>Formule di addizione e sottrazione<\/b><\/span><\/h2>\n<p>&nbsp;<\/p>\n<p><span style=\"font-weight: 400;\">Di seguito sono riportate le formule da memorizzare, ricordando che:<\/span><\/p>\n<p>&nbsp;<\/p>\n<ul>\n<li style=\"font-weight: 400;\"><span style=\"font-weight: 400;\">Nel <\/span><b>seno<\/b><span style=\"font-weight: 400;\"> la struttura della formula \u00e8 sempre <\/span><b>sen cos cos sen<\/b><span style=\"font-weight: 400;\">, gli angoli sono sempre in ordine, ossia<\/span> <span style=\"font-weight: 400;\">, il segno \u00e8 rispettato.<\/span><\/li>\n<\/ul>\n<p>&nbsp;<\/p>\n<ul>\n<li style=\"font-weight: 400;\"><span style=\"font-weight: 400;\">Invece nel\u00a0<\/span><b>coseno<\/b><span style=\"font-weight: 400;\"> la struttura della formula \u00e8 sempre <\/span><b>cos cos sen sen<\/b><span style=\"font-weight: 400;\">, gli angoli sono sempre in ordine, ossia <\/span> <span style=\"font-weight: 400;\">, il segno non \u00e8 rispettato.<\/span><\/li>\n<\/ul>\n<p>&nbsp;<\/p>\n<ul>\n<li style=\"font-weight: 400;\"><span style=\"font-weight: 400;\">Le formule della tangente e della cotangente, anche in questo caso, si ricavano con semplici passaggi matematici.<\/span><\/li>\n<\/ul>\n<p>&nbsp;<\/p>\n<p><span style=\"font-weight: 400;\">Se la memorizzazione delle formule di addizione e sottrazione dovesse portare problemi, naturalmente, la dimostrazione ci risolve tutti i problemi.\u00a0<\/span><\/p>\n<p>&nbsp;<\/p>\n<p><img decoding=\"async\" class=\" wp-image-6639 aligncenter\" src=\"https:\/\/www.ripetizioni.it\/blog\/wp-content\/uploads\/2018\/10\/a-la-signorina-vola.jpg\" alt=\"matematica\" width=\"985\" height=\"564\" \/><\/p>\n<p>&nbsp;<\/p>\n<h3><span style=\"color: #007dc3;\"><strong>Dimostrazione della formula di sottrazione del coseno<\/strong><\/span><\/h3>\n<p>&nbsp;<\/p>\n<p><span style=\"font-weight: 400;\">Siano <\/span><span style=\"font-weight: 400;\">AP<\/span><span style=\"font-weight: 400;\"> = <\/span><span style=\"font-weight: 400;\"> \u00a0\u00a0<\/span><span style=\"font-weight: 400;\">AQ<\/span><span style=\"font-weight: 400;\"> = <\/span><span style=\"font-weight: 400;\"> con <\/span><span style=\"font-weight: 400;\">&gt;<\/span><span style=\"font-weight: 400;\"> i due archi in considerazione e quindi <\/span><span style=\"font-weight: 400;\">QP<\/span><span style=\"font-weight: 400;\"> = <\/span><span style=\"font-weight: 400;\">&#8211;<\/span><span style=\"font-weight: 400;\"> l\u2019arco differenza.<\/span><\/p>\n<p><span style=\"font-weight: 400;\">Dette M,N le proiezioni dei punti P,Q sull\u2019asse x, osserviamo che si ha:<\/span><\/p>\n<p>&nbsp;<\/p>\n<ul>\n<li style=\"font-weight: 400;\"><span style=\"font-weight: 400;\">OM<\/span><span style=\"font-weight: 400;\"> = <\/span><span style=\"font-weight: 400;\">cos <\/span><\/li>\n<li style=\"font-weight: 400;\"><span style=\"font-weight: 400;\">MP<\/span><span style=\"font-weight: 400;\"> = <\/span><span style=\"font-weight: 400;\">sen <\/span><\/li>\n<li style=\"font-weight: 400;\"><span style=\"font-weight: 400;\">ON<\/span><span style=\"font-weight: 400;\"> = <\/span><span style=\"font-weight: 400;\">cos <\/span><\/li>\n<li style=\"font-weight: 400;\"><span style=\"font-weight: 400;\">NQ<\/span><span style=\"font-weight: 