{"id":3927,"date":"2021-08-10T11:04:06","date_gmt":"2021-08-10T11:04:06","guid":{"rendered":"https:\/\/mathemerize.com\/?p=3927"},"modified":"2021-09-11T20:57:32","modified_gmt":"2021-09-11T15:27:32","slug":"trigonometric-equations-solving-strategies","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/trigonometric-equations-solving-strategies\/","title":{"rendered":"Trigonometric Equations Solving Strategies"},"content":{"rendered":"

Here, you will learn trigonometric equations solving strategies i.e by factorisation and reducing it to a quadratic equation etc.<\/p>\n

Let’s begin –<\/p>\n

Different Strategies for Trigonometric Equations Solving<\/h2>\n

(a) Solving trigonometric equations by factorisation<\/strong><\/h4>\n\n\n

Example : <\/span> If \\(1\\over 6\\)sin x, cos x, tan x are in G.P. then the general solution for x is\n <\/p>\n

Solution : <\/span>Since \\(1\\over 6\\)sin x, cos x, tan x are in G.P.

\n \\(\\implies\\)   \\(cos^2x\\) = \\(1\\over 6\\)sin x.tan x

\n \\(\\implies\\)   6\\(cos^3x\\) + \\(cos^2x\\) – 1 = 0

\n \\(\\therefore\\)   (2cos x – 1)(3\\(cos^2x\\) + 2cos x + 1) = 0

\n \\(\\implies\\)   cos x = \\(1\\over 2\\)   (other values of cos x are imaginary)

\n \\(\\implies\\)   cos x = cos\\(\\pi\\over 3\\)   \\(\\implies\\)   x = 2n\\(\\pi\\) \\(\\pm\\) \\(\\pi\\over 3\\)

<\/p>\n\n\n

(b) Solving trigonometric equations by reducing it to a quadratic equation<\/strong><\/h4>\n\n\n

Example : <\/span> Find the number of solutions of tan x + sec x = 2cos x in [0,\\(2\\pi\\)].\n <\/p>\n

Solution : <\/span>Here, tan x + sec x = 2cos x   \\(\\implies\\)   sin x + 1 = 2\\(cos^2x\\)

\n \\(\\implies\\)   2\\(sin^2x\\) + sin x – 1 = 0   \\(\\implies\\)   sin x = \\(1\\over 2\\), -1

\n But sin x = -1   \\(\\implies\\)   x = \\(3\\pi\\over 2\\)   for which tan x + sec x = 2cos x is not defined.

\n Thus sin x = \\(1\\over 2\\)   \\(\\implies\\)   x = \\(\\pi\\over 6\\), \\(5\\pi\\over 6\\)

\n \\(\\implies\\)   number of solutions of tan x + sec x = 2cos x is 2.

<\/p>\n\n\n

(c) Solving trigonometric equations by introducing an auxilliary argument<\/strong><\/h4>\n\n\n

Example : <\/span> Find the number of distinct solutions of sec x + tan x = \\(\\sqrt{3}\\), where 0 \\(\\le\\) x \\(\\le\\) \\(3\\pi\\). \n <\/p>\n

Solution : <\/span>Here, sec x + tan x = \\(\\sqrt{3}\\)   \\(\\implies\\)   sin x + 1 = \\(\\sqrt{3}\\)cos x or \\(\\sqrt{3}\\)cos x – sin x = 1

\n Dividing both sides by \\(\\sqrt{a^2+b^2}\\) i.e. 2, we get

\n \\(\\implies\\)   \\(\\sqrt{3}\\over 2\\)cos x – \\(1\\over 2\\)sin x = \\(1\\over 2\\)

\n \\(\\implies\\)   cos\\(\\pi\\over 6\\)cos x – sin\\(\\pi\\over 6\\)sin x = \\(1\\over 2\\)   \\(\\implies\\)   cos(x + \\(\\pi\\over 6\\)) = \\(1\\over 2\\)

\n As 0 \\(\\le\\) x \\(\\le\\) 3\\(\\pi\\)   \\(\\implies\\)   \\(\\pi\\over 6\\) \\(\\le\\) x + \\(\\pi\\over 6\\) \\(\\le\\) \\(3\\pi + {\\pi\\over 6}\\)

\n \\(\\implies\\)   x + \\(\\pi\\over 6\\) = \\(\\pi\\over 3\\), \\(5\\pi\\over 3\\), \\(7\\pi\\over 3\\)   \\(\\implies\\)   \\(\\pi\\over 6\\), \\(3\\pi\\over 2\\), \\(13\\pi\\over 6\\).

\n But at x = \\(3\\pi\\over 2\\), tan x and sec x is not defined.

\n \\(\\therefore\\)   Total number of solutions are 2.

