{"id":155418,"date":"2022-03-26T00:37:26","date_gmt":"2022-03-25T19:07:26","guid":{"rendered":"https:\/\/infinitylearn.com\/surge\/law-of-sines-triangle-definition-formula-proof-problems\/"},"modified":"2022-12-31T15:01:48","modified_gmt":"2022-12-31T09:31:48","slug":"law-of-sines-triangle-definition-formula-proof-problems","status":"publish","type":"post","link":"https:\/\/infinitylearn.com\/surge\/maths\/law-of-sines-triangle\/","title":{"rendered":"Law of Sines Triangle &#8211; Definition, Formula, Proof &#038; Problems"},"content":{"rendered":"<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_37 counter-hierarchy ez-toc-counter ez-toc-grey ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\">\n<p class=\"ez-toc-title\">Table of Contents<\/p>\n<span class=\"ez-toc-title-toggle\"><a href=\"#\" class=\"ez-toc-pull-right ez-toc-btn ez-toc-btn-xs ez-toc-btn-default ez-toc-toggle\" style=\"display: none;\"><label for=\"item\" aria-label=\"Table of Content\"><span style=\"display: flex;align-items: center;width: 35px;height: 30px;justify-content: center;\"><svg style=\"fill: #999;color:#999\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"list-377408\" width=\"20px\" height=\"20px\" viewBox=\"0 0 24 24\" fill=\"none\"><path d=\"M6 6H4v2h2V6zm14 0H8v2h12V6zM4 11h2v2H4v-2zm16 0H8v2h12v-2zM4 16h2v2H4v-2zm16 0H8v2h12v-2z\" fill=\"currentColor\"><\/path><\/svg><svg style=\"fill: #999;color:#999\" class=\"arrow-unsorted-368013\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10px\" height=\"10px\" viewBox=\"0 0 24 24\" version=\"1.2\" baseProfile=\"tiny\"><path d=\"M18.2 9.3l-6.2-6.3-6.2 6.3c-.2.2-.3.4-.3.7s.1.5.3.7c.2.2.4.3.7.3h11c.3 0 .5-.1.7-.3.2-.2.3-.5.3-.7s-.1-.5-.3-.7zM5.8 14.7l6.2 6.3 6.2-6.3c.2-.2.3-.5.3-.7s-.1-.5-.3-.7c-.2-.2-.4-.3-.7-.3h-11c-.3 0-.5.1-.7.3-.2.2-.3.5-.3.7s.1.5.3.7z\"\/><\/svg><\/span><\/label><input type=\"checkbox\" id=\"item\"><\/a><\/span><\/div>\n<nav><ul class='ez-toc-list ez-toc-list-level-1' style='display:block'><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/law-of-sines-triangle\/#Law_of_Sine\" title=\"Law of Sine\">Law of Sine<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/law-of-sines-triangle\/#Law_of_Sine_Definition\" title=\"Law of Sine Definition\">Law of Sine Definition<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/law-of-sines-triangle\/#Law_of_Sines_Formula\" title=\"Law of Sines Formula\">Law of Sines Formula<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/law-of-sines-triangle\/#Law_of_Sine_Proof\" title=\"Law of Sine Proof\">Law of Sine Proof<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/law-of-sines-triangle\/#Law_of_Sine_-_Ambiguous_Case\" title=\"Law of Sine &#8211; Ambiguous Case\">Law of Sine &#8211; Ambiguous Case<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-6\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/law-of-sines-triangle\/#When_to_Use_the_Law_of_Sines\" title=\"When to Use the Law of Sines?\">When to Use the Law of Sines?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-7\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/law-of-sines-triangle\/#Law_of_sine_problem\" title=\"Law of sine problem\">Law of sine problem<\/a><\/li><\/ul><\/nav><\/div>\n<h2><span class=\"ez-toc-section\" id=\"Law_of_Sine\"><\/span>Law of Sine<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>The law of sine states that the sine of an angle is equal to the ratio of the length of the opposite side to the length of the hypotenuse.