{"id":188459,"date":"2025-02-07T04:07:33","date_gmt":"2025-02-07T04:07:33","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=188459"},"modified":"2025-02-07T04:07:35","modified_gmt":"2025-02-07T04:07:35","slug":"how-do-you-solve-for-sec2pi-3-tan2pi-3","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/02\/07\/how-do-you-solve-for-sec2pi-3-tan2pi-3\/","title":{"rendered":"How do you solve for sec(2pi\/3) * tan(2pi\/3)"},"content":{"rendered":"\n<p>1.How do you solve for sec(2pi\/3) * tan(2pi\/3)<\/p>\n\n\n\n<p>2.2X=7<\/p>\n\n\n\n<p>3.What is the smallest number (positive integer) when spelled out, the letters appear in alphabetical order? The word is explain it briefly<\/p>\n\n\n\n<p><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-6-color\"><strong>The Correct Answer and Explanation is :<\/strong><\/mark><\/p>\n\n\n\n<h3 class=\"wp-block-heading\">1. Solving for (\\sec\\left(\\frac{2\\pi}{3}\\right) \\times \\tan\\left(\\frac{2\\pi}{3}\\right)):<\/h3>\n\n\n\n<p>To solve this, we&#8217;ll break it down into two steps: first find (\\sec\\left(\\frac{2\\pi}{3}\\right)) and (\\tan\\left(\\frac{2\\pi}{3}\\right)), then multiply them together.<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Step 1: Find (\\sec\\left(\\frac{2\\pi}{3}\\right))<\/strong> (\\sec(\\theta) = \\frac{1}{\\cos(\\theta)}) The cosine of (\\frac{2\\pi}{3}) is in the second quadrant, where cosine is negative. We know that (\\cos\\left(\\frac{2\\pi}{3}\\right) = -\\frac{1}{2}). Therefore,<br>[<br>\\sec\\left(\\frac{2\\pi}{3}\\right) = \\frac{1}{-\\frac{1}{2}} = -2.<br>]<\/li>\n\n\n\n<li><strong>Step 2: Find (\\tan\\left(\\frac{2\\pi}{3}\\right))<\/strong> (\\tan(\\theta) = \\frac{\\sin(\\theta)}{\\cos(\\theta)}) We know that (\\sin\\left(\\frac{2\\pi}{3}\\right) = \\frac{\\sqrt{3}}{2}) and (\\cos\\left(\\frac{2\\pi}{3}\\right) = -\\frac{1}{2}). Therefore,<br>[<br>\\tan\\left(\\frac{2\\pi}{3}\\right) = \\frac{\\frac{\\sqrt{3}}{2}}{-\\frac{1}{2}} = -\\sqrt{3}.<br>]<\/li>\n\n\n\n<li><strong>Step 3: Multiply (\\sec\\left(\\frac{2\\pi}{3}\\right)) and (\\tan\\left(\\frac{2\\pi}{3}\\right))<\/strong> Now, multiply the results:<br>[<br>\\sec\\left(\\frac{2\\pi}{3}\\right) \\times \\tan\\left(\\frac{2\\pi}{3}\\right) = (-2) \\times (-\\sqrt{3}) = 2\\sqrt{3}.<br>]<\/li>\n<\/ul>\n\n\n\n<p>So, the answer is (2\\sqrt{3}).<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">2. Solving (2X = 7):<\/h3>\n\n\n\n<p>To solve for (X), divide both sides of the equation by 2:<\/p>\n\n\n\n<p>[<br>X = \\frac{7}{2} = 3.5.<br>]<\/p>\n\n\n\n<p>So, (X = 3.5).<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">3. Smallest number whose spelled-out letters appear in alphabetical order:<\/h3>\n\n\n\n<p>The number you&#8217;re looking for is <strong>40<\/strong>.<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>The word for 40 is &#8220;forty.&#8221;<\/li>\n\n\n\n<li>When spelled out, the letters are: f, o, r, t, y.<\/li>\n\n\n\n<li>The letters are in alphabetical order: <strong>f &lt; o &lt; r &lt; t &lt; y<\/strong>.<\/li>\n<\/ul>\n\n\n\n<p>Now, let&#8217;s explain why <strong>40<\/strong> is the smallest number with this property.<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">Explanation:<\/h4>\n\n\n\n<p>When spelled out, the letters of a number should appear in increasing alphabetical order. To determine the smallest number where this occurs, we start by examining numbers in increasing order and checking their spellings.<\/p>\n\n\n\n<p>For instance:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>The word for <strong>1<\/strong> is &#8220;one&#8221;, which does not have letters in alphabetical order.<\/li>\n\n\n\n<li>The word for <strong>10<\/strong> is &#8220;ten&#8221;, which does not have letters in alphabetical order.<\/li>\n\n\n\n<li>As we move through numbers, we continue checking their spellings until we reach <strong>40<\/strong>, which is &#8220;forty.&#8221;<\/li>\n<\/ul>\n\n\n\n<p>In &#8220;forty&#8221;, the letters are arranged in alphabetical order: f, o, r, t, and y. None of the smaller numbers (like 1, 10, 12, 20, etc.) have their letters in alphabetical order, making <strong>40<\/strong> the smallest number with this property.<\/p>\n\n\n\n<p>While numbers larger than 40 may also satisfy this condition, <strong>40<\/strong> is the first number where it happens. This pattern continues for larger numbers, but none are smaller than 40 with this specific alphabetical order.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>1.How do you solve for sec(2pi\/3) * tan(2pi\/3) 2.2X=7 3.What is the smallest number (positive integer) when spelled out, the letters appear in alphabetical order? The word is explain it briefly The Correct Answer and Explanation is : 1. Solving for (\\sec\\left(\\frac{2\\pi}{3}\\right) \\times \\tan\\left(\\frac{2\\pi}{3}\\right)): To solve this, we&#8217;ll break it down into two steps: first [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-188459","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/188459","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=188459"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/188459\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=188459"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=188459"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=188459"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}