<\/p>\n\n\n

(d) Solving trigonometric equations by transforming sum of trigonometric functions into product<\/strong><\/h4>\n\n\n

Example : <\/span> Solve : cos x + cos 3x + cos 5x + cos 7x = 0\n <\/p>\n

Solution : <\/span>We have cos x + cos 7x + cos 3x + cos 5x = 0

\n \\(\\implies\\)   2cos 4x cos 3x + 2cos 4x cos x = 0   \\(\\implies\\)   cos 4x(cos 3x + cos x) = 0

\n \\(\\implies\\)   cos 4x(2cos 2x cos x) = 0

\n \\(\\implies\\)   Either cos x = 0   \\(\\implies\\)   x = (\\(2n_1 + 1\\))\\(\\pi\\over 2\\), \\(n_1\\) \\(\\in\\) I

\n or   cos 2x = 0   \\(\\implies\\)   x = (\\(2n_2 + 1\\))\\(\\pi\\over 4\\), \\(n_2\\) \\(\\in\\) I

\n or   cos 4x = 0   \\(\\implies\\)   x = (\\(2n_3 + 1\\))\\(\\pi\\over 8\\), \\(n_3\\) \\(\\in\\) I

<\/p>\n\n\n

(e) Solving trigonometric equations by transforming a product into sum<\/strong><\/h4>\n\n\n

Example : <\/span> Solve : cosx cos2x cos 3x = \\(1\\over 4\\)\n <\/p>\n

Solution : <\/span>(2cosx cos3x)cos2x = \\(1\\over 2\\)   \\(\\implies\\)   (cos2x + cos4x)cos2x = \\(1\\over 2\\)

\n \\(\\implies\\)   \\(1\\over 2\\)[\\(2cos^22x\\) + 2cos4x cos2x] = \\(1\\over 2\\)   \\(\\implies\\)   1 + cos4x + 2cos4xcos2x = 1

\n \\(\\therefore\\)   cos 4x(1 + 2cos2x) = 0

\n \\(\\implies\\)   cos 4x = 0   or   (1 + 2cos2x) = 0

\n Now from the first equation : cos 4x = 0 = cos(\\(\\pi\\over 2\\))

\n \\(\\therefore\\)   4x = (2n + 1)\\(\\pi\\over 2\\)   \\(\\implies\\)   x = (2n + 1)\\(\\pi\\over 8\\), n \\(\\in\\) I

\n for   n = 0, x = \\(\\pi\\over 8\\); n = 1, x = \\(3\\pi\\over 8\\); n = 2, x = \\(5\\pi\\over 8\\); n = 3, x = \\(7\\pi\\over 8\\);

\n and from the second equation : cos 2x = -\\(1\\over 2\\) = -cos(\\(\\pi\\over 3\\)) = cos(\\(\\pi-{\\pi\\over 3}\\)) = cos(\\(2\\pi\\over 3\\))

\n \\(\\therefore\\)   2x = 2k\\(\\pi\\) \\(\\pm\\) \\(2\\pi\\over 3\\)   \\(\\therefore\\)   x = k\\(\\pi\\) \\(\\pm\\) \\(\\pi\\over 3\\), k \\(\\in\\) I

\n again for k = 0, x = \\(\\pi\\over 3\\); k = 1, x = \\(2\\pi\\over 3\\)   (\\(\\because\\) 0 \\(\\le\\) x \\(\\le\\) \\(\\pi\\))

\n \\(\\therefore\\)   x = \\(\\pi\\over 8\\), \\(\\pi\\over 3\\), \\(3\\pi\\over 8\\), \\(5\\pi\\over 8\\), \\(2\\pi\\over 3\\), \\(7\\pi\\over 8\\)

<\/p>\n\n\n

Trigonometric Inequalities<\/h2>\n

There is no general rule to solve trigonometric inequalities and the same rules of algebra are valid provided the domain and range of trigonometric function should be kept in mind.<\/p>\n\n\n

Example : <\/span> Find the solution set of inequality sin x > 1\/2.\n <\/p>\n

Solution : <\/span>when sin x = \\(1\\over 2\\), the two values of x between 0 and \\(2\\pi\\) are \\(pi\\over 6\\) and \\(5pi\\over 6\\).

\n From the graph of y = sinx, it is obvious that it is between 0 and \\(2\\pi\\),

\n sinx > \\(1\\over 2\\) for \\(\\pi\/6\\) < x < \\(5\\pi\/6\\)

\n Hence, sinx > 1\/2

\n \\(\\implies\\) \\(2n\\pi\\) + \\(\\pi\\over 6\\) < x < \\(2n\\pi\\) + \\(5\\pi\\over 6\\), n \\(\\in\\) I

<\/p>\n\n\n\n

\n
Previous – General Solution of Trigonometric Equation<\/a><\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

Here, you will learn trigonometric equations solving strategies i.e by factorisation and reducing it to a quadratic equation etc. Let’s begin – Different Strategies for Trigonometric Equations Solving (a) Solving trigonometric equations by factorisation Example : If \\(1\\over 6\\)sin x, cos x, tan x are in G.P. then the general solution for x is Solution …<\/p>\n

Trigonometric Equations Solving Strategies<\/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":[29],"tags":[],"yoast_head":"\nTrigonometric Equations Solving Strategies - Mathemerize<\/title>\n<meta name=\"description\" content=\"In this post, you will learn trigonometric equations solving strategies i.e by factorisation and reducing it to a quadratic equation etc.\" \/>\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\/trigonometric-equations-solving-strategies\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Trigonometric Equations Solving Strategies - 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