{"id":3766,"date":"2021-08-08T07:26:00","date_gmt":"2021-08-08T07:26:00","guid":{"rendered":"https:\/\/mathemerize.com\/?p=3766"},"modified":"2021-11-30T19:32:30","modified_gmt":"2021-11-30T14:02:30","slug":"conditional-trigonometric-identities","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/conditional-trigonometric-identities\/","title":{"rendered":"Conditional Trigonometric Identities – Maximum & Minimum Value"},"content":{"rendered":"

Here, you will learn conditional trigonometric identities and maximum and minimum value in trigonometry.<\/p>\n

Let’s begin –<\/p>\n

Maximum and Minimum values in Trigonometry Expressions :<\/h2>\n

(i)\u00a0 acos\\(\\theta\\) + bcos\\(\\theta\\) will always lie in the interval [-\\(\\sqrt{a^2+b^2}\\), \\(\\sqrt{a^2+b^2}\\)] i.e. the maximum and minimum values are \\(\\sqrt{a^2+b^2}\\), -\\(\\sqrt{a^2+b^2}\\) respectively.<\/p>\n

(ii)\u00a0 Minimum value of \\(a^2tan^2\\theta\\) + \\(b^2\\tan^2\\theta\\) = 2ab where a,b > 0<\/p>\n

(iii)\u00a0 -\\(\\sqrt{a^2 + b^2 + 2abcos(\\alpha – \\beta)}\\) \\(\\le\\) acos(\\(\\alpha + \\theta\\)) + bcos(\\(\\beta + \\theta\\)) \\(\\le\\) \\(\\sqrt{a^2 + b^2 + 2abcos(\\alpha – \\beta)}\\) where \\(\\alpha\\) and \\(\\beta\\) are known angles.<\/p>\n

(iv)\u00a0 In case a quadratic in sin\\(\\theta\\) & cos\\(\\theta\\) is given then the maximum and minimum values can be obtained by making perfect square.<\/p>\n\n\n

Example : <\/span> Find the maximum value of 1 + \\(sin({\\pi\\over 4} + \\theta)\\) + 2\\(cos({\\pi\\over 4} — \\theta)\\)<\/p>\n

Solution : <\/span>We have 1 + \\(sin({\\pi\\over 4} + \\theta)\\) + 2\\(cos({\\pi\\over 4} — \\theta)\\)

\n = 1 + \\(1\\over sqrt{2}\\)(cos\\(\\theta\\) + sin\\(\\theta\\)) + \\(\\sqrt{2}\\)(cos\\(\\theta\\) + sin\\(\\theta\\)) = 1 + (\\({1\\over \\sqrt{2}} + \\sqrt{2}\\)) (cos\\(\\theta\\) + sin\\(\\theta\\))

\n = 1 + (\\({1\\over \\sqrt{2}} + \\sqrt{2}\\)) . \\(\\sqrt{2}\\) = 4

<\/p>\n\n\n

Conditional Trigonometric Identities :<\/h2>\n

If A + B + C = \\(180^{\\circ}\\),then<\/p>\n

(i)  tanA + tanB + tanC = tanA tanB tanC<\/p>\n

(ii)  cotA cotB + cotB cotC + cotC cotA = 1<\/p>\n

(iii)  \\(tan{A\\over 2}\\) \\(tan{B\\over 2}\\) + \\(tan{B\\over 2}\\) \\(tan{C\\over 2}\\) + \\(tan{C\\over 2}\\) \\(tan{A\\over 2}\\) = 1<\/p>\n

(iv)  \\(cot{A\\over 2}\\) + \\(cot{B\\over 2}\\) + \\(cot{C\\over 2}\\) = \\(cot{A\\over 2}\\) \\(cot{B\\over 2}\\) \\(cot{C\\over 2}\\)<\/p>\n

(v)   sin2A + sin2B + sin2C = 4sinA sinB sinC<\/p>\n

(vi)  cos2A + cos2B + cos2C = 1 – 4cosA cosB cosC<\/p>\n

(vii)  sinA + sinB + sinC = 4\\(cos{A\\over 2}\\) \\(cos{B\\over 2}\\) \\(cos{C\\over 2}\\)<\/p>\n

(viii)  cosA + cosB + cosC = 1 + 4\\(sin{A\\over 2}\\) \\(sin{B\\over 2}\\) \\(sin{C\\over 2}\\)<\/p>\n

Some Important results :<\/strong><\/p>\n

(i)  sinA sin(\\(60^{\\circ}\\) – A) sin(\\(60^{\\circ}\\) + A) = \\(1\\over 4\\)sin3A<\/p>\n

(ii)  cosA cos(\\(60^{\\circ}\\) – A) cos(\\(60^{\\circ}\\) + A) = \\(1\\over 4\\)cos3A<\/p>\n

(iii)  tanA tan(\\(60^{\\circ}\\) – A) tan(\\(60^{\\circ}\\) + A) = tan3A<\/p>\n

(iv)  cotA cot(\\(60^{\\circ}\\) – A) cot(\\(60^{\\circ}\\) + A) = cot3A<\/p>\n

(v)  \\(sin^2A\\) + \\(sin^2(60^{\\circ}\\) – A) + \\(sin^2(60^{\\circ}\\) + A) = \\(3\\over 2\\)<\/p>\n

(vi)  \\(cos^2A\\) + \\(cos^2(60^{\\circ}\\) – A) + \\(cos^2(60^{\\circ}\\) + A) = \\(3\\over 2\\)<\/p>\n

(vii)  tanA + tan(\\(60^{\\circ}\\) + A) + tan(\\(120^{\\circ}\\) + A) = 3tan3A<\/p>\n\n\n

\n
Previous – Graph of Trigonometric Functions \u2013 Domain & Range<\/a><\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

Here, you will learn conditional trigonometric identities and maximum and minimum value in trigonometry. Let’s begin – Maximum and Minimum values in Trigonometry Expressions : (i)\u00a0 acos\\(\\theta\\) + bcos\\(\\theta\\) will always lie in the interval [-\\(\\sqrt{a^2+b^2}\\), \\(\\sqrt{a^2+b^2}\\)] i.e. the maximum and minimum values are \\(\\sqrt{a^2+b^2}\\), -\\(\\sqrt{a^2+b^2}\\) respectively. (ii)\u00a0 Minimum value of \\(a^2tan^2\\theta\\) + \\(b^2\\tan^2\\theta\\) = …<\/p>\n

Conditional Trigonometric Identities – Maximum & Minimum Value<\/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":"default","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":""},"categories":[28],"tags":[685,684],"yoast_head":"\nConditional Trigonometric Identities - Maximum & Minimum Value<\/title>\n<meta name=\"description\" content=\"In this post, you will learn conditional trigonometric identities, some formulas and maximum and minimum value in trigonometry.\" \/>\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\/conditional-trigonometric-identities\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Conditional Trigonometric Identities - 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