{"id":5751,"date":"2014-09-15T15:37:42","date_gmt":"2014-09-15T10:07:42","guid":{"rendered":"https:\/\/www.askiitians.com\/blog\/?p=5751"},"modified":"2014-09-15T15:37:42","modified_gmt":"2014-09-15T10:07:42","slug":"back-benchers-tip-learn-trigonometric-identities","status":"publish","type":"post","link":"https:\/\/www.askiitians.com\/blog\/back-benchers-tip-learn-trigonometric-identities\/","title":{"rendered":"The Back Bencher\u00c3\u00a2\u00e2\u201a\u00ac\u00e2\u201e\u00a2s Tip to learn the Trigonometric Identities"},"content":{"rendered":"<p style=\"text-align: left;\"><a href=\"https:\/\/www.askiitians.com\/blog\/wp-content\/uploads\/2014\/09\/trigonometry_word_cloud_70.gif\"><img fetchpriority=\"high\" decoding=\"async\" class=\" wp-image-5752 aligncenter\" title=\"Conquering Trigonometric Identities: Simple tips to learn the lengthy identities!\" src=\"https:\/\/www.askiitians.com\/blog\/wp-content\/uploads\/2014\/09\/trigonometry_word_cloud_70.gif\" alt=\"Conquering Trigonometric Identities: Simple tips to learn the lengthy identities!\" width=\"560\" height=\"320\" \/><\/a>All right! So, now you are planning to study the monster called trigonometry! I am sure majority of you would agree with me on this that trigonometry is a hard nut to crack! The concepts involved in it are not very hard but what makes it all the more challenging is that it is over flooded with formulae.<\/p>\n<p style=\"text-align: left;\">You learn first five formulae, reach the sixth and \u00c3\u00a2\u00e2\u201a\u00ac\u00c2\u00a6. you can\u00c3\u00a2\u00e2\u201a\u00ac\u00e2\u201e\u00a2t recall the first one! If this is what you are experiencing, then you are one of the numerous students suffering from the trigonometry phobia! Well, you should not be cursing yourself for this because, after all you are human with limited memory and you cannot expect your brain to be a memory card which stores everything permanently!<\/p>\n<p style=\"text-align: left;\">Don\u00c3\u00a2\u00e2\u201a\u00ac\u00e2\u201e\u00a2t worry my friend, we bring you some of the interesting ways of learning these formulae and befriending the monster called trigonometry! Get ready to explore the interesting world of trigonometry!<\/p>\n<p>There are certain basic concepts which you will have to learn (and we can\u00c3\u00a2\u00e2\u201a\u00ac\u00e2\u201e\u00a2t really help you in that!) like periodicity of functions and Pythagorean identity. Now, let us begin our tour:<\/p>\n<p>We begin with the basic trigonometric functions i.e. sin, cos and tan.