{"id":143836,"date":"2022-04-16T14:43:07","date_gmt":"2022-04-16T09:13:07","guid":{"rendered":"https:\/\/www.aakash.ac.in\/blog\/?p=143836"},"modified":"2023-04-02T18:52:16","modified_gmt":"2023-04-02T13:22:16","slug":"jee-main-maths-concept-notes-from-trigonometry-for-exam-revision","status":"publish","type":"post","link":"https:\/\/www.aakash.ac.in\/blog\/jee-main-maths-concept-notes-from-trigonometry-for-exam-revision\/","title":{"rendered":"JEE Main 2022 Maths: Concept notes from Trigonometry for exam revision"},"content":{"rendered":"<p><a href=\"https:\/\/www.aakash.ac.in\/jee-mains-results?utm_source=seobanner&amp;utm_medium=jeemain&amp;utm_campaign=jeemain2023result\" target=\"_blank\" rel=\"noopener\"><img decoding=\"async\" src=\"https:\/\/d20x1nptavktw0.cloudfront.net\/wordpress_media\/2023\/02\/1300x420-1140x368.jpg\" alt=\"jee main exam\" width=\"100%\" data-entity-type=\"file\" data-entity-uuid=\"d4e023ef-9ff8-4b8f-b582-b9892a7d2953\" \/><\/a><br \/>\n<span style=\"font-weight: 400;\">In Mathematics, trigonometry is one of the most fundamentally used concepts. The usage of trigonometry is crucial in real life apart from the textbook. The JEE Main 2022 aspirants are motivated by their tutors to study this chapter well to obtain the best possible marks. This chapter might seem hard, but once the students are used to the formulae and identities, it will be a simple task.<\/span><\/p>\n<p><span style=\"font-weight: 400;\">In this article, we will be talking about<a href=\"https:\/\/www.aakash.ac.in\/important-concepts\/maths\/trigonometry\" target=\"_blank\" rel=\"noopener\"> trigonometry<\/a> in general. We will have a detailed understanding of all the related concepts and formulae and revision notes for students to secure maximum marks<\/span><\/p>\n<h3>Trigonometry &#8211; Definition<\/h3>\n<p><span style=\"font-weight: 400;\">Trigonometry is a term that comes from a Greek word, which means measurement. In simple words, it can be described as the term used in measuring right-angled triangles. There are certain numbers of trigonometric ratios which can be obtained from the sides of the triangle.\u00a0<\/span><\/p>\n<p><span style=\"font-weight: 400;\">Multiple trigonometric ratios can be easily formed by utilising both the length and angle of the right-angled triangles. Theta is the angle between two sides and can be represented as. The following are the basic trigonometric ratio forms:<\/span><\/p>\n<ul>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">Sine<\/span><\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">Cosine<\/span><\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">Secant<\/span><\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">Cosecant<\/span><\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">Tangent<\/span><\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">Cotangent<\/span><\/li>\n<\/ul>\n<h3>Basic trigonometric ratios<\/h3>\n<ul>\n<li aria-level=\"1\"><b>Sine:<\/b><\/li>\n<\/ul>\n<p><span style=\"font-weight: 400;\">Sine, abbreviated as sin, is a basic trigonometric function. It can be defined as the ratio of the opposite side of the right-angled triangle to its hypotenuse side. Its derivation is,<\/span><\/p>\n<p><span style=\"font-weight: 400;\">sin<\/span> <span style=\"font-weight: 400;\">\u00a0= <\/span><span style=\"font-weight: 400;\">Opposite side<\/span><span style=\"font-weight: 400;\">Hypotenuse side<\/span><\/p>\n<ul>\n<li aria-level=\"1\"><b>Cosine:<\/b><\/li>\n<\/ul>\n<p><span style=\"font-weight: 400;\">The second basic form of a trigonometric function is cosine, which can be represented as cos. The sine and cosine functions are considered the fundamentals which are capable of deriving the rest of the basic trigonometric functions such as secant, cosecant, tangent and cotangent.