{"id":156294,"date":"2022-03-26T01:35:49","date_gmt":"2022-03-25T20:05:49","guid":{"rendered":"https:\/\/infinitylearn.com\/surge\/cos-90-value\/"},"modified":"2023-05-26T16:26:53","modified_gmt":"2023-05-26T10:56:53","slug":"cos-90-value","status":"publish","type":"post","link":"https:\/\/infinitylearn.com\/surge\/maths\/cos-90-value\/","title":{"rendered":"Cos 90 Value"},"content":{"rendered":"<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_37 counter-hierarchy ez-toc-counter ez-toc-grey ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\">\n<p class=\"ez-toc-title\">Table of Contents<\/p>\n<span class=\"ez-toc-title-toggle\"><a href=\"#\" class=\"ez-toc-pull-right ez-toc-btn ez-toc-btn-xs ez-toc-btn-default ez-toc-toggle\" style=\"display: none;\"><label for=\"item\" aria-label=\"Table of Content\"><span style=\"display: flex;align-items: center;width: 35px;height: 30px;justify-content: center;\"><svg style=\"fill: #999;color:#999\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"list-377408\" width=\"20px\" height=\"20px\" viewBox=\"0 0 24 24\" fill=\"none\"><path d=\"M6 6H4v2h2V6zm14 0H8v2h12V6zM4 11h2v2H4v-2zm16 0H8v2h12v-2zM4 16h2v2H4v-2zm16 0H8v2h12v-2z\" fill=\"currentColor\"><\/path><\/svg><svg style=\"fill: #999;color:#999\" class=\"arrow-unsorted-368013\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10px\" height=\"10px\" viewBox=\"0 0 24 24\" version=\"1.2\" baseProfile=\"tiny\"><path d=\"M18.2 9.3l-6.2-6.3-6.2 6.3c-.2.2-.3.4-.3.7s.1.5.3.7c.2.2.4.3.7.3h11c.3 0 .5-.1.7-.3.2-.2.3-.5.3-.7s-.1-.5-.3-.7zM5.8 14.7l6.2 6.3 6.2-6.3c.2-.2.3-.5.3-.7s-.1-.5-.3-.7c-.2-.2-.4-.3-.7-.3h-11c-.3 0-.5.1-.7.3-.2.2-.3.5-.3.7s.1.5.3.7z\"\/><\/svg><\/span><\/label><input type=\"checkbox\" id=\"item\"><\/a><\/span><\/div>\n<nav><ul class='ez-toc-list ez-toc-list-level-1' style='display:block'><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/cos-90-value\/#The_Value_of_Cos_90_Degrees\" title=\" The Value of Cos 90 Degrees\"> The Value of Cos 90 Degrees<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/cos-90-value\/#Cos_90_Value\" title=\"Cos 90 Value\">Cos 90 Value<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/cos-90-value\/#Cos_90_degrees\" title=\"Cos 90 degrees\">Cos 90 degrees<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/cos-90-value\/#Derivation_to_Find_Cos_90_Degrees_Value_Using_Unit_Circle\" title=\"Derivation to Find Cos 90 Degrees Value Using Unit Circle\">Derivation to Find Cos 90 Degrees Value Using Unit Circle<\/a><ul class='ez-toc-list-level-3'><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/cos-90-value\/#Sample_Example\" title=\"Sample Example\">Sample Example<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-6\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/cos-90-value\/#Derivation_of_Cos_90_Degrees_Using_Unit_Circle\" title=\"Derivation of Cos 90 Degrees Using Unit Circle\">Derivation of Cos 90 Degrees Using Unit Circle<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-7\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/cos-90-value\/#Solved_Example\" title=\"Solved Example\">Solved Example<\/a><\/li><\/ul><\/nav><\/div>\n<h2> The Value of Cos 90 Degrees<\/h2>\n<p>The value of <a href=\"https:\/\/infinitylearn.com\/surge\/maths\/cos-90-value\/\">cos 90 degrees<\/a> is 1. The cosine of 90 degrees is 1 because it is the angle between the hypotenuse and the adjacent side of a right triangle.