<br \/>\nFor remembering this (although, it\u00c3\u00a2\u00e2\u201a\u00ac\u00e2\u201e\u00a2s too simple), write the functions in this order<\/p>\n<p><a href=\"https:\/\/www.askiitians.com\/blog\/wp-content\/uploads\/2014\/09\/12.jpg\"><img decoding=\"async\" class=\"size-full wp-image-5753 alignleft\" title=\"Basic Trigonometric Functions | askIITians\" src=\"https:\/\/www.askiitians.com\/blog\/wp-content\/uploads\/2014\/09\/12.jpg\" alt=\"Basic Trigonometric Functions | askIITians\" width=\"327\" height=\"154\" \/><\/a><\/p>\n<p>sin \u00c3\u201a\u00c2\u00a0 \u00c3\u201a\u00c2\u00a0 \u00c3\u201a\u00c2\u00a0 \u00c3\u201a\u00c2\u00a0 \u00c3\u201a\u00c2\u00a0 cos\u00c3\u201a\u00c2\u00a0\u00c3\u201a\u00c2\u00a0\u00c3\u201a\u00c2\u00a0\u00c3\u201a\u00c2\u00a0\u00c3\u201a\u00c2\u00a0\u00c3\u201a\u00c2\u00a0\u00c3\u201a\u00c2\u00a0\u00c3\u201a\u00c2\u00a0\u00c3\u201a\u00c2\u00a0\u00c3\u201a\u00c2\u00a0 tan<br \/>\nOH\u00c3\u201a\u00c2\u00a0\u00c3\u201a\u00c2\u00a0\u00c3\u201a\u00c2\u00a0\u00c3\u201a\u00c2\u00a0\u00c3\u201a\u00c2\u00a0\u00c3\u201a\u00c2\u00a0\u00c3\u201a\u00c2\u00a0\u00c3\u201a\u00c2\u00a0\u00c3\u201a\u00c2\u00a0\u00c3\u201a\u00c2\u00a0 AH\u00c3\u201a\u00c2\u00a0\u00c3\u201a\u00c2\u00a0\u00c3\u201a\u00c2\u00a0\u00c3\u201a\u00c2\u00a0\u00c3\u201a\u00c2\u00a0\u00c3\u201a\u00c2\u00a0\u00c3\u201a\u00c2\u00a0\u00c3\u201a\u00c2\u00a0\u00c3\u201a\u00c2\u00a0 OA<br \/>\ncosec\u00c3\u201a\u00c2\u00a0\u00c3\u201a\u00c2\u00a0\u00c3\u201a\u00c2\u00a0\u00c3\u201a\u00c2\u00a0\u00c3\u201a\u00c2\u00a0\u00c3\u201a\u00c2\u00a0\u00c3\u201a\u00c2\u00a0 sec\u00c3\u201a\u00c2\u00a0\u00c3\u201a\u00c2\u00a0\u00c3\u201a\u00c2\u00a0\u00c3\u201a\u00c2\u00a0\u00c3\u201a\u00c2\u00a0\u00c3\u201a\u00c2\u00a0\u00c3\u201a\u00c2\u00a0\u00c3\u201a\u00c2\u00a0\u00c3\u201a\u00c2\u00a0\u00c3\u201a\u00c2\u00a0 cot<\/p>\n<p>So, just remember oh ah oa\u00c3\u00a2\u00e2\u201a\u00ac\u00c2\u00a6. or rather you can learn it as <b><i>SOH <\/i><\/b><i>CAH <b>TOA<\/b> <\/i>which says sine means OH (opposite\/hypotenuse), cos means AH (adjacent side\/hypotenuse), tan for OA (opposite side\/adjacent side).<\/p>\n<ul>\n<li>Now, once you know these six basic formulae, next important point is the sign of trigonometric functions. So here comes the technique to remember the signs of trigonometric functions in various quadrants.<\/li>\n<\/ul>\n<p><a href=\"https:\/\/www.askiitians.com\/blog\/wp-content\/uploads\/2014\/09\/22.jpg\"><img decoding=\"async\" class=\"size-full wp-image-5759 alignleft\" title=\"Sign of trigonometric functions in various quadrants\" src=\"https:\/\/www.askiitians.com\/blog\/wp-content\/uploads\/2014\/09\/22.jpg\" alt=\"Sign of trigonometric functions in various quadrants\" width=\"225\" height=\"225\" \/><\/a>We know that there are four quadrants. In the first quadrant, all trigonometric functions are positive. In the second quadrant, sine and cosec are positive and all remaining functions are negative. Similarly only tangent and cotangent are positive in the third quadrant while sec and cosec are positive in the fourth quadrant.<br \/>\nStudents often get confused in memorizing this. This should be better remembered as <b><i>A<\/i><\/b><i>fter <b>S<\/b>chool <b>T<\/b>o <b>C<\/b>ollege<\/i> or <b><i>A<\/i><\/b><i>dd <b>S<\/b>ugar <b>T<\/b>o <b>T<\/b>ea<\/i>. The initials stand for the functions which are positive in the respective quadrants. Once you learn it this way, you are sure to remember it throughout your life without getting confused.