{"id":9449,"date":"2022-01-14T16:22:42","date_gmt":"2022-01-14T10:52:42","guid":{"rendered":"https:\/\/mathemerize.com\/?p=9449"},"modified":"2022-01-14T16:22:45","modified_gmt":"2022-01-14T10:52:45","slug":"what-is-the-general-solution-of-sin-theta-sin-alpha","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/what-is-the-general-solution-of-sin-theta-sin-alpha\/","title":{"rendered":"What is the General Solution of \\(sin \\theta\\) = \\(sin \\alpha\\) ?"},"content":{"rendered":"<h2>Solution :<\/h2>\n<p>The general solution of <strong>\\(sin \\theta\\) = \\(sin \\alpha\\)<\/strong> is given by <strong>\\(\\theta\\) = \\(n\\pi + (-1)^n \\alpha\\),&nbsp; n \\(\\in\\) Z.<\/strong><\/p>\n<p><strong>Proof :<\/strong><\/p>\n<p>We have,&nbsp; \\(sin \\theta\\) = \\(sin \\alpha\\)<\/p>\n<p>\\(\\implies\\)&nbsp; \\(sin \\theta\\) &#8211; \\(sin \\alpha\\) = 0<\/p>\n<p>\\(\\implies\\)&nbsp; &nbsp;\\(2 sin ({\\theta &#8211; \\alpha\\over 2}) cos({\\theta + \\alpha\\over 2})\\) = 0<\/p>\n<p>\\(\\implies\\)&nbsp; \\(sin ({\\theta &#8211; \\alpha\\over 2})\\) = 0&nbsp; &nbsp;or,&nbsp; &nbsp;\\(cos({\\theta + \\alpha\\over 2})\\) = 0<\/p>\n<p>\\(\\implies\\)&nbsp; \\({\\theta &#8211; \\alpha\\over 2}\\) = \\(m\\pi\\)&nbsp; &nbsp;or&nbsp; \\({\\theta + \\alpha\\over 2}\\) = \\((2m + 1){\\pi\\over 2}\\),&nbsp; m \\(\\in\\) Z<\/p>\n<p>\\(\\implies\\)&nbsp; &nbsp;\\(\\theta\\)&nbsp; =&nbsp; \\(2m\\pi + \\alpha\\), \\(\\in\\)&nbsp; Z&nbsp; &nbsp;or,&nbsp; \\(\\theta\\)&nbsp; =&nbsp; \\((2m + 1)\\pi &#8211; \\alpha\\),&nbsp; m \\(\\in\\) Z.<\/p>\n<p>\\(\\implies\\)&nbsp; &nbsp;\\(\\theta\\) = (any even multiple of \\(\\pi\\)) + \\(\\alpha\\)&nbsp; or,&nbsp; \\(\\theta\\) = (any odd multiple of \\(\\pi\\)) &#8211; \\(\\alpha\\)<\/p>\n<p>\\(\\implies\\)&nbsp; &nbsp; \\(\\theta\\) = \\(n\\pi + (-1)^n \\alpha\\),&nbsp; where n \\(\\in\\) Z.<\/p>\n<p><strong>Remark :&nbsp;<\/strong>The equation \\(cosec \\theta\\) = \\(cosec \\alpha\\) is equivalent to \\(sin \\theta\\) = \\(sin \\alpha\\). Thus, \\(cosec \\theta\\) = \\(cosec \\alpha\\) and \\(sin \\theta\\) = \\(sin \\alpha\\) have the same general solution.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Solution : The general solution of \\(sin \\theta\\) = \\(sin \\alpha\\) is given by \\(\\theta\\) = \\(n\\pi + (-1)^n \\alpha\\),&nbsp; n \\(\\in\\) Z. Proof : We have,&nbsp; \\(sin \\theta\\) = \\(sin \\alpha\\) \\(\\implies\\)&nbsp; \\(sin \\theta\\) &#8211; \\(sin \\alpha\\) = 0 \\(\\implies\\)&nbsp; &nbsp;\\(2 sin ({\\theta &#8211; \\alpha\\over 2}) cos({\\theta + \\alpha\\over 2})\\) = 0 \\(\\implies\\)&nbsp; \\(sin &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathemerize.com\/what-is-the-general-solution-of-sin-theta-sin-alpha\/\"> <span class=\"screen-reader-text\">What is the General Solution of \\(sin \\theta\\) = \\(sin \\alpha\\) ?<\/span> Read More &raquo;<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"default","ast-global-header-display":"","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":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":""},"categories":[43,59],"tags":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v20.12 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>What is the General Solution of \\(sin \\theta\\) = \\(sin \\alpha\\) ?<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/mathemerize.com\/what-is-the-general-solution-of-sin-theta-sin-alpha\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"What is the General Solution of \\(sin \\theta\\) = \\(sin \\alpha\\) ?\" \/>\n<meta property=\"og:description\" content=\"Solution : The general solution of (sin theta) = (sin alpha) is given by (theta) = (npi + (-1)^n alpha),&nbsp; n (in) Z. 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Proof : We have,&nbsp; (sin theta) = (sin alpha) (implies)&nbsp; (sin theta) &#8211; (sin alpha) = 0 (implies)&nbsp; &nbsp;(2 sin ({theta &#8211; alphaover 2}) cos({theta + alphaover 2})) = 0 (implies)&nbsp; (sin &hellip; What is the General Solution of (sin theta) = (sin alpha) ? 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