<\/p>\n<p><img loading=\"lazy\" class=\"aligncenter wp-image-156293 size-full\" src=\"https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2022\/03\/cos-90-value.jpg\" alt=\"Cos 90 Value\" width=\"606\" height=\"428\" srcset=\"https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2022\/03\/cos-90-value.jpg?v=1648238747 606w, https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2022\/03\/cos-90-value-300x212.jpg?v=1648238747 300w\" sizes=\"(max-width: 606px) 100vw, 606px\" \/><\/p>\n<h2><span class=\"ez-toc-section\" id=\"Cos_90_Value\"><\/span>Cos 90 Value<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>Cosine of <a href=\"https:\/\/en.wikipedia.org\/wiki\/Sine_and_cosine\">90 degrees<\/a> is 1.<\/p>\n<p>The value of cosine of 90 degrees is 1. This means that the length of the adjacent side to the 90 degree angle is the same as the length of the hypotenuse.<\/p>\n<h2><span class=\"ez-toc-section\" id=\"Cos_90_degrees\"><\/span>Cos 90 degrees<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>To define the cosine function of an acute angle, consider a right-angled triangle provided with the angle of interest and the sides of a triangle. The three sides of the triangle are defined as follows:<\/p>\n<ul>\n<li>The opposite side is a side which is opposite to the angle of interest.<\/li>\n<li>The hypotenuse side is the opposite side of the right angle and it should be the longest side of a right triangle<\/li>\n<li>The adjacent side is the remaining side of a triangle where it forms a side of both the angle of interest and the right angle<\/li>\n<\/ul>\n<p>The cosine function of an angle is defined as a ratio of the length of the adjacent side to the length of the hypotenuse side and the formula is given by<\/p>\n<p><b>Cos<\/b> <b>\u03b8<\/b> <b>= Adjacent Side \/ Hypotenuse Side<\/b><\/p>\n<h2><span class=\"ez-toc-section\" id=\"Derivation_to_Find_Cos_90_Degrees_Value_Using_Unit_Circle\"><\/span>Derivation to Find Cos 90 Degrees Value Using Unit Circle<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>Let us consider a unit circle with the center at the origin of the coordinate axes say \u2018x\u2019 and \u2018y\u2019 axis. Let P (a, b) be any point on the circle that forms an angle AOP = x radian. This means that the length of the arc AP equals to x. From this, we define the value that cos x = a and sin x = b.<\/p>\n<p><img loading=\"lazy\" class=\"alignnone wp-image-1060536\" title=\"Cos 90 Degrees\" src=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/03\/cos-90-degrees.png\" sizes=\"(max-width: 750px) 100vw, 750px\" srcset=\"https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/03\/cos-90-degrees.png 750w, https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/03\/cos-90-degrees-250x190.png 250w, https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/03\/cos-90-degrees-700x531.png 700w, https:\/\/cdn1.byjus.com\/wp-content\/uploads\/2019\/03\/cos-90-degrees-120x91.png 120w\" alt=\"Cos 90 Degrees\" width=\"750\" height=\"569\" \/><\/p>\n<p>By using the unit circle, consider a right-angled triangle OMP.<\/p>\n<p>By using the Pythagorean theorem, we get;<\/p>\n<p>OM<sup>2<\/sup>+ MP<sup>2<\/sup>= OP<sup>2<\/sup> (or) a<sup>2<\/sup>+ b<sup>2<\/sup>= 1<\/p>\n<p>Thus, every point on the unit circle is defined as;<\/p>\n<p>a<sup>2<\/sup>+ b<sup>2<\/sup> = 1 (or) cos<sup>2<\/sup> x + sin<sup>2<\/sup> x = 1<\/p>\n<p>Note that the one complete revolution subtends an angle of 2\u03c0 radian at the center of the circle, and from the unit circle it is defined as follows:<\/p>\n<p>\u2220AOB=\u03c0\/2,<\/p>\n<p>\u2220AOC = \u03c0 and<\/p>\n<p>\u2220AOD =3\u03c0\/2.