{"id":9321,"date":"2022-01-10T00:04:07","date_gmt":"2022-01-09T18:34:07","guid":{"rendered":"https:\/\/mathemerize.com\/?p=9321"},"modified":"2022-01-16T17:13:30","modified_gmt":"2022-01-16T11:43:30","slug":"what-is-the-formula-of-cos-2a","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/what-is-the-formula-of-cos-2a\/","title":{"rendered":"Cos 2A Formula – Proof and Examples"},"content":{"rendered":"

Here you will learn what is the formula of cos 2A in terms of sin and cos and also in terms of tan with proof and examples.<\/p>\n

Let’s begin –<\/p>\n

Cos 2A Formula :<\/h2>\n

(i) In Terms of Cos and Sin<\/h3>\n

Given below are all the formulas for cos 2A.<\/p>\n

\n

(i) cos 2A = \\(cos^2 A\\) – \\(sin^2 A\\)<\/p>\n

(ii) cos 2A = \\(2cos^2 A – 1\\)    or,    1 + cos 2A = \\(2cos^2 A\\)<\/p>\n

(iii) cos 2A = \\(1 – 2sin^2 A\\)    or,    1 – cos 2A = \\(2sin^2 A\\)<\/p>\n<\/blockquote>\n

Proof :<\/strong><\/p>\n

(i) We have,<\/p>\n

Cos (A + B) = cos A cos B – sin A sin B<\/p>\n

Replacing B by A,<\/p>\n

\\(\\implies\\) cos 2A = cos A cos A + sin A sin A<\/p>\n

\\(\\implies\\) cos 2A = \\(cos^2 A\\) – \\(sin^2 A\\)<\/p>\n

(ii) We have,<\/p>\n

cos 2A = \\(cos^2 A\\) – \\(sin^2 A\\)<\/p>\n

\\(\\implies\\) cos 2A = \\(cos^2 A\\) – \\(1 – cos^2 A\\)<\/p>\n

\\(\\implies\\) cos 2A = \\(2cos^2 A – 1\\)<\/p>\n

Again, cos 2A = \\(2cos^2 A – 1\\)<\/p>\n

\\(\\implies\\) 1 + cos 2A = \\(2cos^2 A\\)<\/p>\n

(iii) We have,<\/p>\n

cos 2A = \\(cos^2 A\\) – \\(sin^2 A\\)<\/p>\n

\\(\\implies\\) cos 2A = \\(1 – sin^2 A\\) – \\(sin^2 A\\)<\/p>\n

\\(\\implies\\) cos 2A = \\(1- 2sin^2 A\\)<\/p>\n

Again, cos 2A = \\(1- 2sin^2 A\\)<\/p>\n

\\(\\implies\\) 1 – cos 2A = \\(2ain^2 A\\)<\/p>\n

We can also write above relation in terms of angle A\/2, just replace A by A\/2, we get<\/p>\n

\n

(i) cos A = \\(cos^2 ({A\\over 2})\\) – \\(sin^2 ({A\\over 2})\\)<\/p>\n

(ii) cos A = \\(2cos^2 ({A\\over 2}) – 1\\)    or,    1 + cos A = \\(2cos^2 ({A\\over 2})\\)<\/p>\n

(iii) cos A = \\(1 – 2sin^2 ({A\\over 2})\\)    or,    1 – cos A = \\(2sin^2 ({A\\over 2})\\)<\/p>\n<\/blockquote>\n

(ii) Cos 2A Formula in Terms of Tan :<\/h3>\n
\n

Cos 2A = \\(1 – tan^2 A\\over 1 + tan^2 A\\)<\/p>\n<\/blockquote>\n

Proof :<\/strong><\/p>\n

We have,<\/p>\n

cos 2A = \\(cos^2 A\\) – \\(sin^2 A\\)<\/p>\n

\\(\\implies\\) cos 2A = \\(cos^2 A – sin^2 A\\over sin^2 A + cos^2 A\\)<\/p>\n

[ \\(\\because\\)  \\(sin^2 A + cos^2 A\\) = 1 ]<\/p>\n

Now, Dividing numerator and denominator by \\(cos^2 A\\),<\/p>\n

\\(\\implies\\)  cos 2A = \\({cos^2 A – sin^2 A\\over cos^2 A}\\over {sin^2 A + cos^2 A\\over cos^2 A}\\)<\/p>\n

\\(\\implies\\) cos 2A = \\(1 – tan^2 A\\over 1 + tan^2 A\\)<\/p>\n

We can also write above relation in terms of angle A\/2, just replace A by A\/2, we get<\/p>\n

\n

cos A = \\(1 – tan^2 ({A\\over 2})\\over 1 + tan^2 ({A\\over 2})\\)<\/p>\n<\/blockquote>\n

Example<\/strong><\/span> : Find the value of Cos 120 ?<\/p>\n

Solution<\/span><\/strong> : We Know that sin 60 = \\(\\sqrt{3}\\over 2\\) and cos 60 = \\(1\\over 2\\)<\/p>\n

By using above formula,<\/p>\n

cos 120 = \\(cos^2 60\\) – \\(sin^2 60\\) = \\(1\\over 4\\) – \\(3\\over 4\\)<\/p>\n

\\(\\implies\\)  cos 120 = \\(-1\\over 2\\)<\/p>\n

Example<\/strong><\/span> : If sin A = \\(3\\over 5\\), where 0 < A < 90, find the value of cos 2A ?<\/p>\n

Solution<\/span><\/strong> : We have,<\/p>\n

sin A = \\(3\\over 5\\) where 0 < A < 90 degrees<\/p>\n

\\(\\therefore\\) \\(cos^2 A\\) = 1 – \\(sin^2 A\\)<\/p>\n

\\(\\implies\\) cos A = \\(\\sqrt{1 – sin^2 A}\\) = \\(\\sqrt{1 – {9\\over 25}}\\) = \\(4\\over 5\\)<\/p>\n

By using above formula,<\/p>\n

cos 2A = \\(cos^2 A\\) – \\(sin^2 A\\) = \\(16\\over 25\\) – \\(9\\over 25\\)<\/p>\n

\\(\\implies\\)  cos 2A = \\(7\\over 25\\)<\/p>\n\n\n

<\/p>\n","protected":false},"excerpt":{"rendered":"

Here you will learn what is the formula of cos 2A in terms of sin and cos and also in terms of tan with proof and examples. Let’s begin – Cos 2A Formula : (i) In Terms of Cos and Sin Given below are all the formulas for cos 2A. (i) cos 2A = \\(cos^2 …<\/p>\n

Cos 2A 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":[725,726],"yoast_head":"\nCos 2A Formula - Proof and Examples - Mathemerize<\/title>\n<meta name=\"description\" content=\"In this post you will learn what is the formula of cos 2A in terms of sin and cos and also in terms of tan with proof and examples.\" \/>\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-formula-of-cos-2a\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Cos 2A Formula - 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