\u00a0<\/span><\/p>\n<p><span style=\"font-weight: 400;\">Cosine is the ratio of the right-angled triangle&#8217;s base length to its hypotenuse side. It can be represented as mentioned below:<\/span><\/p>\n<p><span style=\"font-weight: 400;\">cos<\/span> <span style=\"font-weight: 400;\">\u00a0= <\/span><span style=\"font-weight: 400;\">Adjacent side<\/span><span style=\"font-weight: 400;\">Hypotenuse side<\/span><\/p>\n<ul>\n<li aria-level=\"1\"><b>Tangent:<\/b><\/li>\n<\/ul>\n<p><span style=\"font-weight: 400;\">The derivation of the trigonometric function tangent is from the other basic trigonometric functions such as sine and cosine. The abbreviation of this is tan and is expressed as the ratio between sine and cosine.<\/span><\/p>\n<p><span style=\"font-weight: 400;\">Tan can be described as the opposite side of the right-angled triangle to its adjacent side. The following is the mathematical representation of the same:<\/span><\/p>\n<p><span style=\"font-weight: 400;\">tan<\/span> <span style=\"font-weight: 400;\">\u00a0= <\/span><span style=\"font-weight: 400;\">Opposite side<\/span><span style=\"font-weight: 400;\">Adjacent side<\/span><\/p>\n<p><span style=\"font-weight: 400;\">In JEE Main, all these sine, cosine and tangent functions are referred to as basic trigonometric ratios.\u00a0<\/span><\/p>\n<h3>Derived trigonometric ratios<\/h3>\n<p><span style=\"font-weight: 400;\">Derived trigonometric functions are nothing but the ratios which are derived from the basic functions by inverting them. Therefore, the aspirants of JEE should be aware of this topic as well.<\/span><\/p>\n<ul>\n<li aria-level=\"1\"><b>Cosecant:<\/b><\/li>\n<\/ul>\n<p><span style=\"font-weight: 400;\">Cosecant, also called cosec, can be obtained from sine. It is nothing but sine&#8217;s multiplicative inverse. Cosec is written as the ratio between the hypotenuse side and the opposite side of the right-angled triangle.\u00a0<\/span><\/p>\n<p><span style=\"font-weight: 400;\">cosec \u03b8 = <\/span><span style=\"font-weight: 400;\">Hypotenuse side<\/span><span style=\"font-weight: 400;\">Opposite side<\/span><\/p>\n<ul>\n<li aria-level=\"1\"><b>Secant:<\/b><\/li>\n<\/ul>\n<p><span style=\"font-weight: 400;\">While inversing the basic trigonometric function, cosine, another function is formed and is termed as secant. Secant can be abbreviated as sec. It is defined as the ratio between the hypotenuse side and the adjacent side of the right-angled triangle.<\/span><\/p>\n<p><span style=\"font-weight: 400;\">sec<\/span> <span style=\"font-weight: 400;\">\u00a0= <\/span><span style=\"font-weight: 400;\">Hypotenuse side<\/span><span style=\"font-weight: 400;\">Adjacent side<\/span><\/p>\n<ul>\n<li aria-level=\"1\"><b>Cotangent:<\/b><\/li>\n<\/ul>\n<p><span style=\"font-weight: 400;\">Cotangent can be obtained using the multiplicative inverse of the tangent. A cotangent can be represented as a cot. It is nothing but the ratio of the adjacent side to the opposite side of the right-angled triangle.<\/span><\/p>\n<p><span style=\"font-weight: 400;\">cot<\/span> <span style=\"font-weight: 400;\">= <\/span><span style=\"font-weight: 400;\">Adjacent side<\/span><span style=\"font-weight: 400;\">Opposite side<\/span><\/p>\n<h3>Trigonometric table<\/h3>\n<p><span style=\"font-weight: 400;\">To learn <a href=\"https:\/\/www.youtube.com\/watch?v=t0Hsc_iCZNU\" target=\"_blank\" rel=\"noopener\">trigonometry<\/a>, the JEE 2022 aspirants should be able to know the ratios at standard angles. So, the students need to have the table stored in their memory. Apart from mathematical functions, this table is also helpful for solving Chemistry and Physics questions. The students are advised to follow it precisely.