{"id":8406,"date":"2012-04-16T19:39:09","date_gmt":"2012-04-16T18:39:09","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=8406"},"modified":"2022-01-14T01:01:16","modified_gmt":"2022-01-14T01:01:16","slug":"a-partir-das-formulas","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=8406","title":{"rendered":"A partir das f\u00f3rmulas"},"content":{"rendered":"<p><ul id='GTTabs_ul_8406' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_8406' class='GTTabs_curr'><a  id=\"8406_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_8406' ><a  id=\"8406_1\" onMouseOver=\"GTTabsShowLinks('Resolu\u00e7\u00e3o'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Resolu\u00e7\u00e3o<\/a><\/li>\n<\/ul>\n\n<div class='GTTabs_divs GTTabs_curr_div' id='GTTabs_0_8406'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p>A partir das f\u00f3rmulas correspondentes do seno e do cosseno, deduza uma f\u00f3rmula para<\/p>\n<ol>\n<li>$\\operatorname{tg} \\left( {\\alpha\u00a0 + \\beta } \\right)$<\/li>\n<li>$\\operatorname{tg} \\left( {\\alpha\u00a0 &#8211; \\beta } \\right)$<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_8406' onClick='GTTabs_show(1,8406)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_8406'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<blockquote>\n<p>$$\\operatorname{sen} \\left( {\\alpha\u00a0 + \\beta } \\right) = \\operatorname{sen} \\alpha \\cos \\beta\u00a0 + \\cos \\alpha \\operatorname{sen} \\beta $$<\/p>\n<\/blockquote>\n<blockquote>\n<p>$$\\cos \\left( {\\alpha\u00a0 + \\beta } \\right) = \\cos \\alpha \\cos \\beta\u00a0 &#8211; \\operatorname{sen} \\alpha \\operatorname{sen} \\beta $$<\/p>\n<\/blockquote>\n<ol>\n<li>Ora,\u00a0$$\\begin{array}{*{20}{l}}<br \/>\n{\\operatorname{tg} \\left( {\\alpha\u00a0 + \\beta } \\right)}&amp; = &amp;{\\frac{{\\operatorname{sen} \\left( {\\alpha\u00a0 + \\beta } \\right)}}{{\\cos \\left( {\\alpha\u00a0 + \\beta } \\right)}}} \\\\<br \/>\n{}&amp; = &amp;{\\frac{{\\operatorname{sen} \\alpha \\cos \\beta\u00a0 + \\cos \\alpha \\operatorname{sen} \\beta }}{{\\cos \\alpha \\cos \\beta\u00a0 &#8211; \\operatorname{sen} \\alpha \\operatorname{sen} \\beta }}} \\\\<br \/>\n{}&amp; = &amp;{\\frac{{\\frac{{\\operatorname{sen} \\alpha \\cos \\beta }}{{\\cos \\alpha \\cos \\beta }} + \\frac{{\\cos \\alpha \\operatorname{sen} \\beta }}{{\\cos \\alpha \\cos \\beta }}}}{{\\frac{{\\cos \\alpha \\cos \\beta }}{{\\cos \\alpha \\cos \\beta }} &#8211; \\frac{{\\operatorname{sen} \\alpha \\operatorname{sen} \\beta }}{{\\cos \\alpha \\cos \\beta }}}}} \\\\<br \/>\n{}&amp; = &amp;{\\frac{{\\operatorname{tg} \\alpha\u00a0 + \\operatorname{tg} \\beta }}{{1 &#8211; \\operatorname{tg} \\alpha \\operatorname{tg} \\beta }}}<br \/>\n\\end{array}$$<br \/>\nLogo, $$\\operatorname{tg} \\left( {\\alpha\u00a0 + \\beta } \\right) = \\frac{{\\operatorname{tg} \\alpha\u00a0 + \\operatorname{tg} \\beta }}{{1 &#8211; \\operatorname{tg} \\alpha \\operatorname{tg} \\beta }}$$<br \/>\n\u00ad<\/li>\n<li>Tendo em considera\u00e7\u00e3o a rela\u00e7\u00e3o obtida em 1, temos:<br \/>\n$$\\begin{array}{*{20}{l}}<br \/>\n{\\operatorname{tg} \\left( {\\alpha\u00a0 &#8211; \\beta } \\right)}&amp; = &amp;{\\operatorname{tg} \\left( {\\alpha\u00a0 + ( &#8211; \\beta )} \\right)} \\\\<br \/>\n{}&amp; = &amp;{\\frac{{\\operatorname{tg} \\alpha\u00a0 + \\operatorname{tg} \\left( { &#8211; \\beta } \\right)}}{{1 &#8211; \\operatorname{tg} \\alpha \\operatorname{tg} \\left( { &#8211; \\beta } \\right)}}} \\\\<br \/>\n{}&amp; = &amp;{\\frac{{\\operatorname{tg} \\alpha\u00a0 &#8211; \\operatorname{tg} \\beta }}{{1 + \\operatorname{tg} \\alpha \\operatorname{tg} \\beta }}}<br \/>\n\\end{array}$$<br \/>\nLogo, $$\\operatorname{tg} \\left( {\\alpha\u00a0 &#8211; \\beta } \\right) = \\frac{{\\operatorname{tg} \\alpha\u00a0 &#8211; \\operatorname{tg} \\beta }}{{1 + \\operatorname{tg} \\alpha \\operatorname{tg} \\beta }}$$<br \/>\n\u00ad<\/li>\n<\/ol>\n<blockquote>\n<p>$$\\operatorname{tg} \\left( {\\alpha\u00a0 + \\beta } \\right) = \\frac{{\\operatorname{tg} \\alpha\u00a0 + \\operatorname{tg} \\beta }}{{1 &#8211; \\operatorname{tg} \\alpha \\operatorname{tg} \\beta }}$$<\/p>\n<\/blockquote>\n<blockquote>\n<p>$$\\operatorname{tg} \\left( {\\alpha\u00a0 &#8211; \\beta } \\right) = \\frac{{\\operatorname{tg} \\alpha\u00a0 &#8211; \\operatorname{tg} \\beta }}{{1 + \\operatorname{tg} \\alpha \\operatorname{tg} \\beta }}$$<\/p>\n<\/blockquote>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_8406' onClick='GTTabs_show(0,8406)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado A partir das f\u00f3rmulas correspondentes do seno e do cosseno, deduza uma f\u00f3rmula para $\\operatorname{tg} \\left( {\\alpha\u00a0 + \\beta } \\right)$ $\\operatorname{tg} \\left( {\\alpha\u00a0 &#8211; \\beta } \\right)$ Resolu\u00e7\u00e3o &gt;&gt; Resolu\u00e7\u00e3o&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":19170,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[226,97,295],"tags":[427,301],"series":[],"class_list":["post-8406","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-12--ano","category-aplicando","category-funcoes-seno-co-seno-e-tangente","tag-12-o-ano","tag-formulas-trigonometricas"],"views":2048,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2021\/12\/Mat61.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/8406","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=8406"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/8406\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/19170"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=8406"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=8406"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=8406"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=8406"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}