{"id":4478,"date":"2014-12-30T18:09:08","date_gmt":"2014-12-30T16:09:08","guid":{"rendered":"http:\/\/pimedios.es\/?p=4478"},"modified":"2014-12-30T18:09:08","modified_gmt":"2014-12-30T16:09:08","slug":"viete-y-la-suma-de-senos","status":"publish","type":"post","link":"https:\/\/pimedios.jesussoto.es\/?p=4478","title":{"rendered":"Vi\u00e8te y la suma de senos"},"content":{"rendered":"<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-4479\" src=\"http:\/\/pimedios.es\/wp-content\/uploads\/2014\/12\/vieta0.png\" alt=\"vieta0\" width=\"197\" height=\"188\" \/>El pasado d\u00eda vimos como Fran\u00e7ois Vi\u00e8te realiz\u00f3 la demostraci\u00f3n del teorema de Pit\u00e1goras utilizando la circunferencia. Hoy vamos a traer otra de sus famosas f\u00f3rmulas, la suma de dos senos:<\/p>\n<p>$$\\sin x +\\sin y=2 \\sin \\frac{x+y}{2}\\cos\\frac{x-y}{2}.$$<\/p>\n<p>Lo podr\u00edamos resolver de forma sencilla, con los conocimientos que poseemos; pero volvamos al siglo XVI para ver c\u00f3mo lo consigui\u00f3 Vi\u00e8te. Una vez m\u00e1s la clave nos la proporciona la imagen que tenemos.<\/p>\n<p>Si recordamos que la longitud de un arco de circunferencia es proporcional al \u00e1ngulo y el radio, para nuestra circunferencia de la imagen, de radio unidad, $x$ representa tanto la longitud de arco como el \u00e1ngulo de esta imagen<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-4480\" src=\"http:\/\/pimedios.es\/wp-content\/uploads\/2014\/12\/vieta1.png\" alt=\"vieta1\" width=\"197\" height=\"191\" \/><\/p>\n<p>Del mismo modo $y$ nos representa el \u00e1ngulo del arco<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-4481\" src=\"http:\/\/pimedios.es\/wp-content\/uploads\/2014\/12\/vieta2.png\" alt=\"vieta2\" width=\"197\" height=\"191\" \/><\/p>\n<p>Lo que Vi\u00e8te pretend\u00eda hacer era calcular $$\\sin x +\\sin y.$$ Pero estos senos se corresponde con los segmentos dados por la siguiente imagen<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-4482\" src=\"http:\/\/pimedios.es\/wp-content\/uploads\/2014\/12\/vieta3.png\" alt=\"vieta3\" width=\"197\" height=\"190\" \/><\/p>\n<p>Es decir, $$\\sin x +\\sin y=\\overline{AB}+\\overline{CD},$$ o lo que es lo mismo<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-4483\" src=\"http:\/\/pimedios.es\/wp-content\/uploads\/2014\/12\/vieta4.png\" alt=\"vieta4\" width=\"197\" height=\"189\" \/>$$\\sin x +\\sin y=\\overline{AB}+\\overline{CD}=\\overline{AE}.$$<\/p>\n<p>Ahora, si consideramos el tri\u00e1ngulo\u00a0 $\\widehat{EAC}$<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-4485\" src=\"http:\/\/pimedios.es\/wp-content\/uploads\/2014\/12\/vieta5.png\" alt=\"vieta5\" width=\"197\" height=\"190\" \/><\/p>\n<p>resulta $\\overline{AE}=\\overline{AC}\\cos \\angle EAC$. Por tanto<\/p>\n<p>$$\\sin x +\\sin y=\\overline{AC}\\cos \\angle EAC.$$<\/p>\n<p>Por \u00faltimo tenemos el tri\u00e1ngulo $\\widehat{AOC}$, que es is\u00f3sceles y su \u00e1ngulo\u00a0 $\\angle AOC \\equiv x+y$.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-4486\" src=\"http:\/\/pimedios.es\/wp-content\/uploads\/2014\/12\/vieta6.png\" alt=\"vieta6\" width=\"197\" height=\"190\" \/><\/p>\n<p>Como la bisectriz de $\\angle AOC$ divide el segmento $\\overline{AC}$ en dos partes iguales, resultar\u00e1 que $$\\sin \\frac{\\angle AOC}{2}=\\frac{1}{2}\\overline{AC},$$ y, por consiguiente $$\\sin x +\\sin y=2\\sin \\frac{x+y}{2}\\cos \\angle EAC.$$<\/p>\n<p>S\u00f3lo nos resta deducir que $\\cos \\angle EAC=\\cos\\frac{x-y}{2}$, que lo dejaremos como ejercicio para el lector.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>La construcci\u00f3n de la f\u00f3rmula de la suma de senos por Fran\u00e7ois Vi\u00e8te <\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[6],"tags":[145,584],"class_list":["post-4478","post","type-post","status-publish","format-standard","hentry","category-historia","tag-francois-viete","tag-seno","entry"],"_links":{"self":[{"href":"https:\/\/pimedios.jesussoto.es\/index.php?rest_route=\/wp\/v2\/posts\/4478","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pimedios.jesussoto.es\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/pimedios.jesussoto.es\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/pimedios.jesussoto.es\/index.php?rest_route=\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/pimedios.jesussoto.es\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=4478"}],"version-history":[{"count":4,"href":"https:\/\/pimedios.jesussoto.es\/index.php?rest_route=\/wp\/v2\/posts\/4478\/revisions"}],"predecessor-version":[{"id":4489,"href":"https:\/\/pimedios.jesussoto.es\/index.php?rest_route=\/wp\/v2\/posts\/4478\/revisions\/4489"}],"wp:attachment":[{"href":"https:\/\/pimedios.jesussoto.es\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=4478"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/pimedios.jesussoto.es\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=4478"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/pimedios.jesussoto.es\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=4478"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}