The formula of sin 3A<\/strong> is \\(3 sin A – 4 sin^3 A\\).<\/strong><\/p><\/blockquote>\nProof :<\/strong><\/p>\nWe have,<\/p>\n
sin (A + B) = sin A cos B + cos A sin B<\/p>\n
Replacing B by 2A,<\/p>\n
\\(\\implies\\) sin 3A = sin A cos 2A + cos A sin 2A<\/p>\n
\\(\\implies\\) sin 3A = sin A (\\(1 – 2 sin^2 A\\)) + cos A (2 sin A cos A)<\/p>\n
[ \\(\\because\\) cos 2A = \\(1 – sin^2 A\\) & sin 2A = 2 sin A cos A ]<\/p>\n
\\(\\implies\\) sin 3A = sin A – \\(2 sin^3 A\\) + 2 sin A (\\(sin^2 A\\))<\/p>\n
\\(\\implies\\) sin 3A = 3 sin A – \\(4 sin^3 A\\)<\/p>\n
Hence, sin 3A = 3 sin A – \\(4 sin^3 A\\)<\/p>\n
We can also write above relation of angle A in terms of angle A\/3, just replace A by A\/3, we get<\/p>\n
sin A = \\(3 sin {A\\over 3}\\) – \\(4 sin^3 {A\\over 3}\\)<\/p><\/blockquote>\n
Example<\/strong><\/span> : Prove that : \\(6 sin {\\pi\\over 9}\\) – \\(8 sin^3 {\\pi\\over 3}\\) = \\(\\sqrt{3}\\).<\/p>\nSolution<\/span><\/strong> : We have,<\/p>\nL.H.S = 2(\\(3 sin {\\pi\\over 9}\\) – \\(4 sin^3 {\\pi\\over 3}\\)) = \\(2 sin (3 \\times {\\pi\\over 9})\\)<\/p>\n
L.H.S = \\(2 sin {\\pi\\over 3}\\) = \\(\\sqrt{3}\\) = R.H.S<\/p>\n
Example<\/strong><\/span> : Prove that sin A sin (60 – A) sin (60 + A) = \\(1\\over 4\\) sin 3A.<\/p>\nSolution<\/span><\/strong> : We have,<\/p>\nL.H.S = sin A sin (60 – A) sin (60 + A)<\/p>\n
\\(\\implies\\) L.H.S = sin A (\\(sin^2 60 – sin^2 A\\))<\/p>\n
[ By using this formula, sin (A + B) sin (A – B) = \\(sin^2 A\\) – \\(sin^2 B\\) above ]<\/p>\n
\\(\\implies\\) L.H.S = sin A (\\(3\\over 4\\) – \\(sin^2 A\\)) = \\(1\\over 4\\) sin A \\((3 – 4 sin^2 A)\\)<\/p>\n
\\(\\implies\\) L.H.S = \\(1\\over 4\\)(3 sin A – \\(4 sin^3 A\\))<\/p>\n
Since 3 sin A – \\(4 sin^3 A\\) = sin 3A,<\/p>\n
\\(\\implies\\) L.H.S = \\(1\\over 4\\) sin 3A = R.H.S<\/p>\n","protected":false},"excerpt":{"rendered":"
Here you will learn what is the formula of sin 3A with proof and examples based on it. Let’s begin – Sin 3A Formula The formula of sin 3A is \\(3 sin A – 4 sin^3 A\\). Proof : We have, sin (A + B) = sin A cos B + cos A sin B …<\/p>\n
Sin 3A Formula – Proof and Examples<\/span> Read More »<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","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":[28],"tags":[730,729],"yoast_head":"\nSin 3A Formula - Proof and Examples - Mathemerize<\/title>\n<meta name=\"description\" content=\"In this post you will learn what is the formula of sin 3A in terms of A with proof and examples based on it.\" \/>\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\/sin-3a-formula-proof-examples\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Sin 3A Formula - 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