{"id":2567,"date":"2021-07-12T21:56:17","date_gmt":"2021-07-12T21:56:17","guid":{"rendered":"https:\/\/mathemerize.com\/?p=2567"},"modified":"2022-01-13T23:04:37","modified_gmt":"2022-01-13T17:34:37","slug":"general-solution-of-trigonometric-equation","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/general-solution-of-trigonometric-equation\/","title":{"rendered":"How to Find General Solution of Trigonometric Equation"},"content":{"rendered":"

Here, you will learn what is trigonometric equation and how to find general solution of trigonometric equation with examples.<\/p>\n

Let’s begin –<\/p>\n

An equation involving one or more trigonometrical ratios of unknown angles is called a trigonometrical equation.<\/p>\n

Solution of Trigonometric Equation<\/h2>\n

A value of the unknown angle which satisfies the given equation is called a solution of the trigonometric equation.<\/p>\n

(a) Principal solution :-<\/strong> The solution of the trigonometric equation lying in the interval [0, \\(2\\pi\\)).<\/p>\n

(b) General Solution :-\u00a0<\/strong>Since all the trigonometric functions are many one & periodic, hence there are infinite values values of 0 for which trigonometric functions have the same value. All such possible values of 0 for which the given trigonometric function is satisfied is given by a general formula. Such a general formula is called general solution of trigonometric equation.<\/p>\n

(c) Particular Solution<\/strong> :- The solution of the trigonometric equation lying in the given interval.<\/p>\n

General Solution of Trigonometric Equation<\/h2>\n
\n

(a)<\/strong> If sin \\(\\theta\\) = 0, then \\(\\theta\\) = n\\(\\pi\\), n \\(\\in\\) I (set of integers)<\/p>\n

(b)<\/strong> If cos \\(\\theta\\) = 0, then \\(\\theta\\) = (2n+1)\\(\\pi\\over 2\\), n \\(\\in\\) I<\/p>\n

(c)<\/strong> If tan \\(\\theta\\) = 0, then \\(\\theta\\) = n\\(\\pi\\), n \\(\\in\\) I<\/p>\n

(d)\u00a0<\/strong>If cot \\(\\theta\\) = 0, then \\(\\theta\\) = (2n+1)\\(\\pi\\over 2\\), n \\(\\in\\) I<\/p>\n

Note :<\/strong> Since sec \\(\\theta\\) \\(\\ge\\) 1 or sec \\(\\theta\\) \\(\\le\\) 1, therefore sec \\(\\theta\\) = 0 does not have any solution.<\/p>\n

Similarly, cosec \\(\\theta\\) = 0 has no solution.<\/p>\n<\/blockquote>\n\n\n

Example : <\/span>Find the general solution of trigonometric equation :
\n(i) \\(sin2\\theta\\) = 0
\n(ii) \\(tan{3\\theta\\over 4}\\) = 0<\/p>\n

Solution : <\/span>
\n(i) \\(sin2\\theta\\) = 0

\n\\(\\implies\\) \\(2\\theta\\) = \\(n\\pi\\), where n \\(\\in\\) Z     [sin \\(\\theta\\) = 0, then \\(\\theta\\) = n\\(\\pi\\)]

\n\\(\\implies\\) \\(\\theta\\) = \\(n\\pi\\over 2\\), where n \\(\\in\\) Z

\n(ii) \\(tan{3\\theta\\over 4}\\) = 0

\n\\(\\implies\\) \\(3\\theta\\over 4\\) = \\(n\\pi\\), where n \\(\\in\\) Z     [tan \\(\\theta\\) = 0, then \\(\\theta\\) = n\\(\\pi\\)]

\n\\(\\implies\\) \\(\\theta\\) = \\(4n\\pi\\over 3\\), where n \\(\\in\\) Z<\/p>\n\n\n

\n

(d)<\/strong> If sin \\(\\theta\\) = sin \\(\\alpha\\), then \\(\\theta\\) = n\\(\\pi\\) + \\({(-1)}^n\\alpha\\), where \\(\\alpha\\) \\(\\in\\) [-\\(\\pi\\over 2\\), \\(\\pi\\over 2\\)], n \\(\\in\\) I<\/p>\n