400;\"> = <\/span><span style=\"font-weight: 400;\">sen <\/span><\/li>\n<\/ul>\n<p>&nbsp;<\/p>\n<p><span style=\"font-weight: 400;\">Si unisca Q con O e si prolunga detto segmento fino a incontrare in Q\u2019 la circonferenza. Il triangolo Q\u2019PQ, poich\u00e9 \u00a0\u00e8 inserito in mezza circonferenza, \u00e8 rettangolo: <\/span><span style=\"font-weight: 400;\">Q&#8217;Q<\/span><span style=\"font-weight: 400;\"> \u00e8 l\u2019ipotenusa, <\/span><span style=\"font-weight: 400;\">QP<\/span><span style=\"font-weight: 400;\"> e <\/span><span style=\"font-weight: 400;\">PQ&#8217;<\/span><span style=\"font-weight: 400;\"> ne sono i cateti.<\/span><\/p>\n<p>&nbsp;<\/p>\n<p><span style=\"font-weight: 400;\">Detta R la proiezione del punto P \u00a0sull&#8217;ipotenusa, osserviamo che si ha:<\/span><\/p>\n<p>&nbsp;<\/p>\n<ul>\n<li style=\"list-style-type: none;\">\n<ul>\n<li style=\"list-style-type: none;\">\n<ul>\n<li style=\"font-weight: 400;\"><span style=\"font-weight: 400;\">OR<\/span><span style=\"font-weight: 400;\"> = <\/span><span style=\"font-weight: 400;\">cos (<\/span><span style=\"font-weight: 400;\">&#8211;<\/span><span style=\"font-weight: 400;\">)<\/span><\/li>\n<li style=\"font-weight: 400;\"><span style=\"font-weight: 400;\">MP<\/span><span style=\"font-weight: 400;\"> \u00a0= <\/span><span style=\"font-weight: 400;\">sen (<\/span><span style=\"font-weight: 400;\">&#8211;<\/span><span style=\"font-weight: 400;\">)<\/span><\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p>&nbsp;<\/p>\n<p><span style=\"font-weight: 400;\">Adesso si calcola la distanza fra i punti PQ, ossia:<\/span><\/p>\n<p>&nbsp;<\/p>\n<p><span style=\"font-weight: 400;\">PQ <\/span><span style=\"font-weight: 400;\">2<\/span><span style=\"font-weight: 400;\">=<\/span><span style=\"font-weight: 400;\">(cos <\/span><span style=\"font-weight: 400;\"> -cos <\/span><span style=\"font-weight: 400;\">)<\/span><span style=\"font-weight: 400;\">2<\/span><span style=\"font-weight: 400;\">+<\/span><span style=\"font-weight: 400;\">(sen <\/span><span style=\"font-weight: 400;\"> &#8211; sen<\/span><span style=\"font-weight: 400;\">)<\/span><span style=\"font-weight: 400;\">2<\/span><span style=\"font-weight: 400;\">=<\/span><\/p>\n<p><span style=\"font-weight: 400;\">= <\/span><span style=\"font-weight: 400;\">cos<\/span><span style=\"font-weight: 400;\">2<\/span><span style=\"font-weight: 400;\"> + <\/span><span style=\"font-weight: 400;\">cos<\/span><span style=\"font-weight: 400;\">2<\/span><span style=\"font-weight: 400;\"> -2<\/span><span style=\"font-weight: 400;\">cos <\/span><span style=\"font-weight: 400;\">cos <\/span><span style=\"font-weight: 400;\"> + <\/span><span style=\"font-weight: 400;\">sen<\/span><span style=\"font-weight: 400;\">2<\/span><span style=\"font-weight: 400;\"> + <\/span><span style=\"font-weight: 400;\">sen<\/span><span style=\"font-weight: 400;\">2<\/span> <span style=\"font-weight: 400;\"> &#8211; 2<\/span><span style=\"font-weight: 400;\">sen <\/span><span style=\"font-weight: 400;\">sen <\/span><span style=\"font-weight: 400;\"> =<\/span><\/p>\n<p><span style=\"font-weight: 400;\">Quindi = (<\/span><span style=\"font-weight: 400;\">cos<\/span><span style=\"font-weight: 400;\">2<\/span> <span