<\/span><\/p>\n<p><span style=\"font-weight: 400;\">The below-mentioned is the table for the same:<\/span><\/p>\n<table>\n<tbody>\n<tr>\n<td><b>Angles (In Degrees)<\/b><\/td>\n<td><b>0 \u00b0<\/b><\/td>\n<td><b>30 \u00b0<\/b><\/td>\n<td><b>45 \u00b0<\/b><\/td>\n<td><b>60 \u00b0<\/b><\/td>\n<td><b>90 \u00b0<\/b><\/td>\n<td><b>180 \u00b0<\/b><\/td>\n<td><b>270 \u00b0<\/b><\/td>\n<td><b>360 \u00b0<\/b><\/td>\n<\/tr>\n<tr>\n<td><b>Angles (In Radians)<\/b><\/td>\n<td><b>0<\/b><\/td>\n<td><b>\u03c0\/6<\/b><\/td>\n<td><b>\u03c0\/4<\/b><\/td>\n<td><b>\u03c0\/3<\/b><\/td>\n<td><b>\u03c0\/2<\/b><\/td>\n<td><b>\u03c0<\/b><\/td>\n<td><b>3\u03c0\/2<\/b><\/td>\n<td><b>2\u03c0<\/b><\/td>\n<\/tr>\n<tr>\n<td><span style=\"font-weight: 400;\">sin<\/span><\/td>\n<td><span style=\"font-weight: 400;\">0<\/span><\/td>\n<td><span style=\"font-weight: 400;\">1\/2<\/span><\/td>\n<td><span style=\"font-weight: 400;\">1\/\u221a2<\/span><\/td>\n<td><span style=\"font-weight: 400;\">\u221a3\/2<\/span><\/td>\n<td><span style=\"font-weight: 400;\">1<\/span><\/td>\n<td><span style=\"font-weight: 400;\">0<\/span><\/td>\n<td><span style=\"font-weight: 400;\">-1<\/span><\/td>\n<td><span style=\"font-weight: 400;\">0<\/span><\/td>\n<\/tr>\n<tr>\n<td><span style=\"font-weight: 400;\">cos<\/span><\/td>\n<td><span style=\"font-weight: 400;\">1<\/span><\/td>\n<td><span style=\"font-weight: 400;\">\u221a3\/2<\/span><\/td>\n<td><span style=\"font-weight: 400;\">1\/\u221a2<\/span><\/td>\n<td><span style=\"font-weight: 400;\">1\/2<\/span><\/td>\n<td><span style=\"font-weight: 400;\">0<\/span><\/td>\n<td><span style=\"font-weight: 400;\">-1<\/span><\/td>\n<td><span style=\"font-weight: 400;\">0<\/span><\/td>\n<td><span style=\"font-weight: 400;\">1<\/span><\/td>\n<\/tr>\n<tr>\n<td><span style=\"font-weight: 400;\">tan<\/span><\/td>\n<td><span style=\"font-weight: 400;\">0<\/span><\/td>\n<td><span style=\"font-weight: 400;\">1\/\u221a3<\/span><\/td>\n<td><span style=\"font-weight: 400;\">1<\/span><\/td>\n<td><span style=\"font-weight: 400;\">\u221a3<\/span><\/td>\n<td><span style=\"font-weight: 400;\">\u221e<\/span><\/td>\n<td><span style=\"font-weight: 400;\">0<\/span><\/td>\n<td><span style=\"font-weight: 400;\">\u221e<\/span><\/td>\n<td><span style=\"font-weight: 400;\">0<\/span><\/td>\n<\/tr>\n<tr>\n<td><span style=\"font-weight: 400;\">cot<\/span><\/td>\n<td><span style=\"font-weight: 400;\">\u221e<\/span><\/td>\n<td><span style=\"font-weight: 400;\">\u221a3<\/span><\/td>\n<td><span style=\"font-weight: 400;\">1<\/span><\/td>\n<td><span style=\"font-weight: 400;\">1\/\u221a3<\/span><\/td>\n<td><span style=\"font-weight: 400;\">0<\/span><\/td>\n<td><span style=\"font-weight: 400;\">\u221e<\/span><\/td>\n<td><span style=\"font-weight: 400;\">0<\/span><\/td>\n<td><span style=\"font-weight: 400;\">\u221e<\/span><\/td>\n<\/tr>\n<tr>\n<td><span style=\"font-weight: 400;\">cosec<\/span><\/td>\n<td><span style=\"font-weight: 400;\">\u221e<\/span><\/td>\n<td><span style=\"font-weight: 400;\">2<\/span><\/td>\n<td><span style=\"font-weight: 400;\">\u221a2<\/span><\/td>\n<td><span style=\"font-weight: 400;\">2\/\u221a3<\/span><\/td>\n<td><span style=\"font-weight: 400;\">1<\/span><\/td>\n<td><span style=\"font-weight: 400;\">\u221e<\/span><\/td>\n<td><span style=\"font-weight: 400;\">-1<\/span><\/td>\n<td><span style=\"font-weight: 400;\">\u221e<\/span><\/td>\n<\/tr>\n<tr>\n<td><span style=\"font-weight: 400;\">sec<\/span><\/td>\n<td><span style=\"font-weight: 400;\">1<\/span><\/td>\n<td><span style=\"font-weight: 400;\">2\/\u221a3<\/span><\/td>\n<td><span style=\"font-weight: 400;\">\u221a2<\/span><\/td>\n<td><span style=\"font-weight: 400;\">2<\/span><\/td>\n<td><span style=\"font-weight: 400;\">\u221e<\/span><\/td>\n<td><span style=\"font-weight: 400;\">-1<\/span><\/td>\n<td><span style=\"font-weight: 400;\">\u221e<\/span><\/td>\n<td><span style=\"font-weight: 400;\">1<\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h3>Basic trigonometric identities<\/h3>\n<p><span style=\"font-weight: 400;\">The basic