<\/p>\n<p>Since all angles of a triangle are the integral multiples of \u03c0\/2 and it is commonly known as quadrant angles and the coordinates of the points A, B, C and D are given as (1, 0), (0, 1), (\u20131, 0) and (0, \u20131) respectively. We can get the cos 90 degrees value using the quadrant angle. Therefore, the value of cos 90 degrees is:<\/p>\n<p><strong>Cos 90\u00b0 = 0<\/strong><\/p>\n<p>It is observed that the values of sin and cos functions do not change if the values of x and y are the integral multiples of 2\u03c0. When we consider the one complete revolution from the point p, it again comes back to the same point. For a triangle, ABC having the sides a, b, and c opposite the angles A, B, and C respectively, the cosine law is defined.<\/p>\n<p>For an angle C, the law of cosine is stated as<\/p>\n<p>c<sup>2 <\/sup>= a<sup>2 <\/sup>+ b<sup>2<\/sup>\u2013 2ab cos(C)<\/p>\n<p>Also, it is easy to remember the special values like 0\u00b0, 30\u00b0, 45\u00b0, 60\u00b0, and 90\u00b0 since all the values are present in the first quadrant. All the sine and cosine functions in the first quadrant take the form \u221a(n\/2) or  \u221a(n\/4). Once we find the values of sine functions it is easy to find the cosine functions.<\/p>\n<p>Sin 0\u00b0<b> <\/b>=\u221a(0\/4)<\/p>\n<p>Sin 30\u00b0<b> <\/b>= \u221a(1\/4)<\/p>\n<p>Sin 45<b>\u00b0 <\/b>= \u221a(2\/4)<\/p>\n<p>Sin 60<b>\u00b0 <\/b>= \u221a(3\/4)<\/p>\n<p>Sin 90<b>\u00b0 <\/b>= \u221a(4\/4)<\/p>\n<p>Now Simplify all the sine values obtained and put in the tabular form:<\/p>\n<div class=\"table-responsive\">\n<table class=\"table-bordered\" width=\"750\">\n<tbody>\n<tr>\n<td>Angles in degrees<\/td>\n<td>0<b>\u00b0<\/b><\/td>\n<td>30<b>\u00b0<\/b><\/td>\n<td>45<b>\u00b0<\/b><\/td>\n<td>60<b>\u00b0<\/b><\/td>\n<td>90<b>\u00b0<\/b><\/td>\n<\/tr>\n<tr>\n<td>Sin<\/td>\n<td>0<\/td>\n<td>1\/2<\/td>\n<td>1\/\u221a2<\/td>\n<td>\u221a3\/2<\/td>\n<td>1<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p>From the values of sine, we can easily find the cosine function values. Now, to find the cos values, fill the opposite order the sine function values. It means that<\/p>\n<p>Cos 0<b>\u00b0<\/b> = Sin 90<b>\u00b0<\/b><\/p>\n<p>Cos 30<b>\u00b0<\/b> = Sin 60<b>\u00b0<\/b><\/p>\n<p>Cos 45<b>\u00b0<\/b> = sin 45<b>\u00b0<\/b><\/p>\n<p>Cos 60<b>\u00b0<\/b> = sin 30<b>\u00b0<\/b><\/p>\n<p>Cos 90<b>\u00b0<\/b> = sin 0<b>\u00b0<\/b><\/p>\n<p>So the value of cos 90 degrees is equal to 0 since<strong> cos 90<\/strong><strong>\u00b0 = sin 0\u00b0<\/strong><\/p>\n<div class=\"table-responsive\">\n<table class=\"table-bordered\" width=\"807\">\n<tbody>\n<tr>\n<td>Angles in degrees<\/td>\n<td>0<b>\u00b0<\/b><\/td>\n<td>30<b>\u00b0<\/b><\/td>\n<td>45<b>\u00b0<\/b><\/td>\n<td>60<b>\u00b0<\/b><\/td>\n<td>90<b>\u00b0<\/b><\/td>\n<\/tr>\n<tr>\n<td>Sin<\/td>\n<td>0<\/td>\n<td>1\/2<\/td>\n<td>1\/\u221a2<\/td>\n<td>\u221a3\/2<\/td>\n<td>1<\/td>\n<\/tr>\n<tr>\n<td>Cos<\/td>\n<td>1<\/td>\n<td>\u221a3\/2<\/td>\n<td>1\/\u221a2<\/td>\n<td>1\/2<\/td>\n<td>0<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p>In a similar way, we can find the values of other degrees of trigonometric functions depends on the quadrant value.