style=\"font-weight: 400;\">+<\/span><span style=\"font-weight: 400;\">sen<\/span><span style=\"font-weight: 400;\">2<\/span> <span style=\"font-weight: 400;\">)+(<\/span><span style=\"font-weight: 400;\">cos<\/span><span style=\"font-weight: 400;\">2 <\/span><span style=\"font-weight: 400;\"> +<\/span><span style=\"font-weight: 400;\">sen<\/span><span style=\"font-weight: 400;\">2<\/span> <span style=\"font-weight: 400;\"> ) -2<\/span><span style=\"font-weight: 400;\">( cos <\/span><span style=\"font-weight: 400;\">cos <\/span><span style=\"font-weight: 400;\">+sen <\/span><span style=\"font-weight: 400;\">sen <\/span><span style=\"font-weight: 400;\"> )= <\/span><\/p>\n<p><span style=\"font-weight: 400;\">= 1 +1 &#8211; 2<\/span><span style=\"font-weight: 400;\">( cos <\/span><span style=\"font-weight: 400;\">cos <\/span><span style=\"font-weight: 400;\">+sen <\/span><span style=\"font-weight: 400;\">sen <\/span><span style=\"font-weight: 400;\"> )=<\/span><\/p>\n<p><span style=\"font-weight: 400;\">2 &#8211; 2<\/span><span style=\"font-weight: 400;\">( cos <\/span><span style=\"font-weight: 400;\">cos <\/span><span style=\"font-weight: 400;\">+sen <\/span><span style=\"font-weight: 400;\">sen <\/span><span style=\"font-weight: 400;\"> ).<\/span><\/p>\n<p>&nbsp;<\/p>\n<p><span style=\"font-weight: 400;\">Ricordando che <\/span><span style=\"font-weight: 400;\">Q&#8217;Q<\/span><span style=\"font-weight: 400;\">=2<\/span><span style=\"font-weight: 400;\"> ,in quanto diametro della circonferenza trigonometrica di raggio unitario, e <\/span><span style=\"font-weight: 400;\">RQ <\/span><span style=\"font-weight: 400;\">= <\/span><span style=\"font-weight: 400;\">OQ<\/span><span style=\"font-weight: 400;\">&#8211;<\/span><span style=\"font-weight: 400;\">OR<\/span><span style=\"font-weight: 400;\">=1-cos (<\/span><span style=\"font-weight: 400;\">&#8211;<\/span><span style=\"font-weight: 400;\">)<\/span><span style=\"font-weight: 400;\">.<\/span><\/p>\n<p>&nbsp;<\/p>\n<p><span style=\"font-weight: 400;\">Adesso, semplicemente, si calcola la lunghezza \u00a0<\/span><span style=\"font-weight: 400;\">PQ <\/span><span style=\"font-weight: 400;\">2<\/span><span style=\"font-weight: 400;\"> utilizzando il primo teorema di <\/span><b>Euclide <\/b><span style=\"font-weight: 400;\">\u00a0e si eseguono le relative sostituzioni, ossia:<\/span><\/p>\n<p>&nbsp;<\/p>\n<p><span style=\"font-weight: 400;\">PQ <\/span><span style=\"font-weight: 400;\">2<\/span><span style=\"font-weight: 400;\">= <\/span><span style=\"font-weight: 400;\">QQ&#8217;<\/span><span style=\"font-weight: 400;\">RQ<\/span><span style=\"font-weight: 400;\"> \u00a0\u00a0\u00a0<\/span><\/p>\n<p>&nbsp;<\/p>\n<p><span style=\"font-weight: 400;\">2 &#8211; 2<\/span><span style=\"font-weight: 400;\">( cos <\/span><span style=\"font-weight: 400;\">cos <\/span><span style=\"font-weight: 400;\">+sen <\/span><span style=\"font-weight: 400;\">sen <\/span><span style=\"font-weight: 400;\"> )=2<\/span><span style=\"font-weight: 400;\">(1-cos (<\/span><span style=\"font-weight: 400;\">&#8211;<\/span><span style=\"font-weight: 400;\">))<\/span><\/p>\n<p>&nbsp;<\/p>\n<p><span style=\"font-weight: 400;\"> 2 &#8211; 2<\/span><span style=\"font-weight: 400;\">( cos <\/span><span style=\"font-weight: 400;\">cos <\/span><span style=\"font-weight: 400;\">+sen <\/span><span style=\"font-weight: 400;\">sen <\/span><span style=\"font-weight: 400;\"> )=2-2<\/span><span style=\"font-weight: 400;\">cos (<\/span><span style=\"font-weight: 400;\">&#8211;<\/span><span style=\"font-weight: 400;\">)<\/span><\/p>\n<p>&nbsp;<\/p>\n<p>(cos cos +sen sen )=cos (-)<\/p>\n<p>&nbsp;<\/p>\n<p><img decoding=\"async\" class=\"size-full wp-image-6161 aligncenter\" src=\"https:\/\/www.ripetizioni.it\/blog\/wp-content\/uploads\/2018\/09\/prof-ricevimento.jpg\" alt=\"matematica\" width=\"1200\" height=\"800\" \/><\/p>\n<p>&nbsp;<\/p>\n<h2><span style=\"color: #007dc3;\"><strong>Formule di duplicazione<\/strong><\/span><\/h2>\n<p>&nbsp;<\/p>\n<p><span style=\"font-weight: 400;\">Come accennato nel primo paragrafo, le formule di duplicazione si ricavano semplicemente utilizzando le identit\u00e0 di addizione \u00a0ponendo <\/span><span style=\"font-weight: 400;\">=<\/span><span style=\"font-weight: 400;\">.<\/span><span style=\"font-weight: 400;\"> Vediamo adesso come si ricava la formula di duplicazione del seno e del coseno.<\/span><\/p>\n<p>&nbsp;<\/p>\n<h3><span style=\"color: #007dc3;\"><b>Seno<\/b><\/span><\/h3>\n<p>&nbsp;<\/p>\n<p><span style=\"font-weight: 400;\">Ricordando che:<\/span><\/p>\n<p>&nbsp;<\/p>\n<p><span style=\"font-weight: 400;\">sen (<\/span><span style=\"font-weight: 400;\">+<\/span><span style=\"font-weight: 400;\">) \u00a0=sen <\/span><span style=\"font-weight: 400;\">cos <\/span><span style=\"font-weight: 400;\"> + cos <\/span><span style=\"font-weight: 400;\">sen <\/span><\/p>\n<p><span style=\"font-weight: 400;\">[ponendo <\/span><span style=\"font-weight: 400;\">= <\/span><span style=\"font-weight: 400;\">] <\/span><\/p>\n<p><span style=\"font-weight: 400;\">sen (<\/span><span style=\"font-weight: 400;\">+<\/span><span style=\"font-weight: 400;\">) = sen <\/span><span style=\"font-weight: 400;\">cos <\/span><span style=\"font-weight: 400;\"> + cos <\/span><span style=\"font-weight: 400;\">sen <\/span><\/p>\n<p><span style=\"font-weight: 400;\">Quindi: sen (2<\/span><span style=\"font-weight: 400;\">) =2<\/span><span style=\"font-weight: 400;\"> sen <\/span><span style=\"font-weight: 400;\">cos <\/span><\/p>\n<p>&nbsp;<\/p>\n<h3><span style=\"color: #007dc3;\"><b>Coseno<\/b><\/span><\/h3>\n<p>&nbsp;<\/p>\n<p><span style=\"font-weight: 400;\">Ricordando che<\/span><\/p>\n<p>&nbsp;<\/p>\n<p><span style=\"font-weight: 400;\">cos (<\/span><span style=\"font-weight: 400;\">+<\/span><span style=\"font-weight: 400;\">) =cos <\/span><span style=\"font-weight: 400;\">cos <\/span><span style=\"font-weight: 400;\"> &#8211; sen <\/span><span style=\"font-weight: 400;\">sen <\/span><\/p>\n<p><span style=\"font-weight: 400;\">[ponendo <\/span><span style=\"font-weight: 400;\">= <\/span><span style=\"font-weight: 400;\">] <\/span><\/p>\n<p><span style=\"font-weight: 400;\">cos (<\/span><span style=\"font-weight: 400;\">+<\/span><span style=\"font-weight: 400;\">) <\/span><span style=\"font-weight: 400;\">=cos <\/span><span style=\"font-weight: 400;\">cos <\/span><span style=\"font-weight: 400;\"> &#8211; sen <\/span><span style=\"font-weight: 400;\">sen <\/span><span style=\"font-weight: 