trigonometric identities are usually asked indirectly in the JEE Main examinations. The problems can sometimes be complex in nature. However, a regular practice on this topic would be beneficial in remembering during the time of examination. The following are the identities derived from basic trigonometric ratios:<\/span><\/p>\n<ul>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">sin <\/span><span style=\"font-weight: 400;\">2 (a)<\/span> <span style=\"font-weight: 400;\">\u00a0+cos 2 (a)<\/span> <span style=\"font-weight: 400;\">\u00a0= 1<\/span><\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">1 + cot<\/span> <span style=\"font-weight: 400;\">\u00a02 <\/span><span style=\"font-weight: 400;\">a<\/span> <span style=\"font-weight: 400;\">\u00a0= cosec 2 <\/span><span style=\"font-weight: 400;\">a<\/span><\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">1 +tan 2 <\/span><span style=\"font-weight: 400;\">a<\/span> <span style=\"font-weight: 400;\">\u00a0=sec 2 <\/span><span style=\"font-weight: 400;\">a<\/span><\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">cosec <\/span><span style=\"font-weight: 400;\">a<\/span><span style=\"font-weight: 400;\"> &#8211; cot<\/span> <span style=\"font-weight: 400;\">\u00a0<\/span><span style=\"font-weight: 400;\">a<\/span> <span style=\"font-weight: 400;\">\u00a0<\/span><span style=\"font-weight: 400;\">= 1 cosec <\/span><span style=\"font-weight: 400;\">a<\/span><span style=\"font-weight: 400;\"> + cot<\/span> <span style=\"font-weight: 400;\">\u00a0<\/span><span style=\"font-weight: 400;\">a<\/span> <span style=\"font-weight: 400;\">\u00a0<\/span><\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">sin <\/span><span style=\"font-weight: 400;\">4 (a)<\/span> <span style=\"font-weight: 400;\">\u00a0+cos\u00a0 2 (a)<\/span> <span style=\"font-weight: 400;\">\u00a0= 1 &#8211; 2 cos 2 <\/span><span style=\"font-weight: 400;\">a<\/span><span style=\"font-weight: 400;\"> sin<\/span> <span style=\"font-weight: 400;\">2 (a)<\/span><\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">sec<\/span> <span style=\"font-weight: 400;\">\u00a0(a)<\/span> <span style=\"font-weight: 400;\">\u00a0&#8211; tan<\/span> <span style=\"font-weight: 400;\">\u00a0<\/span><span style=\"font-weight: 400;\">a<\/span> <span style=\"font-weight: 400;\">\u00a0= 1 sec <\/span><span style=\"font-weight: 400;\">a<\/span> <span style=\"font-weight: 400;\">\u00a0<\/span><span style=\"font-weight: 400;\">+ tan<\/span> <span style=\"font-weight: 400;\">\u00a0(a)<\/span><\/li>\n<\/ul>\n<h3>Trigonometric identities &#8211; Compound angles<\/h3>\n<p><span style=\"font-weight: 400;\">Consider an angle which is the sum or difference of the other two angles, whose values are unknown. In this case, to calculate the trigonometric ratio value, the aspirants must memorise the following identities, that consist of compound angles. It is used to solve any question simply. The below-mentioned are the standard trigonometric identities:<\/span><\/p>\n<ul>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">sin<\/span> <span style=\"font-weight: 400;\">\u00a0(K \u00b1 L)<\/span> <span style=\"font-weight: 400;\">\u00a0=sin K cos<\/span> <span style=\"font-weight: 400;\">L<\/span> <span style=\"font-weight: 400;\">\u00a0\u00b1 cos K sin<\/span> <span style=\"font-weight: 400;\">L<\/span><\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">cos<\/span> <span style=\"font-weight: 400;\">\u00a0<\/span><span style=\"font-weight: 400;\">K \u00b1 L<\/span> <span style=\"font-weight: 400;\">\u00a0=cos K cos<\/span> <span style=\"font-weight: 400;\">L<\/span> <span style=\"font-weight: 400;\">\u00a0\u2213 sin K<\/span><span style=\"font-weight: 400;\"> sin<\/span><span style=\"font-weight: 400;\"> L<\/span><\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">tan <\/span><span style=\"font-weight: 