<br \/>\nNext is the core of trigonometry i.e. the trigonometric identities. Frankly speaking, there is no substitute for learning the trigonometric identities. One of the options is that you can derive them using the Euler\u00c3\u00a2\u00e2\u201a\u00ac\u00e2\u201e\u00a2s formula. But that is possible only if you have sufficient time. Demoivre\u00c3\u00a2\u00e2\u201a\u00ac\u00e2\u201e\u00a2s theorem is also useful in this context but that also takes time. Hence, while the sad part is that one has to learn the identities as there is no other alternative, the good part is that we are here to simplify this process for you!<\/p>\n<p>Consider the figure given below. This figure is called the hexagon of trigonometric identities. Some of the basic identities can be remembered with the help of this hexagon.<\/p>\n<p><a href=\"https:\/\/www.askiitians.com\/blog\/wp-content\/uploads\/2014\/09\/3.png\"><img loading=\"lazy\" decoding=\"async\" class=\"alignleft\" title=\"central 1 diagonally gives us the reciprocal functions\" src=\"https:\/\/www.askiitians.com\/blog\/wp-content\/uploads\/2014\/09\/3.png\" alt=\"central 1 diagonally gives us the reciprocal functions\" width=\"220\" height=\"190\" \/><\/a>Reading across the central 1 diagonally gives us the reciprocal functions i.e.<\/p>\n<p>cosec x = 1\/sin x<\/p>\n<p>sec x = 1\/cos x<\/p>\n<p>cot x = 1\/tan x<\/p>\n<p>Next, we can also derive the standard identities reading down any triangle in clockwise direction i.e.<\/p>\n<p>sin<sup>2<\/sup>x + cos<sup>2<\/sup>x = 1<\/p>\n<p>1 + cot<sup>2<\/sup>x = cosec<sup>2<\/sup>x<\/p>\n<p>tan<sup>2<\/sup>x + 1 = sec<sup>2<\/sup>x<\/p>\n<p>Hence, these basic concepts can be easily remembered with the help of this hexagon.<\/p>\n<ul>\n<li>We now move on to the sine and cosine laws in triangle:<\/li>\n<\/ul>\n<p>In any triangle ABC having sides a, b and c, we have<\/p>\n<p>1.The sine law<\/p>\n<p>sin A \/ a = sin B \/ b = sin C \/ c<\/p>\n<p>So, here you can just remember that sine of an angle divided by same side is equal to the rest of the angles divided by the corresponding sides.<\/p>\n<p>2 The cosine laws<\/p>\n<p>a<sup> 2<\/sup> = b<sup> 2<\/sup> + c<sup> 2<\/sup> &#8211; 2 bc cos A<\/p>\n<p>b<sup> 2<\/sup> = a<sup> 2<\/sup> + c<sup> 2<\/sup> &#8211; 2 ac cos B<\/p>\n<p>c<sup> 2<\/sup> = a<sup> 2<\/sup> + b<sup> 2<\/sup> &#8211; 2 a b cos C<\/p>\n<p>We take the first one. You can learn this as one side a = (b-c)<sup>2<\/sup> and the 2bc term has cosine of the angle corresponding to the main side i.e. a.