400;\"> \u00a0<\/span><\/p>\n<p><span style=\"font-weight: 400;\">Quindi: cos (2<\/span><span style=\"font-weight: 400;\">) =<\/span><span style=\"font-weight: 400;\">cos<\/span><span style=\"font-weight: 400;\">2<\/span> <span style=\"font-weight: 400;\"> &#8211; <\/span><span style=\"font-weight: 400;\">sen<\/span><span style=\"font-weight: 400;\">2 <\/span><span style=\"font-weight: 400;\"> \u00a0<\/span><\/p>\n<p>&nbsp;<\/p>\n<p><span style=\"font-weight: 400;\">Seguendo questo procedimento si ricavano facilmente tutte le altre identit\u00e0.<\/span><\/p>\n<p>&nbsp;<\/p>\n<h2><span style=\"color: #007dc3;\"><b>Formule di bisezione<\/b><\/span><\/h2>\n<p>&nbsp;<\/p>\n<p><span style=\"font-weight: 400;\">Per ricavare in modo semplice queste identit\u00e0, esprimiamo le formule di duplicazione del seno e del coseno nel solo coseno, ossia:<\/span><\/p>\n<p>&nbsp;<\/p>\n<p><span style=\"font-weight: 400;\">cos (2<\/span><span style=\"font-weight: 400;\">) =<\/span><span style=\"font-weight: 400;\">cos<\/span><span style=\"font-weight: 400;\">2<\/span> <span style=\"font-weight: 400;\"> &#8211; <\/span><span style=\"font-weight: 400;\">sen<\/span><span style=\"font-weight: 400;\">2 <\/span><span style=\"font-weight: 400;\"> \u00a0<\/span><\/p>\n<p><span style=\"font-weight: 400;\">Quindi: cos (2<\/span><span style=\"font-weight: 400;\">) =(1-<\/span><span style=\"font-weight: 400;\">sen<\/span><span style=\"font-weight: 400;\">2<\/span><span style=\"font-weight: 400;\">) &#8211; <\/span><span style=\"font-weight: 400;\">sen<\/span><span style=\"font-weight: 400;\">2 <\/span><span style=\"font-weight: 400;\"> \u00a0<\/span><\/p>\n<p><span style=\"font-weight: 400;\">cos (2<\/span><span style=\"font-weight: 400;\">) =1-2<\/span><span style=\"font-weight: 400;\">sen<\/span><span style=\"font-weight: 400;\">2 <\/span><span style=\"font-weight: 400;\"> \u00a0<\/span><\/p>\n<p>&nbsp;<\/p>\n<p><span style=\"font-weight: 400;\">ricordando che <\/span> <span style=\"font-weight: 400;\">sen<\/span><span style=\"font-weight: 400;\">2 <\/span><span style=\"font-weight: 400;\"> = 1 &#8211; <\/span><span style=\"font-weight: 400;\">cos<\/span><span style=\"font-weight: 400;\">2<\/span><span style=\"font-weight: 400;\"> \u00a0<\/span><\/p>\n<p>&nbsp;<\/p>\n<p><span style=\"font-weight: 400;\">si ha:<\/span><\/p>\n<p>&nbsp;<\/p>\n<p><span style=\"font-weight: 400;\">cos (2<\/span><span style=\"font-weight: 400;\">) =1-2<\/span><span style=\"font-weight: 400;\">(1 &#8211; <\/span><span style=\"font-weight: 400;\">cos<\/span><span style=\"font-weight: 400;\">2<\/span><span style=\"font-weight: 400;\">) <\/span><\/p>\n<p><span style=\"font-weight: 400;\">Quindi: cos (2<\/span><span style=\"font-weight: 400;\">) =1-2 +2<\/span> <span style=\"font-weight: 400;\">cos<\/span><span style=\"font-weight: 400;\">2<\/span><\/p>\n<p><span style=\"font-weight: 400;\">cos (2<\/span><span style=\"font-weight: 400;\">) =2<\/span> <span style=\"font-weight: 400;\">cos<\/span><span style=\"font-weight: 400;\">2<\/span><span style=\"font-weight: 400;\"> &#8211; 1 <\/span><\/p>\n<p>&nbsp;<\/p>\n<p><span style=\"font-weight: 400;\">da questa si ottiene:<\/span><\/p>\n<p>&nbsp;<\/p>\n<p><span style=\"font-weight: 400;\">2<\/span> <span style=\"font-weight: 