400;\">K + L<\/span> <span style=\"font-weight: 400;\">\u00a0<\/span><span style=\"font-weight: 400;\">=tan K<\/span> <span style=\"font-weight: 400;\">\u00a0\u00b1 tan L 1<\/span> <span style=\"font-weight: 400;\">\u00a0\u00b1 tan K<\/span> <span style=\"font-weight: 400;\">\u00a0\u2219 tan L<\/span><\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">cot <\/span><span style=\"font-weight: 400;\">K + L<\/span> <span style=\"font-weight: 400;\">\u00a0=cot K<\/span> <span style=\"font-weight: 400;\">\u00a0\u2219 cot L F<\/span> <span style=\"font-weight: 400;\">\u2213 1 cot L<\/span> <span style=\"font-weight: 400;\">\u00a0\u00b1 cot K<\/span><\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">sin<\/span> <span style=\"font-weight: 400;\">\u00a0<\/span><span style=\"font-weight: 400;\">K + L<\/span><span style=\"font-weight: 400;\"> sin<\/span> <span style=\"font-weight: 400;\">K &#8211; L<\/span> <span style=\"font-weight: 400;\">\u00a0=sin 2\u00a0 K <\/span><span style=\"font-weight: 400;\">\u00a0<\/span><span style=\"font-weight: 400;\">&#8211; sin 2 L <\/span><span style=\"font-weight: 400;\">\u00a0<\/span><span style=\"font-weight: 400;\">= cos 2\u00a0 L<\/span> <span style=\"font-weight: 400;\">\u00a0&#8211; cos 2 K<\/span><\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">cos<\/span> <span style=\"font-weight: 400;\">\u00a0<\/span><span style=\"font-weight: 400;\">K + L<\/span> <span style=\"font-weight: 400;\">\u00a0cos<\/span> <span style=\"font-weight: 400;\">\u00a0(K &#8211; L)<\/span> <span style=\"font-weight: 400;\">\u00a0= cos<\/span> <span style=\"font-weight: 400;\">\u00a02 K <\/span><span style=\"font-weight: 400;\">\u00a0<\/span><span style=\"font-weight: 400;\">&#8211; sin 2 L <\/span><span style=\"font-weight: 400;\">\u00a0<\/span><span style=\"font-weight: 400;\">=cos 2 L<\/span> <span style=\"font-weight: 400;\">\u00a0&#8211; sin<\/span> <span style=\"font-weight: 400;\">\u00a02 K<\/span><\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">sin<\/span> <span style=\"font-weight: 400;\">\u00a0(K + L + M) <\/span><span style=\"font-weight: 400;\">\u00a0<\/span><span style=\"font-weight: 400;\">= sin K cos<\/span> <span style=\"font-weight: 400;\">L cos<\/span> <span style=\"font-weight: 400;\">M <\/span><span style=\"font-weight: 400;\">\u00a0<\/span><span style=\"font-weight: 400;\">+ sin L cos<\/span> <span style=\"font-weight: 400;\">K cos<\/span> <span style=\"font-weight: 400;\">M<\/span> <span style=\"font-weight: 400;\">\u00a0+sin M<\/span><span style=\"font-weight: 400;\"> cos<\/span><span style=\"font-weight: 400;\"> K cos<\/span> <span style=\"font-weight: 400;\">L <\/span><span style=\"font-weight: 400;\">\u00a0<\/span><span style=\"font-weight: 400;\">&#8211; sin<\/span> <span style=\"font-weight: 400;\">\u00a0K<\/span><span style=\"font-weight: 400;\"> sin<\/span><span style=\"font-weight: 400;\"> L sin<\/span> <span style=\"font-weight: 400;\">M<\/span> <span style=\"font-weight: 400;\">\u00a0<\/span><\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">cos<\/span> <span style=\"font-weight: 400;\">\u00a0<\/span><span style=\"font-weight: 400;\">K + L + M<\/span> <span style=\"font-weight: 400;\">\u00a0=cos K<\/span><span style=\"font-weight: 400;\"> cos<\/span><span style=\"font-weight: 400;\"> L cos<\/span> <span style=\"font-weight: 400;\">M<\/span> <span style=\"font-weight: 400;\">&#8211; cos K<\/span><span style=\"font-weight: 400;\"> sin<\/span><span style=\"font-weight: 400;\"> L<\/span> <span style=\"font-weight: 400;\">sin M<\/span> <span style=\"font-weight: 400;\">\u00a0&#8211; cos<\/span> <span style=\"font-weight: 400;\">L<\/span><span style=\"font-weight: 400;\"> sin<\/span><span style=\"font-weight: 400;\"> K sin<\/span> <span style=\"font-weight: 400;\">M<\/span> <span style=\"font-weight: 400;\">\u00a0-cos M