(e)<\/strong> cos \\(\\theta\\) = cos \\(\\alpha\\), then \\(\\theta\\) = 2n\\(\\pi\\) \\(\\pm\\) \\(\\alpha\\), where \\(\\alpha\\) \\(\\in\\) [0,\\(\\pi\\)], n \\(\\in\\) I<\/p>\n

(f)<\/strong> tan \\(\\theta\\) = tan \\(\\alpha\\), then \\(\\theta\\) = n\\(\\pi\\) + \\(\\alpha\\), where \\(\\alpha\\) \\(\\in\\) (-\\(\\pi\\over 2\\), \\(\\pi\\over 2\\)), n \\(\\in\\) I<\/p>\n<\/blockquote>\n\n\n

Example : <\/span>Find the general solution of trigonometric equation :
\n(i) \\(sin\\theta\\) = \\(\\sqrt{3}\\over 2\\)
\n(ii) \\(cos3\\theta\\) = \\(-1\\over 2\\)<\/p>\n

Solution : <\/span>
\n(i) A value of \\(\\theta\\) satisfying \\(sin\\theta\\) = \\(\\sqrt{3}\\over 2\\) is \\(\\pi\\over 3\\)

\n\\(sin\\theta\\) = \\(\\sqrt{3}\\over 2\\)

\n \\(\\implies\\) \\(sin\\theta\\) = \\(sin\\pi\\over 3\\) \\(\\implies\\) \\(\\theta\\) = n\\(\\pi\\) + \\({(-1)}^n{\\pi\\over 3}\\),

\n(ii) \\(cos3\\theta\\) = \\(-1\\over 2\\)

\n\\(\\implies\\) \\(cos3\\theta\\) = \\(cos{2\\pi\\over 3}\\) \\(\\implies\\) \\(3\\theta\\) = 2n\\(\\pi\\) \\(\\pm\\) \\(2\\pi\\over 3\\)

\n\\(\\implies\\) \\(\\theta\\) = 2n\\(\\pi\\over 3\\) \\(\\pm\\) \\(2\\pi\\over 9\\)<\/p>\n\n\n

\n

(g) <\/strong>If sin \\(\\theta\\) = 1, then \\(\\theta\\) = 2n\\(\\pi\\) + \\(\\pi\\over 2\\) = (4n + 1)\\(\\pi\\over 2\\), n \\(\\in\\) I<\/p>\n

(h)\u00a0<\/strong>If cos \\(\\theta\\) = 1, then \\(\\theta\\) = 2n\\(\\pi\\), n \\(\\in\\) I<\/p>\n

(i) <\/strong>If \\(sin^2\\theta\\) = \\(sin^2\\alpha\\) or \\(cos^2\\theta\\) = \\(cos^2\\alpha\\) or \\(tan^2\\theta\\) = \\(tan^2\\alpha\\), then \\(\\theta\\) = n\\(\\pi\\) \\(\\pm\\) \\(\\alpha\\), n \\(\\in\\) I<\/p>\n<\/blockquote>\n\n\n

Example : <\/span>Find the general solution of trigonometric equation \\(7cos^2\\theta\\) + \\(3sin^2\\theta\\) = 4
\n<\/p>\n

Solution : <\/span>We have,

\n\\(7cos^2\\theta\\) + \\(3sin^2\\theta\\) = 4

\n\\(7(1-sin^2\\theta)\\) + \\(3sin^2\\theta\\) = 4

\n\\(4sin^2\\theta\\) = 3

\n\\(sin^2\\theta\\) = \\(3\\over 4\\) = \\(({\\sqrt{3}\\over 2})^2\\)

\n\\(sin^2\\theta\\) = \\(sin^2{\\pi\\over 3}\\) \\(\\implies\\) \\(\\theta\\) = \\(n\\pi\\pm{\\pi\\over 3}\\).<\/p>\n\n\n\n

\n
Next – Trigonometric Equations Solving Strategies<\/a><\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

Here, you will learn what is trigonometric equation and how to find general solution of trigonometric equation with examples. Let’s begin – An equation involving one or more trigonometrical ratios of unknown angles is called a trigonometrical equation. Solution of Trigonometric Equation A value of the unknown angle which satisfies the given equation is called …<\/p>\n

How to Find General Solution of Trigonometric Equation<\/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":[682,683],"yoast_head":"\nHow to Find General Solution of Trigonometric Equation<\/title>\n<meta name=\"description\" content=\"Learn what is trigonometric equation and how to find general solution of trigonometric equation with 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