400;\">cos<\/span><span style=\"font-weight: 400;\">2<\/span><span style=\"font-weight: 400;\"> = 1 + cos 2<\/span><\/p>\n<p>&nbsp;<\/p>\n<p><span style=\"font-weight: 400;\">Con analogo ragionamento si ricava facilmente anche la seguente identit\u00e0, la quale sar\u00e0 utile per ricavare le formule di bisezione, ossia: <\/span><\/p>\n<p>&nbsp;<\/p>\n<p><span style=\"font-weight: 400;\">2<\/span> <span style=\"font-weight: 400;\">sen<\/span><span style=\"font-weight: 400;\">2<\/span><span style=\"font-weight: 400;\"> = 1 &#8211; cos 2<\/span><\/p>\n<p>&nbsp;<\/p>\n<p><span style=\"font-weight: 400;\">Adesso, \u00a0ponendo semplicemente <\/span><span style=\"font-weight: 400;\">=<\/span><span style=\"font-weight: 400;\">\/2<\/span><span style=\"font-weight: 400;\">, si ottiene:<\/span><\/p>\n<p>&nbsp;<\/p>\n<p><span style=\"font-weight: 400;\">2<\/span> <span style=\"font-weight: 400;\">sen<\/span><span style=\"font-weight: 400;\">2<\/span><span style=\"font-weight: 400;\">\/2 = 1 &#8211; cos 2<\/span><span style=\"font-weight: 400;\">\/2 <\/span><\/p>\n<p><span style=\"font-weight: 400;\">2<\/span> <span style=\"font-weight: 400;\">sen<\/span><span style=\"font-weight: 400;\">2<\/span><span style=\"font-weight: 400;\">\/2 = 1 &#8211; cos <\/span><\/p>\n<p><span style=\"font-weight: 400;\">sen<\/span><span style=\"font-weight: 400;\">2<\/span><span style=\"font-weight: 400;\">\/2 =1\/2<\/span><span style=\"font-weight: 400;\"> (1 &#8211; cos <\/span><span style=\"font-weight: 400;\">) <\/span><\/p>\n<p><span style=\"font-weight: 400;\">Quindi: sen <\/span><span style=\"font-weight: 400;\">\/2 = <\/span> <span style=\"font-weight: 400;\"> (1 &#8211; cos <\/span><span style=\"font-weight: 400;\">)<\/span><span style=\"font-weight: 400;\">2<\/span><\/p>\n<p>&nbsp;<\/p>\n<p><span style=\"font-weight: 400;\">Con analogo procedimento si ottengono le altre identit\u00e0.<\/span><\/p>\n<p>&nbsp;<\/p>\n<p><img decoding=\"async\" class=\"size-full wp-image-6079\" src=\"https:\/\/www.ripetizioni.it\/blog\/wp-content\/uploads\/2018\/09\/index-1-1.jpg\" alt=\"matematica\" width=\"1600\" height=\"1067\" \/><\/p>\n<p>&nbsp;<\/p>\n<h2><span style=\"color: #007dc3;\"><b>Formule di prostaferesi<\/b><\/span><\/h2>\n<p>&nbsp;<\/p>\n<p><span style=\"font-weight: 400;\">Di seguito si riportano le formule di prostaferesi, le quali si ricavano sommando e sottraendo membro a membro le formule di addizione e sottrazione. <\/span><\/p>\n<p>&nbsp;<\/p>\n<p><span style=\"font-weight: 400;\">sen(<\/span><span style=\"font-weight: 400;\">+<\/span><span style=\"font-weight: 400;\">) +sen(<\/span><span style=\"font-weight: 400;\">&#8211;<\/span><span style=\"font-weight: 400;\">) = 2<\/span><span style=\"font-weight: 400;\">sen <\/span><span style=\"font-weight: 400;\">cos <\/span><span style=\"font-weight: 400;\"> \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0(*)<\/span><\/p>\n<p><span style=\"font-weight: 400;\">sen(<\/span><span style=\"font-weight: 400;\">+<\/span><span style=\"font-weight: 400;\">) -sen(<\/span><span style=\"font-weight: 400;\">&#8211;<\/span><span style=\"font-weight: 400;\">) = 2<\/span><span style=\"font-weight: 400;\">sen <\/span><span style=\"font-weight: 400;\">cos <\/span><span style=\"font-weight: 