The formula of cos 3A<\/strong> is \\(4 cos^3 A – 3 cos A\\).<\/strong><\/p><\/blockquote>\nProof :<\/strong><\/p>\nWe have,<\/p>\n
cos (A + B) = cos A cos B – sin A sin B<\/p>\n
Replacing B by 2A,<\/p>\n
\\(\\implies\\) cos 3A = cos A cos 2A – sin A sin 2A<\/p>\n
\\(\\implies\\) cos 3A = cos A (\\(2 cos^2 A – 1\\)) + sin A (2 sin A cos A)<\/p>\n
[ \\(\\because\\) cos 2A = \\(2 cos^2 A – 1\\) & sin 2A = 2 sin A cos A ]<\/p>\n
\\(\\implies\\) cos 3A = \\(2 cos^3 A\\) – cos A + 2 cos A (\\(sin^2 A\\))<\/p>\n
\\(\\implies\\) cos 3A = \\(2 cos^3 A\\) – cos A + 2 cos A (\\(1 – cos^2 A\\))<\/p>\n
Hence, cos 3A = \\(4 cos^3 A\\) – 3 cos 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
cos A = \\(4 cos^3 {A\\over 3}\\) – \\(3 cos {A\\over 3}\\)<\/p><\/blockquote>\n
Example<\/strong><\/span> : Prove that : \\(8 cos^3 {\\pi\\over 3}\\) – \\(6 sin {\\pi\\over 9}\\) = 1.<\/p>\nSolution<\/span><\/strong> : We have,<\/p>\nL.H.S = 2(\\(8 cos^3 {\\pi\\over 3}\\) – \\(6 sin {\\pi\\over 9}\\)) = \\(2 cos (3 \\times {\\pi\\over 9})\\)<\/p>\n
L.H.S = \\(2 cos {\\pi\\over 3}\\) = 1 = R.H.S<\/p>\n
Example<\/strong><\/span> : Prove that cos A cos (60 – A) cos (60 + A) = \\(1\\over 4\\) cos 3A.<\/p>\nSolution<\/span><\/strong> : We have,<\/p>\nL.H.S = cos A cos (60 – A) cos (60 + A)<\/p>\n
\\(\\implies\\) L.H.S = cos A (\\(cos^2 60 – sin^2 A\\))<\/p>\n
[ By using this formula, cos (A + B) cos (A – B) = \\(cos^2 A\\) – \\(sin^2 B\\) above ]<\/p>\n
\\(\\implies\\) L.H.S = cos A (\\(1\\over 4\\) – \\(sin^2 A\\)) = cos A \\(({1\\over 4} – (1 – cos^2 A))\\)<\/p>\n
\\(\\implies\\) L.H.S = cos A (\\({-3\\over 4} + cos^2 A\\))<\/p>\n
L.H.S = \\(1\\over 4\\) cos A (\\(-3 + 4 cos^2 A\\)) = \\(1\\over 4\\)(\\(4 cos^3 A\\) – 3 cos A)<\/p>\n
Since \\(4 cos^3 A\\) – 3 cos A = cos 3A,<\/p>\n
\\(\\implies\\) L.H.S = \\(1\\over 4\\) cos 3A = R.H.S<\/p>\n","protected":false},"excerpt":{"rendered":"
Here you will learn what is the formula of cos 3A with proof and examples based on it. Let’s begin – Cos 3A Formula The formula of cos 3A is \\(4 cos^3 A – 3 cos A\\). Proof : We have, cos (A + B) = cos A cos B – sin A sin B …<\/p>\n
Cos 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":[731,732],"yoast_head":"\nCos 3A Formula - Proof and Examples - Mathemerize<\/title>\n<meta name=\"description\" content=\"In this post you will learn what is the formula of cos 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\/cos-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=\"Cos 3A Formula - 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