{"id":7768,"date":"2022-09-18T05:17:55","date_gmt":"2022-09-18T05:17:55","guid":{"rendered":"https:\/\/mdr.foobrdigital.com\/?p=7768"},"modified":"2022-09-18T05:17:55","modified_gmt":"2022-09-18T05:17:55","slug":"solving-trigonometric-equations","status":"publish","type":"post","link":"https:\/\/mudassirbackup.infinitycodestudio.com\/index.php\/2022\/09\/18\/solving-trigonometric-equations\/","title":{"rendered":"Solving Trigonometric Equations"},"content":{"rendered":"\n<p>Unlike normal solutions of algebraic equations with the number of solutions based on the degree of the variable, in trigonometric equations, the solutions are of two types, based on the different value of angle for the\u00a0trigonometric function, for the same solution. For example, for a simple trigonometric equation 2Cos\u03b8 &#8211; 1 = 0, the solution is given by, Cos\u03b8 = 1\/2 and, the \u03b8 values are \u03c0\/3, 5\u03c0\/3, 7\u03c0\/3, 11\u03c0\/3, and so on as the values of the\u00a0cosine function\u00a0repeat after every 2\u03c0 radians and cos x is positive in the first and fourth quadrants. We have two types of solutions to the trigonometric equations:<\/p>\n\n\n\n<ul><li><strong>Principal Solution:<\/strong>\u00a0The initial values of angles for the trigonometric functions are referred to as principal solutions. The solution of Sinx and Cosx repeat after an interval of 2\u03c0, and the solution of Tanx repeat after an interval of \u03c0. The solutions of these trigonometric equations, for which x lies between 0 and 2\u03c0, are called principal solutions.<\/li><li><strong>General Solution:<\/strong>\u00a0The values of the\u00a0angles\u00a0for the same answer of the trigonometric function are referred to as the general solution of the trigonometric function. The solutions of trigonometric equations beyond 2\u03c0 are all consolidated and expressed as a general solution of the trigonometric equations. The general solutions of Sin\u03b8, Cos\u03b8, Tan\u03b8 are as follows.<ul><li>Sin\u03b8 = Sin\u03b1, and the general solution is \u03b8 = n\u03c0 + (-1)<sup>n<\/sup>\u03b1, where n \u2208 Z<\/li><li>Cos\u03b8 = Cos\u03b1, and the general solution is \u03b8 = 2n\u03c0\u00a0+\u00a0\u03b1, where n \u2208 Z<\/li><li>Tan\u03b8 = Tan\u03b1, and the general solution is \u03b8 = n\u03c0 + \u03b1, where n \u2208 Z<\/li><\/ul><\/li><\/ul>\n\n\n\n<h2 class=\"wp-block-heading\">Steps to Solve Trigonometric Equations<\/h2>\n\n\n\n<p>The following steps are to be followed, for solving a trigonometric equation.<\/p>\n\n\n\n<ul><li>Transform the given trigonometric equation into an equation with a single trigonometric ratio (sin, cos, tan)<\/li><li>Change the equation with the trigonometric equation, having multiple angles, or submultiple angles into a simple angle.<\/li><li>Now represent the equation as a\u00a0polynomial equation, quadratic equation, or linear equation.<\/li><li>Solve the trigonometric equation similar to normal equations, and find the value of the trigonometric ratio.<\/li><li>The angle of the trigonometric ratio or the value of the trigonometric ratio represents the solution of the trigonometric equation.<\/li><\/ul>\n\n\n\n<h2 class=\"wp-block-heading\">Examples of Solving Trigonometric Equations<\/h2>\n\n\n\n<p><strong>Example 1:&nbsp;<\/strong>Find the principal solutions of the trigonometric equation sin x = \u221a3\/2.