400;\"> \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0(**)<\/span><\/p>\n<p><span style=\"font-weight: 400;\">cos(<\/span><span style=\"font-weight: 400;\">+<\/span><span style=\"font-weight: 400;\">) +cos(<\/span><span style=\"font-weight: 400;\">&#8211;<\/span><span style=\"font-weight: 400;\">) = 2<\/span><span style=\"font-weight: 400;\">cos <\/span><span style=\"font-weight: 400;\">cos <\/span><span style=\"font-weight: 400;\"> \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0(***)<\/span><\/p>\n<p><span style=\"font-weight: 400;\">E cos(<\/span><span style=\"font-weight: 400;\">+<\/span><span style=\"font-weight: 400;\">) -cos(<\/span><span style=\"font-weight: 400;\">&#8211;<\/span><span style=\"font-weight: 400;\">) = -2<\/span><span style=\"font-weight: 400;\">sen <\/span><span style=\"font-weight: 400;\">cos <\/span><span style=\"font-weight: 400;\"> \u00a0\u00a0(****).<\/span><\/p>\n<p>&nbsp;<\/p>\n<p><span style=\"font-weight: 400;\">Ponendo poi <\/span><span style=\"font-weight: 400;\">+<\/span><span style=\"font-weight: 400;\"> = p <\/span><span style=\"font-weight: 400;\"> \u00a0e <\/span><span style=\"font-weight: 400;\">&#8211;<\/span><span style=\"font-weight: 400;\"> = q<\/span><span style=\"font-weight: 400;\"> e facendo la semisomma e semidifferenza membro a membro, si ha: <\/span><\/p>\n<p><span style=\"font-weight: 400;\"> = (p+q)\/2 \u00a0\u00a0\u00a0e <\/span><span style=\"font-weight: 400;\"> = (p-q)\/2.<\/span><\/p>\n<p>&nbsp;<\/p>\n<p><span style=\"font-weight: 400;\">Adesso sostituendo nelle formule precedenti si ottiene:<\/span><\/p>\n<p>&nbsp;<\/p>\n<p><span style=\"font-weight: 400;\">sen((p+q)\/2+(p-q)\/2) +sen((p+q)\/2-(p-q)\/2) = 2<\/span><span style=\"font-weight: 400;\">sen \u00a0(p+q)\/2<\/span><span style=\"font-weight: 400;\">cos (p-q)\/2<\/span><\/p>\n<p><span style=\"font-weight: 400;\">sen p +sen q= 2<\/span><span style=\"font-weight: 400;\">sen \u00a0(p+q)\/2<\/span><span style=\"font-weight: 400;\">cos (p-q)\/2.<\/span><\/p>\n<p>&nbsp;<\/p>\n<p><span style=\"font-weight: 400;\">Analogamente si ricavano tutte le altre. Ponendo adesso <\/span><span style=\"font-weight: 400;\">p=<\/span><span style=\"font-weight: 400;\"> \u00a0e q = <\/span><span style=\"font-weight: 400;\"> si ottengono le formule di prostaferesi.<\/span><\/p>\n<p>&nbsp;<\/p>\n<h2><span style=\"color: #007dc3;\"><b>Formule di Werner<\/b><\/span><\/h2>\n<p>&nbsp;<\/p>\n<p><span style=\"font-weight: 400;\">Vogliamo ancora dare un gruppo di formule, dette di <\/span><b>Werner<\/b><span style=\"font-weight: 400;\">, le<\/span><b> \u00a0<\/b><span style=\"font-weight: 400;\">quali, come promesso, si ricavano dalle formule (*),(**),(***) e (****) dividendo ambo i membri per 2. \u00a0Utilizzando la (***) ricaviamo la prima delle seguenti identit\u00e0:<\/span><\/p>\n<p>&nbsp;<\/p>\n<p><span style=\"font-weight: 400;\">Ricordando la (***)<\/span><\/p>\n<p>&nbsp;<\/p>\n<p><span style=\"font-weight: 400;\">2<\/span><span style=\"font-weight: 400;\">cos <\/span><span style=\"font-weight: 400;\">cos <\/span><span style=\"font-weight: 400;\"> = cos(<\/span><span style=\"font-weight: 400;\">+<\/span><span style=\"font-weight: 400;\">) +cos(<\/span><span style=\"font-weight: 400;\">&#8211;<\/span><span