sin<\/span> <span style=\"font-weight: 400;\">K sin<\/span> <span style=\"font-weight: 400;\">L<\/span><\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">tan <\/span><span style=\"font-weight: 400;\">(K + L + M)<\/span> <span style=\"font-weight: 400;\">\u00a0=tan K + tan L<\/span> <span style=\"font-weight: 400;\">+tan M <\/span><span style=\"font-weight: 400;\">\u00a0<\/span><span style=\"font-weight: 400;\">&#8211; tan K<\/span><span style=\"font-weight: 400;\"> tan<\/span><span style=\"font-weight: 400;\"> L<\/span><span style=\"font-weight: 400;\"> tan<\/span><span style=\"font-weight: 400;\"> M 1 -tan K<\/span> <span style=\"font-weight: 400;\">\u00a0\u2219 tan L &#8211; tan L<\/span> <span style=\"font-weight: 400;\">\u00a0\u2219 tan M<\/span> <span style=\"font-weight: 400;\">-tan M \u2219 tan K<\/span><\/li>\n<\/ul>\n<h3>Trigonometric identities &#8211; Multiple angles<\/h3>\n<p><span style=\"font-weight: 400;\">In the JEE 2022 exam, there are a few questions which can be asked based on multiple angles. The following are the formulae:<\/span><\/p>\n<ul>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">sin 2 K = 2 sin K<\/span> <span style=\"font-weight: 400;\">cos<\/span> <span style=\"font-weight: 400;\">\u00a0K<\/span> <span style=\"font-weight: 400;\">\u00a0=<\/span><span style=\"font-weight: 400;\">\u00a0<\/span><span style=\"font-weight: 400;\">2K<\/span> <span style=\"font-weight: 400;\">1 + tan<\/span> <span style=\"font-weight: 400;\">\u00a02 K<\/span> <span style=\"font-weight: 400;\">\u00a0<\/span><\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">sin<\/span> <span style=\"font-weight: 400;\">3 K<\/span> <span style=\"font-weight: 400;\">\u00a0= 3 sin<\/span> <span style=\"font-weight: 400;\">\u00a0K<\/span> <span style=\"font-weight: 400;\">\u2013 4 sin 3 K<\/span><\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">cos <\/span><span style=\"font-weight: 400;\">2 K<\/span> <span style=\"font-weight: 400;\">\u00a0= 2cos<\/span> <span style=\"font-weight: 400;\">\u00a02 K<\/span> <span style=\"font-weight: 400;\">\u2013 1 = 1 \u2013 2 sin<\/span> <span style=\"font-weight: 400;\">\u00a02 K<\/span> <span style=\"font-weight: 400;\">\u00a0=(1 &#8211; tan<\/span> <span style=\"font-weight: 400;\">\u00a02 K)\/(1<\/span> <span style=\"font-weight: 400;\">\u00a0+ tan<\/span> <span style=\"font-weight: 400;\">2 K)=cos<\/span> <span style=\"font-weight: 400;\">\u00a02 K<\/span> <span style=\"font-weight: 400;\">\u00a0\u2013 <\/span><span style=\"font-weight: 400;\">sin<\/span> <span style=\"font-weight: 400;\">\u00a02 K<\/span><\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">cos<\/span> <span style=\"font-weight: 400;\">\u00a03 K<\/span> <span style=\"font-weight: 400;\">\u00a0= 4cos<\/span> <span style=\"font-weight: 400;\">\u00a03 K<\/span> <span style=\"font-weight: 400;\">\u00a0\u2013 3cos<\/span> <span style=\"font-weight: 400;\">\u00a0K<\/span><\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">tan<\/span> <span style=\"font-weight: 400;\">\u00a02 K<\/span> <span style=\"font-weight: 400;\">\u00a0=<\/span><span style=\"font-weight: 400;\">2 tan<\/span> <span style=\"font-weight: 400;\">\u00a0K<\/span> <span style=\"font-weight: 400;\">1 &#8211; 2 K<\/span><\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">tan<\/span> <span style=\"font-weight: 400;\">\u00a03 K<\/span> <span style=\"font-weight: 400;\">\u00a0= 3 tan<\/span> <span style=\"font-weight: 400;\">\u00a0K<\/span> <span style=\"font-weight: 400;\">\u00a0&#8211; <\/span><span style=\"font-weight: 400;\">tan 3 K<\/span> <span style=\"font-weight: 400;\">1 &#8211; 3 tan<\/span> <span style=\"font-weight: 400;\">\u00a02 K<\/span><\/li>\n<\/ul>\n<h3>Transformation formulae<\/h3>\n<ul>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">sin<\/span> <span style=\"font-weight: 400;\">\u00a0M<\/span> <span