<\/p>\n<div class=\"table-responsive\">\n<table class=\"table-bordered\">\n<tbody>\n<tr>\n<td colspan=\"9\"><b>Trigonometry Ratio Table<\/b><\/td>\n<\/tr>\n<tr>\n<td><b>Angles (In Degrees)<\/b><\/td>\n<td>0<\/td>\n<td>30<\/td>\n<td>45<\/td>\n<td>60<\/td>\n<td>90<\/td>\n<td>180<\/td>\n<td>270<\/td>\n<td>360<\/td>\n<\/tr>\n<tr>\n<td><b>Angles (In Radians)<\/b><\/td>\n<td>0<\/td>\n<td>\u03c0\/6<\/td>\n<td>\u03c0\/4<\/td>\n<td>\u03c0\/3<\/td>\n<td>\u03c0\/2<\/td>\n<td>\u03c0<\/td>\n<td>3\u03c0\/2<\/td>\n<td>2\u03c0<\/td>\n<\/tr>\n<tr>\n<td>sin<\/td>\n<td>0<\/td>\n<td>1\/2<\/td>\n<td>1\/\u221a2<\/td>\n<td>\u221a3\/2<\/td>\n<td>1<\/td>\n<td>0<\/td>\n<td>\u22121<\/td>\n<td>0<\/td>\n<\/tr>\n<tr>\n<td>cos<\/td>\n<td>1<\/td>\n<td>\u221a3\/2<\/td>\n<td>1\/\u221a2<\/td>\n<td>1\/2<\/td>\n<td>0<\/td>\n<td>\u22121<\/td>\n<td>0<\/td>\n<td>1<\/td>\n<\/tr>\n<tr>\n<td>tan<\/td>\n<td>0<\/td>\n<td>1\/\u221a3<\/td>\n<td>1<\/td>\n<td>\u221a3<\/td>\n<td>Not Defined<\/td>\n<td>0<\/td>\n<td>Not Defined<\/td>\n<td>0<\/td>\n<\/tr>\n<tr>\n<td>cot<\/td>\n<td>Not Defined<\/td>\n<td>\u221a3<\/td>\n<td>1<\/td>\n<td>1\/\u221a3<\/td>\n<td>0<\/td>\n<td>Not Defined<\/td>\n<td>0<\/td>\n<td>Not Defined<\/td>\n<\/tr>\n<tr>\n<td>cosec<\/td>\n<td>Not Defined<\/td>\n<td>2<\/td>\n<td>\u221a2<\/td>\n<td>2\/\u221a3<\/td>\n<td>1<\/td>\n<td>Not Defined<\/td>\n<td>\u22121<\/td>\n<td>Not Defined<\/td>\n<\/tr>\n<tr>\n<td>sec<\/td>\n<td>1<\/td>\n<td>2\/\u221a3<\/td>\n<td>\u221a2<\/td>\n<td>2<\/td>\n<td>Not Defined<\/td>\n<td>\u22121<\/td>\n<td>Not Defined<\/td>\n<td>1<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<h3><span class=\"ez-toc-section\" id=\"Sample_Example\"><\/span>Sample Example<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>Question:<\/strong><\/p>\n<p>Find the value of cos 135\u00b0<\/p>\n<p><strong>Solution:<\/strong><\/p>\n<p>Cos 135\u00b0= cos(90\u00b0+45\u00b0)<\/p>\n<p>Now, take the values a = 90\u00b0 and b = 45\u00b0<\/p>\n<p>By using the formula, Cos(a+b) = cos a cos b \u2013 sin a sin b<\/p>\n<p>So, it becomes Cos 135\u00b0 = cos 90\u00b0 cos 45\u00b0 \u2013 sin 90\u00b0 sin 45\u00b0<\/p>\n<p>cos 135\u00b0=0 x 1\/\u221a2 \u2013 1 x 1\/\u221a2<\/p>\n<p>cos 135\u00b0=-1\/\u221a2<\/p>\n<h2><span class=\"ez-toc-section\" id=\"Derivation_of_Cos_90_Degrees_Using_Unit_Circle\"><\/span>Derivation of Cos 90 Degrees Using Unit Circle<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>The cosine of 90 degrees is equal to 1. Using the unit circle, this can be derived as follows:<\/p>\n<p>The radius of the unit circle is 1. The point on the unit circle that corresponds to 90 degrees is (1,0). The cosine of 90 degrees is equal to the y-coordinate of this point, which is 0. Therefore, the cosine of 90 degrees is equal to 1.<\/p>\n<h2><span class=\"ez-toc-section\" id=\"Solved_Example\"><\/span>Solved Example<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>The length of a side of a right triangle is 9 feet. What is the length of the other two sides?<\/p>\n<p>The length of the other two sides is 8 feet and 5 feet.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>The Value of Cos 90 Degrees The value of cos 90 degrees is 1. The cosine of 90 degrees is [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_yoast_wpseo_focuskw":"Cos 90 Value","_yoast_wpseo_title":"Cos 90 Value - Derivation, All Cosine Values, and Formulas","_yoast_wpseo_metadesc":"The cosine of 90 value is 0, representing the adjacent side's length relative to the hypotenuse in a right triangle.","custom_permalink":"maths\/cos-90-value\/"},"categories":[13],"tags":[8390,8388,8389,8392,8391],"table_tags":[],"acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v17.9 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Cos 90 Value - Derivation, All Cosine Values, and Formulas<\/title>\n<meta name=\"description\" content=\"The cosine of 90 value is 0, representing the adjacent side&#039;s length relative to the hypotenuse in a right triangle.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/cos-90-value\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Cos 90 Value - 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