style=\"font-weight: 400;\">) <\/span><span style=\"font-weight: 400;\"> \u00a0<\/span><\/p>\n<p><span style=\"font-weight: 400;\">Dividendo per 2 ambo i membri, si ottiene la prima delle formule di <\/span><b>Werner<\/b><span style=\"font-weight: 400;\">, ossia:<\/span><\/p>\n<p>&nbsp;<\/p>\n<p><span style=\"font-weight: 400;\">cos <\/span><span style=\"font-weight: 400;\">cos <\/span><span style=\"font-weight: 400;\"> =1\/2<\/span><span style=\"font-weight: 400;\"> [cos(<\/span><span style=\"font-weight: 400;\">+<\/span><span style=\"font-weight: 400;\">) +cos(<\/span><span style=\"font-weight: 400;\">&#8211;<\/span><span style=\"font-weight: 400;\">)] <\/span><span style=\"font-weight: 400;\"> \u00a0<\/span><\/p>\n<p>&nbsp;<\/p>\n<p><span style=\"font-weight: 400;\">Analogamente si procedere per ricavare le restanti identit\u00e0.<\/span><\/p>\n<p>&nbsp;<\/p>\n<p><span style=\"font-weight: 400;\">Ing. Antonio Pugliese.<\/span><\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: center;\"><a href=\"https:\/\/www.ripetizioni.it\/lezioni-private\/matematica?utm_source=blog-ripetizioni&amp;utm_medium=refarral&amp;utm_campaign=formule-trigonometriche-trucchi-per-impararle\"><b><i>Sei interessato ad approfondire questi argomenti? Prendi ripetizioni di trigonometria tramite il portale di Skuola.net | Ripetizioni e inizia subito a migliorare i tuoi voti!<\/i><\/b><\/a><\/p>\n<p>&nbsp;<\/p>\n<p><a href=\"https:\/\/www.ripetizioni.it\/lezioni-private\/matematica?utm_source=blog-ripetizioni&amp;utm_medium=refarral&amp;utm_campaign=formule-trigonometriche-trucchi-per-impararle\"><img decoding=\"async\" class=\"size-full wp-image-6011 aligncenter\" src=\"https:\/\/www.ripetizioni.it\/blog\/wp-content\/uploads\/2018\/09\/ripetizionni.png\" alt=\"Ripetizioni Skuola.net\" width=\"851\" height=\"315\" \/><\/a><br \/>\n<script src=\"https:\/\/skuolanet.activehosted.com\/f\/embed.php?id=35\" type=\"text\/javascript\" charset=\"utf-8\"><\/script><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Se stai leggendo questo articolo sicuramente sarai alle prese con gli odiosissimi esercizi di trigonometria. Bene, allora posso consolarti nel dirti che tutti i tuoi dubbi a breve cesseranno di esistere. E\u2019 noto che la risoluzione degli esercizi di trigonometria, nella maggior parte dei casi, si risolvono riducendo le equazioni utilizzando una serie di formule&nbsp; <a href=\"https:\/\/www.ripetizioni.it\/blog\/formule-trigonometriche-trucchi-per-impararle\/\">&hellip; Read more <i class=\"glyphicon glyphicon-arrow-right\"><\/i><\/a><\/p>\n","protected":false},"author":15,"featured_media":5526,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"wl_entities_gutenberg":"","footnotes":""},"categories":[70],"tags":[],"wl_entity_type":[321],"class_list":["post-7546","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-matematica","wl_entity_type-article"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.3 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Formule trigonometriche: trucchi per impararle | Il blog delle Ripetizioni e lezioni private: consigli utili<\/title>\n<meta name=\"description\" content=\"Hai bisogno di una mano nello studio della matematica? 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