<\/p>\n\n\n\n<p><strong>Solution:&nbsp;<\/strong>To find the principal solutions of sin x = \u221a3\/2, we know that sin \u03c0\/3 = \u221a3\/2 and sin (\u03c0 &#8211; \u03c0\/3) = \u221a3\/2<\/p>\n\n\n\n<p>\u21d2 sin \u03c0\/3 = sin 2\u03c0\/3 = \u221a3\/2<\/p>\n\n\n\n<p>We can find other values of x such that sin x = \u221a3\/2, but we need to find only those values of x such that x lies in [0, 2\u03c0] because a principal solution lies between 0 and 2\u03c0.<\/p>\n\n\n\n<p>So, the principal solutions of sin x = \u221a3\/2 are x = \u03c0\/3 and 2\u03c0\/3.<\/p>\n\n\n\n<p><strong>Example 2:&nbsp;<\/strong>Find the solution of cos x = 1\/2.<\/p>\n\n\n\n<p><strong>Solution:&nbsp;<\/strong>In this case, we will find the general solution of cos x = 1\/2. We know that cos \u03c0\/3 = 1\/2, so we have<\/p>\n\n\n\n<p>cos x = 1\/2<\/p>\n\n\n\n<p>\u21d2 cos x = cos \u03c0\/3<\/p>\n\n\n\n<p>\u21d2 x = 2n\u03c0&nbsp;+&nbsp;(\u03c0\/3), where n \u2208 Z &#8212;- [Using Cos\u03b8 = Cos\u03b1, and the general solution is \u03b8 = 2n\u03c0&nbsp;+&nbsp;\u03b1, where n \u2208 Z]<\/p>\n\n\n\n<p>Therefore, the general solution of cos x = 1\/2 is x = 2n\u03c0&nbsp;+&nbsp;(\u03c0\/3), where n \u2208 Z.<\/p>\n\n\n\n<p><strong>Important Notes on Trigonometric Equations<\/strong><\/p>\n\n\n\n<ul><li>For any real numbers x and y, sin x = sin y implies x = n\u03c0 + (-1)<sup>n<\/sup>y, where n \u2208 Z.<\/li><li>For any real numbers x and y, cos x = cos y implies x = 2n\u03c0 \u00b1 y, where n \u2208 Z.<\/li><li>If x and y are not odd multiples of \u03c0\/2, then tan x = tan y implies x = n\u03c0 + y, where n \u2208 Z.<\/li><li>sin A = 0 implies A = n\u03c0 and cos A = 0 implies A = (2n + 1)\u03c0\/2, where n \u2208 Z<\/li><\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Unlike normal solutions of algebraic equations with the number of solutions based on the degree of the variable, in trigonometric equations, the solutions are of two types, based on the different value of angle for the\u00a0trigonometric function, for the same solution. For example, for a simple trigonometric equation 2Cos\u03b8 &#8211; 1 = 0, the solution [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[384],"tags":[],"_links":{"self":[{"href":"https:\/\/mudassirbackup.infinitycodestudio.com\/index.php\/wp-json\/wp\/v2\/posts\/7768"}],"collection":[{"href":"https:\/\/mudassirbackup.infinitycodestudio.com\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mudassirbackup.infinitycodestudio.com\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mudassirbackup.infinitycodestudio.com\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/mudassirbackup.infinitycodestudio.com\/index.php\/wp-json\/wp\/v2\/comments?post=7768"}],"version-history":[{"count":0,"href":"https:\/\/mudassirbackup.infinitycodestudio.com\/index.php\/wp-json\/wp\/v2\/posts\/7768\/revisions"}],"wp:attachment":[{"href":"https:\/\/mudassirbackup.infinitycodestudio.com\/index.php\/wp-json\/wp\/v2\/media?parent=7768"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mudassirbackup.infinitycodestudio.com\/index.php\/wp-json\/wp\/v2\/categories?post=7768"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mudassirbackup.infinitycodestudio.com\/index.php\/wp-json\/wp\/v2\/tags?post=7768"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}