<\/p>\n<ul>\n<li><em><b>Sum and Difference Formulae:<\/b><\/em><\/li>\n<\/ul>\n<p>1. sin (A \u00c3\u201a\u00c2\u00b1 B) = sin A cos B \u00c3\u201a\u00c2\u00b1 cos A sin B<\/p>\n<p>First of all, remember that the formula of sin (A \u00c3\u201a\u00c2\u00b1 B) contains both the functions sin and cos. Now, to memorize the formula, learn \u00c3\u00a2\u00e2\u201a\u00ac\u00c5\u201c<b><i>SC<\/i><\/b><i> student studies <b>CS <\/b>(computer science)\u00c3\u00a2\u00e2\u201a\u00ac\u00c2\u009d<\/i>, where <i>SC\u00c3\u00a2\u00e2\u20ac\u00a0\u00e2\u20ac\u2122CS<\/i> imply sin cos and then cos sin. The sign between the terms remains the same i.e. the formula for sin (A + B) will contain a positive sign in between terms and that of sin (A &#8211; B) will contain a negative sign.<\/p>\n<p>2. cos (A \u00c3\u201a\u00c2\u00b1 B) = cos A cos B \u00c3\u00a2\u00cb\u2020\u00e2\u20ac\u0153 sin A sin B<\/p>\n<p>This formula of cos contains both the trigonometric functions in pairs. You can learn it as <b><i>C<\/i><\/b><i>aste <b>C<\/b>onflict is a form of <b>S<\/b>ocial <b>S<\/b>in<\/i> i.e. CC\u00c3\u00a2\u00e2\u20ac\u00a0\u00e2\u20ac\u2122SS. Now this statement is in negative sign and hence the sign would be negative i.e. while computing cos (A + B), the sign between the terms is negative and vice-versa.<\/p>\n<p>Once you remember this, you can easily derive the remaining formulae.<\/p>\n<ul>\n<li><em><b>Multiple Angle Formulae:<\/b><\/em><\/li>\n<\/ul>\n<p>1. sin 2A = 2 sin A cos A<\/p>\n<p>Remember that both the sides contain 2. Further you can learn it as SSC exam i.e. S\u00c3\u00a2\u00e2\u20ac\u00a0\u00e2\u20ac\u2122SC, which implies that sine of an angle = twice sine and cosine of half of that angle.<\/p>\n<p>2. cos 2A = cos<sup>2<\/sup>A \u00c3\u00a2\u00e2\u201a\u00ac\u00e2\u20ac\u0153 sin <sup>2<\/sup>A<\/p>\n<p>= 1 &#8211; 2sin<sup>2<\/sup>A<\/p>\n<p>= 2 cos<sup>2<\/sup>A -1<\/p>\n<p>In this case, there are three formulae for cosine of 2A. But, one should try to learn just one as others can be derived from it using the relationship of sine and cosine of angles.<\/p>\n<p>3. sin 3A = 3 sin A \u00c3\u00a2\u00e2\u201a\u00ac\u00e2\u20ac\u0153 4sin<sup>3<\/sup>A<\/p>\n<p>In this again, remember that every term contains 3. Secondly, all the terms contain the same trigonometric ratio i.e. sine.<\/p>\n<p>4. cos 3A = 4cos<sup>3<\/sup>A &#8211; 3 cos A<\/p>\n<p>This term is completely based on the previous formula. By just interchanging both the terms in the previous case and replacing sine by cos we obtain this formula.<\/p>\n<ul>\n<li><em><strong>Some general tips to help you learn the identities:<\/strong><\/em><\/li>\n<\/ul>\n<p>1. Remember that sin x = (e<sup>ix<\/sup> \u00c3\u00a2\u00e2\u201a\u00ac\u00e2\u20ac\u0153 e<sup>-ix<\/sup>)\/2i and cos x = (e<sup>ix<\/sup> + e<sup>-ix<\/sup>)\/2 and use the exponent rules to derive the identities.<\/p>\n<p>2. Usually cos carries a positive sign while sine carries a negative sign.<\/p>\n<p>3. One cannot escape from learning the formulae of reciprocal functions, periodicity of functions and Pythagorean identities.<\/p>\n<p>4. Instead of attempting to cram the identities, try to establish relations between terms so as to minimize the chances of forgetting the formulae.