style=\"font-weight: 400;\">\u00a0+sin N<\/span> <span style=\"font-weight: 400;\">\u00a0= 2 sin<\/span> <span style=\"font-weight: 400;\">\u00a0(M + N 2) <\/span><span style=\"font-weight: 400;\">\u00a0<\/span><span style=\"font-weight: 400;\">cos<\/span> <span style=\"font-weight: 400;\">\u00a0(M &#8211; N 2)<\/span><\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">sin<\/span> <span style=\"font-weight: 400;\">\u00a0M<\/span> <span style=\"font-weight: 400;\">\u00a0\u2013 sin N<\/span> <span style=\"font-weight: 400;\">\u00a0= 2 cos<\/span> <span style=\"font-weight: 400;\">\u00a0(M + N 2)<\/span> <span style=\"font-weight: 400;\">sin<\/span> <span style=\"font-weight: 400;\">\u00a0(M &#8211; N 2)<\/span> <span style=\"font-weight: 400;\">\u00a0<\/span><\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">2 sin L cos K = sin (K + L) \u2013 sin (K \u2013 L)\u00a0<\/span><\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">2 cos K cos L=cos (K + L) + cos (K \u2013 L)\u00a0<\/span><\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">2 sin K sin L = cos (K \u2013 L) \u2013 cos (K + L)\u00a0<\/span><\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">2 cos<\/span> <span style=\"font-weight: 400;\">\u00a0K<\/span><span style=\"font-weight: 400;\"> cos<\/span><span style=\"font-weight: 400;\"> L<\/span> <span style=\"font-weight: 400;\">\u00a0=cos <\/span><span style=\"font-weight: 400;\">K + L<\/span> <span style=\"font-weight: 400;\">\u00a0<\/span><span style=\"font-weight: 400;\">+ cos <\/span><span style=\"font-weight: 400;\">K \u2013 L<\/span><\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">cos M \u2013 cos N = 2 cos (M + N 2) cos (M &#8211; N 2)\u00a0<\/span><\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">cos M \u2013 cos N = 2 sin (M + N 2) sin (N &#8211; M 2)<\/span><\/li>\n<\/ul>\n<h3>Trigonometric equation<\/h3>\n<p><span style=\"font-weight: 400;\">In trigonometric equations, n belongs to Z. Here Z means integers. The following the tabulation for the same:<\/span><\/p>\n<table>\n<tbody>\n<tr>\n<td><span style=\"font-weight: 400;\">Equation\u00a0<\/span><\/td>\n<td><span style=\"font-weight: 400;\">General solution<\/span><\/td>\n<\/tr>\n<tr>\n<td><span style=\"font-weight: 400;\">sin<\/span> <span style=\"font-weight: 400;\">\u00a0(a)<\/span> <span style=\"font-weight: 400;\">\u00a0= 0<\/span><\/td>\n<td><span style=\"font-weight: 400;\">n \u03c0<\/span><\/td>\n<\/tr>\n<tr>\n<td><span style=\"font-weight: 400;\">cos (a) <\/span><span style=\"font-weight: 400;\">\u00a0<\/span><span style=\"font-weight: 400;\">= 0<\/span><\/td>\n<td><span style=\"font-weight: 400;\">2 n + 1<\/span><span style=\"font-weight: 400;\"> \u03c0 2<\/span><\/td>\n<\/tr>\n<tr>\n<td><span style=\"font-weight: 400;\">tan (a)<\/span> <span style=\"font-weight: 400;\">\u00a0= 0<\/span><\/td>\n<td><span style=\"font-weight: 400;\">n \u03c0<\/span><\/td>\n<\/tr>\n<tr>\n<td><span style=\"font-weight: 400;\">sin (a)<\/span> <span style=\"font-weight: 400;\">\u00a0= sin (b)<\/span><\/td>\n<td><span style=\"font-weight: 400;\">n \u03c0 + <\/span><span style=\"font-weight: 400;\">&#8211; 1<\/span><span style=\"font-weight: 400;\"> n \u2219 b<\/span><\/td>\n<\/tr>\n<tr>\n<td><span style=\"font-weight: 400;\">cos<\/span> <span style=\"font-weight: 400;\">\u00a0<\/span><span style=\"font-weight: 400;\">a<\/span> <span style=\"font-weight: 400;\">\u00a0<\/span><span style=\"font-weight: 400;\">=cos\u00a0 (b)<\/span><\/td>\n<td><span style=\"font-weight: 400;\">2 n \u03c0 \u00b1 b<\/span><\/td>\n<\/tr>\n<tr>\n<td><span style=\"font-weight: 400;\">tan<\/span> <span style=\"font-weight: 400;\">\u00a0(a)<\/span> <span style=\"font-weight: 400;\">\u00a0= tan (b)<\/span><\/td>\n<td><span style=\"font-weight: 400;\">n \u03c0 + b<\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h3>Applications