<\/p>\n<p><img loading=\"lazy\" class=\"aligncenter wp-image-155417 size-full\" src=\"https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2022\/03\/law-of-sines-triangle-definition-formula-proof-problems.jpg\" alt=\"\" width=\"606\" height=\"428\" srcset=\"https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2022\/03\/law-of-sines-triangle-definition-formula-proof-problems.jpg?v=1648235243 606w, https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2022\/03\/law-of-sines-triangle-definition-formula-proof-problems-300x212.jpg?v=1648235243 300w\" sizes=\"(max-width: 606px) 100vw, 606px\" \/><\/p>\n<h2><span class=\"ez-toc-section\" id=\"Law_of_Sine_Definition\"><\/span>Law of Sine Definition<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>The law of sine states that for any given angle, the length of the sine function&#8217;s curve is proportional to the length of the opposite side. In mathematical terms, this law states that the sine of an angle is equal to the length of the opposite side divided by the length of the hypotenuse.<\/p>\n<h2><span class=\"ez-toc-section\" id=\"Law_of_Sines_Formula\"><\/span>Law of Sines Formula<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>The law of sines states that the ratio of the length of the sine of an angle to the length of the opposite side is equal to the ratio of the lengths of the other two sides.<\/p>\n<h2><span class=\"ez-toc-section\" id=\"Law_of_Sine_Proof\"><\/span>Law of Sine Proof<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>A proof of the law of sine can be found using the Pythagorean theorem.<\/p>\n<p>In the diagram, A is the angle of the triangle and b is the length of the side opposite A. The length of the other two sides is c and a, respectively.<\/p>\n<p>The Pythagorean theorem states that:<\/p>\n<p>a2 + b2 = c2<\/p>\n<p>In this triangle, a is the length of the hypotenuse and b and c are the lengths of the two other sides. Substituting these values into the Pythagorean theorem, we get:<\/p>\n<p>c2 = a2 + b2<\/p>\n<p>c2 &#8211; a2 = b2<\/p>\n<p>Now, using the law of sine, we can rewrite this equation as:<\/p>\n<p>sin(A) = b\/c<\/p>\n<h2><span class=\"ez-toc-section\" id=\"Law_of_Sine_-_Ambiguous_Case\"><\/span>Law of Sine &#8211; Ambiguous Case<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>In the ambiguous case, the law of sine states that the sine of an angle is the same as the sine of its complementary angle.<\/p>\n<h2><span class=\"ez-toc-section\" id=\"When_to_Use_the_Law_of_Sines\"><\/span>When to Use the Law of Sines?<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>The law of sines can be used to solve problems in geometry, including problems involving triangles.<\/p>\n<h2><span class=\"ez-toc-section\" id=\"Law_of_sine_problem\"><\/span>Law of sine problem<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>A boat travelling due east at 10 mph passes a lighthouse at an angle of 15\u00b0 north of east. Find the speed of the boat when it is nearest to the lighthouse.<\/p>\n<p>The boat is travelling at 11.6 mph when it is nearest to the lighthouse.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Law of Sine The law of sine states that the sine of an angle is equal to the ratio of [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_yoast_wpseo_focuskw":"Law of Sines Triangle","_yoast_wpseo_title":"%%title%% %%page%%","_yoast_wpseo_metadesc":"Learn about Law of Sines Triangle topic of Maths in details explained by subject experts on infinitylearn.com.","custom_permalink":"maths\/law-of-sines-triangle\/"},"categories":[13],"tags":[],"table_tags":[],"acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v17.9 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Law of Sines Triangle - Definition, Formula, Proof &amp; Problems<\/title>\n<meta name=\"description\" content=\"Learn about Law of Sines Triangle topic of Maths in details explained by subject experts on infinitylearn.com.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/law-of-sines-triangle\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Law of Sines Triangle - 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