<\/p>\n<p>5. Learn one of the sum identities and the remaining sum identities can be easily derived using the facts that cos x is an even function and sin x is an odd function.<\/p>\n<p>6. It is crucial to have your basics clear as they can help you in reaching the formulae in case you forget or get confused.<\/p>\n<p>7. Always remember that all the trigonometric functions are periodic i.e. they repeat after a specific interval. This concept also proves useful.<\/p>\n<p>8. Once you remember the sum and difference identities for sine and cos i.e. the formulae of sin (a \u00c3\u201a\u00c2\u00b1 b) and cos (a \u00c3\u201a\u00c2\u00b1 b), the multiple angle formulae can be derived from these identities by putting b = a, we get the formula for cos 2a.<\/p>\n<p>9. Practicing numerous questions based on application of these formulae can help you in mugging them up without putting in extra effort.<\/p>\n<p>We have listed all the important trigonometric identities here so that students don\u00c3\u00a2\u00e2\u201a\u00ac\u00e2\u201e\u00a2t miss out any:<\/p>\n<h3>\u00c3\u201a\u00c2\u00b7 <strong><em>Pythagorean Identities<\/em><\/strong><\/h3>\n<p>sin<sup> 2<\/sup>X + cos<sup> 2<\/sup>X = 1<\/p>\n<p>1 + tan<sup> 2<\/sup>X = sec<sup> 2<\/sup>X<\/p>\n<p>1 + cot<sup> 2<\/sup>X = csc<sup> 2<\/sup>X<\/p>\n<h3>\u00c3\u201a\u00c2\u00b7 \u00c3\u201a\u00c2\u00a0<em><strong>Negative Angle Identities<\/strong><\/em><\/h3>\n<p>sin (-X) = &#8211; sin X, odd function<\/p>\n<p>csc (-X) = &#8211; csc X, odd function<\/p>\n<p>cos (-X) = cos X, even function<\/p>\n<p>sec (-X) = sec X, even function<\/p>\n<p>tan (-X) = &#8211; tan X, odd function<\/p>\n<p>cot (-X) = &#8211; cot X, odd function<\/p>\n<h3>\u00c3\u201a\u00c2\u00b7 <em>Cofunctions Identities<\/em><\/h3>\n<p>sin (\u00c3\u008f\u00e2\u201a\u00ac \/2 &#8211; X) = cos X<\/p>\n<p>cos (\u00c3\u008f\u00e2\u201a\u00ac \/2 &#8211; X) = sin X<\/p>\n<p>tan (\u00c3\u008f\u00e2\u201a\u00ac \/2 &#8211; X) = cot X<\/p>\n<p>cot (\u00c3\u008f\u00e2\u201a\u00ac\/2 &#8211; X) = tan X<\/p>\n<p>sec (\u00c3\u008f\u00e2\u201a\u00ac \/2 &#8211; X) = csc X<\/p>\n<p>csc (\u00c3\u008f\u00e2\u201a\u00ac \/2 &#8211; X) = sec X<\/p>\n<h3>\u00c3\u201a\u00c2\u00b7 <em>Addition Formulas<\/em><\/h3>\n<p>cos (X + Y) = cos X cos Y \u00c3\u00a2\u00e2\u201a\u00ac\u00e2\u20ac\u0153 sin X sin Y<\/p>\n<p>cos (X &#8211; Y) = cos X cos Y + sin X sin Y<\/p>\n<p>sin (X + Y) = sin X cos Y + cos X sin Y<\/p>\n<p>sin (X &#8211; Y) = sin X cosY \u00c3\u00a2\u00e2\u201a\u00ac\u00e2\u20ac\u0153 cos X sin Y<\/p>\n<p>tan (X + Y) = [ tan X + tan Y ] \/ [ 1 \u00c3\u00a2\u00e2\u201a\u00ac\u00e2\u20ac\u0153 tan X tan Y]<\/p>\n<p>tan (X &#8211; Y) = [ tan X \u00c3\u00a2\u00e2\u201a\u00ac\u00e2\u20ac\u0153 tan Y ] \/ [ 1 + tan X tan Y]<\/p>\n<p>cot (X + Y) = [ cot X cot Y &#8211; 1 ] \/ [ cot X + cot Y]<\/p>\n<p>cot (X &#8211; Y) = [ cot X cot Y + 1 ] \/ [ cot Y \u00c3\u00a2\u00e2\u201a\u00ac\u00e2\u20ac\u0153 cot X]<\/p>\n<h3>\u00c3\u201a\u00c2\u00b7 <em>Sum to Product Formulas<\/em><\/h3>\n<p>cos X + cos Y = 2cos[(X + Y)\/ 2] cos[(X &#8211; Y)\/ 2]<\/p>\n<p>sin X + sin Y = 2sin[(X + Y)\/ 2] cos[(X &#8211; Y)\/ 2]<\/p>\n<h3>\u00c3\u201a\u00c2\u00b7 <em>Difference to Product Formulas<\/em><\/h3>\n<p>cos X \u00c3\u00a2\u00e2\u201a\u00ac\u00e2\u20ac\u0153 cos Y = &#8211; 2sin[(X + Y) \/ 2] sin[(X &#8211; Y) \/ 2]<\/p>\n<p>sin X \u00c3\u00a2\u00e2\u201a\u00ac\u00e2\u20ac\u0153 sin Y = 2cos[(X + Y) \/ 2] sin[(X &#8211; Y) \/ 2]<\/p>\n<h3>\u00c3\u201a\u00c2\u00b7 <em>Product to Sum\/Difference Formulas<\/em><\/h3>\n<p>cos X cos Y = (1\/2) [cos (X &#8211; Y) + cos (X + Y)]<\/p>\n<p>sin X cos Y = (1\/2) [sin (X + Y) + sin (X &#8211; Y)]<\/p>\n<p>cos X sin Y = (1\/2) [sin (X + Y) &#8211; sin[ (X &#8211; Y)]<\/p>\n<p>sin X sin Y = (1\/2) [cos (X &#8211; Y) &#8211; cos (X + Y)]<\/p>\n<h3>\u00c3\u201a\u00c2\u00b7 <em>Difference of Squares Formulas<\/em><\/h3>\n<p>sin<sup> 2<\/sup>X &#8211; sin<sup> 2<\/sup>Y = sin (X + Y) sin (X &#8211; Y)<\/p>\n<p>cos<sup> 2<\/sup>X &#8211; cos<sup> 2<\/sup>Y = &#8211; sin (X + Y) sin (X &#8211; Y)<\/p>\n<p>cos<sup> 2<\/sup>X &#8211; sin<sup> 2<\/sup>Y = cos (X + Y) cos (X &#8211; Y)<\/p>\n<h3>\u00c3\u201a\u00c2\u00b7 <em>Double Angle Formulas<\/em><\/h3>\n<p>sin (2X) = 2 sin X cos X<\/p>\n<p>cos (2X) = 1 &#8211; 2sin<sup> 2<\/sup>X = 2cos<sup> 2<\/sup>X &#8211; 1<\/p>\n<p>tan (2X) = 2tan X\/[1 &#8211; tan<sup> 2<\/sup>X]<\/p>\n<h3>\u00c3\u201a\u00c2\u00b7 \u00c3\u201a\u00c2\u00a0<em>Multiple Angle Formulas<\/em><\/h3>\n<p>sin (3X) = 3sin X &#8211; 4sin<sup> 3<\/sup>X<\/p>\n<p>cos (3X) = 4cos<sup> 3<\/sup>X &#8211; 3cos X<\/p>\n<p>sin (4X) = 4sin X cos X &#8211; 8sin<sup> 3<\/sup>X cos X<\/p>\n<p>cos (4X) = 8cos<sup> 4<\/sup>X &#8211; 8cos<sup> 2<\/sup>X + 1<\/p>\n<h3>\u00c3\u201a\u00c2\u00b7 <em>Half Angle Formulas<\/em><\/h3>\n<p>sin (X\/2) = \u00c3\u201a\u00c2\u00b1\u00c3\u00a2\u00cb\u2020\u00c5\u00a1[(1 \u00c3\u00a2\u00e2\u201a\u00ac\u00e2\u20ac\u0153 cos X)\/2]<\/p>\n<p>cos (X\/2) = \u00c3\u201a\u00c2\u00b1\u00c3\u00a2\u00cb\u2020\u00c5\u00a1[(1 + cos X)\/2]<\/p>\n<p>tan (X\/2) = \u00c3\u201a\u00c2\u00b1\u00c3\u00a2\u00cb\u2020\u00c5\u00a1[(1 \u00c3\u00a2\u00e2\u201a\u00ac\u00e2\u20ac\u0153 cos X)\/(1 + cos X)]<\/p>\n<p>= sin X\/(1 + cos X)<\/p>\n<p>= (1 \u00c3\u00a2\u00e2\u201a\u00ac\u00e2\u20ac\u0153 cos X)\/sin X<\/p>\n<h3>\u00c3\u201a\u00c2\u00b7 <em>Power Reducing Formulas<\/em><\/h3>\n<p>sin<sup> 2<\/sup>X = 1\/2 &#8211; (1\/2) cos (2X))<\/p>\n<p>cos<sup> 2<\/sup>X = 1\/2 + (1\/2) cos (2X))<\/p>\n<p>sin<sup> 3<\/sup>X = (3\/4) sin X &#8211; (1\/4) sin (3X)<\/p>\n<p>cos<sup> 3<\/sup>X = (3\/4) cos X + (1\/4) cos (3X)<\/p>\n<p>sin<sup> 4<\/sup>X = (3\/8) &#8211; (1\/2)cos (2X) + (1\/8)cos (4X)<\/p>\n<p>cos<sup> 4<\/sup>X = (3\/8) + (1\/2)cos (2X) + (1\/8)cos (4X)<\/p>\n<p>sin<sup> 5<\/sup>X = (5\/8)sin X &#8211; (5\/16)sin (3X) + (1\/16)sin (5X)<\/p>\n<p>cos<sup> 5<\/sup>X = (5\/8)cos X + (5\/16)cos (3X) + (1\/16)cos (5X)<\/p>\n<p>sin<sup> 6<\/sup>X = 5\/16 &#8211; (15\/32)cos (2X) + (6\/32)cos (4X) &#8211; (1\/32)cos (6X)<\/p>\n<p>cos<sup> 6<\/sup>X = 5\/16 + (15\/32)cos (2X) + (6\/32)cos (4X) + (1\/32)cos (6X)<\/p>\n<h3>\u00c3\u201a\u00c2\u00b7 <em>Trigonometric Functions Periodicity<\/em><\/h3>\n<p>sin (X + 2\u00c3\u008f\u00e2\u201a\u00ac) = sin X, period 2\u00c3\u008f\u00e2\u201a\u00ac<\/p>\n<p>cos (X + 2\u00c3\u008f\u00e2\u201a\u00ac) = cos X, period 2\u00c3\u008f\u00e2\u201a\u00ac<\/p>\n<p>sec (X + 2\u00c3\u008f\u00e2\u201a\u00ac) = sec X, period 2\u00c3\u008f\u00e2\u201a\u00ac<\/p>\n<p>csc (X + 2\u00c3\u008f\u00e2\u201a\u00ac) = csc X, period 2\u00c3\u008f\u00e2\u201a\u00ac<\/p>\n<p>tan (X + \u00c3\u008f\u00e2\u201a\u00ac) = tan X, period \u00c3\u008f\u00e2\u201a\u00ac<\/p>\n<p>cot (X + \u00c3\u008f\u00e2\u201a\u00ac) = cot X, period \u00c3\u008f\u00e2\u201a\u00ac<\/p>\n<p><a href=\"https:\/\/www.askiitians.com\/blog\/back-benchers-tip-learn-periodic-table\/\"><em><strong>The Back Bencher\u00c3\u00a2\u00e2\u201a\u00ac\u00e2\u201e\u00a2s Tip to learn the Periodic Table<\/strong><\/em><\/a><\/p>\n<p><a href=\"http:\/\/www.askiitians.com\/iit-jee-mathematics\/trigonometry.aspx\" target=\"_blank\" rel=\"noopener\"><em><strong>Trigonometry Study Material<\/strong><\/em><\/a><\/p>\n","protected":false},"excerpt":{"rendered":"<p>All right! So, now you are planning to study the monster called trigonometry! I am sure majority of you would agree with me on this that trigonometry is a hard nut to crack! The concepts involved in it are not very hard but what makes it all the more challenging is that it is over [&hellip;]<\/p>\n","protected":false},"author":2,"featured_media":5760,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[5],"tags":[],"class_list":["post-5751","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-engineering-exams"],"_links":{"self":[{"href":"https:\/\/www.askiitians.com\/blog\/wp-json\/wp\/v2\/posts\/5751","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.askiitians.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.askiitians.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.askiitians.com\/blog\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/www.askiitians.com\/blog\/wp-json\/wp\/v2\/comments?post=5751"}],"version-history":[{"count":0,"href":"https:\/\/www.askiitians.com\/blog\/wp-json\/wp\/v2\/posts\/5751\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.askiitians.com\/blog\/wp-json\/wp\/v2\/media\/5760"}],"wp:attachment":[{"href":"https:\/\/www.askiitians.com\/blog\/wp-json\/wp\/v2\/media?parent=5751"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.askiitians.com\/blog\/wp-json\/wp\/v2\/categories?post=5751"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.askiitians.com\/blog\/wp-json\/wp\/v2\/tags?post=5751"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}