{"id":238265,"date":"2025-06-17T22:06:10","date_gmt":"2025-06-17T22:06:10","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=238265"},"modified":"2025-06-17T22:06:12","modified_gmt":"2025-06-17T22:06:12","slug":"sin-12-degree-sin-18-degree-sin-54-degree","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/06\/17\/sin-12-degree-sin-18-degree-sin-54-degree\/","title":{"rendered":"Sin 12 degree sin 18 degree sin 54 degree"},"content":{"rendered":"\n<p>Sin 12 degree sin 18 degree sin 54 degree<\/p>\n\n\n\n<p><strong><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-1-color\">The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n<p>To evaluate the expression:sin\u2061(12\u2218)\u22c5sin\u2061(18\u2218)\u22c5sin\u2061(54\u2218)\\sin(12^\\circ) \\cdot \\sin(18^\\circ) \\cdot \\sin(54^\\circ)sin(12\u2218)\u22c5sin(18\u2218)\u22c5sin(54\u2218)<\/p>\n\n\n\n<p>we can make use of known trigonometric identities and exact values for special angles.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Step-by-step calculation:<\/h3>\n\n\n\n<p>Start by using known sine values for 18\u00b0 and 54\u00b0:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>sin\u2061(18\u2218)=5\u221214\\sin(18^\\circ) = \\dfrac{\\sqrt{5} &#8211; 1}{4}sin(18\u2218)=45\u200b\u22121\u200b<\/li>\n\n\n\n<li>sin\u2061(54\u2218)=5+14\\sin(54^\\circ) = \\dfrac{\\sqrt{5} + 1}{4}sin(54\u2218)=45\u200b+1\u200b<\/li>\n<\/ul>\n\n\n\n<p>Multiplying these two:sin\u2061(18\u2218)\u22c5sin\u2061(54\u2218)=(5\u221214)\u22c5(5+14)\\sin(18^\\circ) \\cdot \\sin(54^\\circ) = \\left( \\dfrac{\\sqrt{5} &#8211; 1}{4} \\right) \\cdot \\left( \\dfrac{\\sqrt{5} + 1}{4} \\right)sin(18\u2218)\u22c5sin(54\u2218)=(45\u200b\u22121\u200b)\u22c5(45\u200b+1\u200b)<\/p>\n\n\n\n<p>Use the identity (a\u2212b)(a+b)=a2\u2212b2(a &#8211; b)(a + b) = a^2 &#8211; b^2(a\u2212b)(a+b)=a2\u2212b2:=5\u2212116=416=14= \\dfrac{5 &#8211; 1}{16} = \\dfrac{4}{16} = \\dfrac{1}{4}=165\u22121\u200b=164\u200b=41\u200b<\/p>\n\n\n\n<p>Now, multiply that by sin\u2061(12\u2218)\\sin(12^\\circ)sin(12\u2218):sin\u2061(12\u2218)\u22c514\\sin(12^\\circ) \\cdot \\dfrac{1}{4}sin(12\u2218)\u22c541\u200b<\/p>\n\n\n\n<p>We now need the value of sin\u2061(12\u2218)\\sin(12^\\circ)sin(12\u2218). This does not have a simple radical form, but using a calculator or table:sin\u2061(12\u2218)\u22480.2079\\sin(12^\\circ) \\approx 0.2079sin(12\u2218)\u22480.2079<\/p>\n\n\n\n<p>Therefore:sin\u2061(12\u2218)\u22c5sin\u2061(18\u2218)\u22c5sin\u2061(54\u2218)\u22480.2079\u22c514=0.2079\u22c50.25=0.051975\\sin(12^\\circ) \\cdot \\sin(18^\\circ) \\cdot \\sin(54^\\circ) \\approx 0.2079 \\cdot \\dfrac{1}{4} = 0.2079 \\cdot 0.25 = 0.051975sin(12\u2218)\u22c5sin(18\u2218)\u22c5sin(54\u2218)\u22480.2079\u22c541\u200b=0.2079\u22c50.25=0.051975<\/p>\n\n\n\n<p>Rounded to four decimal places:0.0520\\boxed{0.0520}0.0520\u200b<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">Explanation<\/h3>\n\n\n\n<p>The given expression involves multiplying three sine values: sin\u2061(12\u2218)\\sin(12^\\circ)sin(12\u2218), sin\u2061(18\u2218)\\sin(18^\\circ)sin(18\u2218), and sin\u2061(54\u2218)\\sin(54^\\circ)sin(54\u2218). These are not standard angles typically found on the unit circle, but some of them, such as 18\u00b0 and 54\u00b0, are linked to the geometry of regular pentagons and the golden ratio. Their exact values can be expressed using square roots.<\/p>\n\n\n\n<p>First, we use known exact values:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>sin\u2061(18\u2218)\\sin(18^\\circ)sin(18\u2218) equals (5\u22121)\/4(\\sqrt{5} &#8211; 1)\/4(5\u200b\u22121)\/4<\/li>\n\n\n\n<li>sin\u2061(54\u2218)\\sin(54^\\circ)sin(54\u2218) equals (5+1)\/4(\\sqrt{5} + 1)\/4(5\u200b+1)\/4<\/li>\n<\/ul>\n\n\n\n<p>When multiplied, these two expressions form a difference of squares:(5\u221214)(5+14)=5\u2212116=14\\left(\\dfrac{\\sqrt{5} &#8211; 1}{4}\\right)\\left(\\dfrac{\\sqrt{5} + 1}{4}\\right) = \\dfrac{5 &#8211; 1}{16} = \\dfrac{1}{4}(45\u200b\u22121\u200b)(45\u200b+1\u200b)=165\u22121\u200b=41\u200b<\/p>\n\n\n\n<p>This reduces our problem to finding the product of sin\u2061(12\u2218)\\sin(12^\\circ)sin(12\u2218) and 1\/41\/41\/4. While sin\u2061(12\u2218)\\sin(12^\\circ)sin(12\u2218) does not have a simple radical form, it can be approximated numerically using a calculator. The value is about 0.2079. Multiplying by 1\/4 gives approximately 0.0520.<\/p>\n\n\n\n<p>This method demonstrates the power of using identities and known exact values to simplify seemingly complex trigonometric products. The final result is a simple decimal, even though the original expression involved non-standard angles.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" src=\"https:\/\/learnexams.com\/blog\/wp-content\/uploads\/2025\/06\/learnexams-banner8-1017.jpeg\" alt=\"\" class=\"wp-image-238266\"\/><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>Sin 12 degree sin 18 degree sin 54 degree The Correct Answer and Explanation is: To evaluate the expression:sin\u2061(12\u2218)\u22c5sin\u2061(18\u2218)\u22c5sin\u2061(54\u2218)\\sin(12^\\circ) \\cdot \\sin(18^\\circ) \\cdot \\sin(54^\\circ)sin(12\u2218)\u22c5sin(18\u2218)\u22c5sin(54\u2218) we can make use of known trigonometric identities and exact values for special angles. Step-by-step calculation: Start by using known sine values for 18\u00b0 and 54\u00b0: Multiplying these two:sin\u2061(18\u2218)\u22c5sin\u2061(54\u2218)=(5\u221214)\u22c5(5+14)\\sin(18^\\circ) \\cdot \\sin(54^\\circ) = [&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-238265","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/238265","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=238265"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/238265\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=238265"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=238265"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=238265"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}