{"id":30392,"date":"2025-06-21T18:09:28","date_gmt":"2025-06-21T18:09:28","guid":{"rendered":"https:\/\/gaviki.com\/blog\/?p=30392"},"modified":"2025-06-21T18:09:29","modified_gmt":"2025-06-21T18:09:29","slug":"use-trigonometric-identities-to-write-sin-x-in-terms-of-cosx-choose-the-correct-expression-equal-to-sin-x","status":"publish","type":"post","link":"https:\/\/gaviki.com\/blog\/use-trigonometric-identities-to-write-sin-x-in-terms-of-cosx-choose-the-correct-expression-equal-to-sin-x\/","title":{"rendered":"Use trigonometric identities to write sin X in terms of cosx Choose the correct expression equal to sin x"},"content":{"rendered":"\n<p>Use trigonometric identities to write sin X in terms of cosx Choose the correct expression equal to sin x sin x = 1- cos X sin x= +v1- cosx sin x = (1 &#8211; cosx)(1 cos X) sin x= cos<\/p>\n\n\n\n<p><strong><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-vivid-cyan-blue-color\">The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n<p>To express <strong>sin(x)<\/strong> in terms of <strong>cos(x)<\/strong> using trigonometric identities, we start from the <strong>Pythagorean identity<\/strong>:sin\u20612(x)+cos\u20612(x)=1\\sin^2(x) + \\cos^2(x) = 1sin2(x)+cos2(x)=1<\/p>\n\n\n\n<p>This can be rearranged to solve for <strong>sin\u00b2(x)<\/strong>:sin\u20612(x)=1\u2212cos\u20612(x)\\sin^2(x) = 1 &#8211; \\cos^2(x)sin2(x)=1\u2212cos2(x)<\/p>\n\n\n\n<p>Now, take the square root of both sides:sin\u2061(x)=1\u2212cos\u20612(x)\\sin(x) = \\sqrt{1 &#8211; \\cos^2(x)}sin(x)=1\u2212cos2(x)\u200b<\/p>\n\n\n\n<p>However, because square roots can be positive or negative, the sign depends on the quadrant in which angle <strong>x<\/strong> lies. If <strong>x<\/strong> is in the first or second quadrant, then <strong>sin(x)<\/strong> is positive. If <strong>x<\/strong> is in the third or fourth quadrant, then <strong>sin(x)<\/strong> is negative.<\/p>\n\n\n\n<p>So the correct identity is:sin\u2061(x)=\u00b11\u2212cos\u20612(x)\\sin(x) = \\pm \\sqrt{1 &#8211; \\cos^2(x)}sin(x)=\u00b11\u2212cos2(x)\u200b<\/p>\n\n\n\n<p>Now look at your answer choices:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>sin x = 1 &#8211; cos x<\/strong> \u2192 Incorrect. This is not a valid trigonometric identity.<\/li>\n\n\n\n<li><strong>sin x = +\u221a(1 &#8211; cos x)<\/strong> \u2192 Incorrect. It should be <strong>cos\u00b2(x)<\/strong> inside the square root, not <strong>cos(x)<\/strong>.<\/li>\n\n\n\n<li><strong>sin x = (1 &#8211; cos x)(1 + cos x)<\/strong> \u2192 This equals <strong>1 &#8211; cos\u00b2(x)<\/strong>, which is <strong>sin\u00b2(x)<\/strong>, not <strong>sin(x)<\/strong>.<\/li>\n\n\n\n<li><strong>sin x = cos<\/strong> \u2192 Incorrect. Sin(x) is not equal to cos(x) in general.<\/li>\n<\/ul>\n\n\n\n<p>So the correct answer should be:sin\u2061(x)=\u00b11\u2212cos\u20612(x)\\sin(x) = \\pm \\sqrt{1 &#8211; \\cos^2(x)}sin(x)=\u00b11\u2212cos2(x)\u200b<\/p>\n\n\n\n<p>Even though it is not listed exactly among the choices, it is important to recognize this as the correct expression derived from identities. It helps solve many problems in trigonometry, especially in calculus or geometry, where expressing one trig function in terms of another is often necessary.<\/p>\n\n\n\n<p>In summary, we used the Pythagorean identity to isolate <strong>sin\u00b2(x)<\/strong> and then took the square root to solve for <strong>sin(x)<\/strong>. The sign depends on the quadrant of angle <strong>x<\/strong>. This identity is fundamental and commonly used in trigonometric simplification and solving equations.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"1024\" src=\"https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/learnexams-banner5-144.jpeg\" alt=\"\" class=\"wp-image-30410\" srcset=\"https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/learnexams-banner5-144.jpeg 1024w, https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/learnexams-banner5-144-300x300.jpeg 300w, https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/learnexams-banner5-144-150x150.jpeg 150w, https:\/\/gaviki.com\/blog\/wp-content\/uploads\/2025\/06\/learnexams-banner5-144-768x768.jpeg 768w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>Use trigonometric identities to write sin X in terms of cosx Choose the correct expression equal to sin x sin x = 1- cos X sin x= +v1- cosx sin x = (1 &#8211; cosx)(1 cos X) sin x= cos The Correct Answer and Explanation is: To express sin(x) in terms of cos(x) using trigonometric [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-30392","post","type-post","status-publish","format-standard","hentry","category-quiz-questions"],"_links":{"self":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/30392","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/comments?post=30392"}],"version-history":[{"count":1,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/30392\/revisions"}],"predecessor-version":[{"id":30411,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/30392\/revisions\/30411"}],"wp:attachment":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/media?parent=30392"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/categories?post=30392"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/tags?post=30392"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}