of trigonometry<\/h3>\n<p><span style=\"font-weight: 400;\">Trigonometry is being used in so many different fields in this world. Its real-world applications are enormous. The following are some of them:<\/span><\/p>\n<ul>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">The usage of trigonometry can be seen in criminology<\/span><\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">Marine biology acts as a great source in the usage of trigonometry<\/span><\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">In the aviation sector, trigonometry plays a major role<\/span><\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">Navigation needs basic trigonometry concepts to perform<\/span><\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\"><span style=\"font-weight: 400;\">It is very much helpful in measuring the height of mountains and buildings<\/span><\/li>\n<\/ul>\n<h3>Conclusion<\/h3>\n<p><span style=\"font-weight: 400;\">To conclude, since it is a fundamental concept, the students are taught trigonometry&#8217;s definition and its basic and derived trigonometric ratios. Moreover, a deep understanding of the trigonometric table is also being taught.\u00a0<\/span><\/p>\n<p><span style=\"font-weight: 400;\">\u00a0In addition to this, basic trigonometric identities, compound angles and multiple angles were also discussed. Lastly, transformation formulae and trigonometric equations are discussed concerning the formulae.<\/span><\/p>\n<p>ALSO READ:<\/p>\n<p><a href=\"https:\/\/www.aakash.ac.in\/blog\/introduction-to-trigonometry-ncert-solutions-for-cbse-class-10-maths-chapter-8\/\" target=\"_blank\" rel=\"noopener\"><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Introduction to Trigonometry, NCERT solutions for CBSE Class 10 Maths Chapter 8&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:14915,&quot;3&quot;:{&quot;1&quot;:0},&quot;4&quot;:{&quot;1&quot;:2,&quot;2&quot;:16777215},&quot;9&quot;:0,&quot;12&quot;:0,&quot;14&quot;:{&quot;1&quot;:2,&quot;2&quot;:3948611},&quot;15&quot;:&quot;Roboto, Arial, Helvetica, sans-serif&quot;,&quot;16&quot;:11}\">Introduction to Trigonometry, NCERT solutions for CBSE Class 10 Maths Chapter 8<\/span><\/a><\/p>\n<p><a href=\"https:\/\/www.aakash.ac.in\/blog\/trigonometry-formulas-identities-for-jee-main-cbse-12th-exam\/\" target=\"_blank\" rel=\"noopener\">Trigonometry Formulas &amp; Identities for JEE Main, CBSE 12th exam<\/a><\/p>\n<p><a href=\"https:\/\/www.aakash.ac.in\/blog\/basic-trigonometric-concept-notes-for-jee-aspirants\/\" target=\"_blank\" rel=\"noopener\">Basic Trigonometric Concept Notes For JEE Aspirants<\/a><\/p>\n","protected":false},"excerpt":{"rendered":"<p>In Mathematics, trigonometry is one of the most fundamentally used concepts. The usage of trigonometry is crucial in real life apart from the textbook. The JEE Main 2022 aspirants are motivated by their tutors to study this chapter well to obtain the best possible marks. This chapter might seem hard, but once the students are [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":137755,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1283],"tags":[1590,1865,57,2067],"class_list":["post-143836","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-aakash-engineering-coaching","tag-aakash-engineering","tag-engineering-exam-preparation","tag-jee-main","tag-jee-main-2022"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v26.0 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>JEE Main 2022 Maths: Concept notes from Trigonometry for exam revision<\/title>\n<meta name=\"description\" content=\"We will have a detailed understanding of all the trigonometry related concepts and formulae and revision notes for students to secure maximum marks.\" \/>\n<meta name=\"robots\" content=\"index, follow, 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