{"id":899,"date":"2023-05-05T10:00:42","date_gmt":"2023-05-05T04:30:42","guid":{"rendered":"https:\/\/www.aplustopper.com\/?p=899"},"modified":"2023-05-06T09:19:01","modified_gmt":"2023-05-06T03:49:01","slug":"trigonometry-for-specific-angles","status":"publish","type":"post","link":"https:\/\/www.aplustopper.com\/trigonometry-for-specific-angles\/","title":{"rendered":"Trigonometric Ratios Of Some Specific Angles"},"content":{"rendered":"<h2><strong>Trigonometric Ratios Of Some Specific Angles<\/strong><\/h2>\n<p>The angles 0\u00b0, 30\u00b0, 45\u00b0, 60\u00b0, 90\u00b0 are angles for which we have values of T.R.<\/p>\n<table style=\"height: 424px;\" border=\"2\" width=\"644\">\n<tbody>\n<tr>\n<td width=\"85\">\n<p style=\"text-align: center;\"><strong>\u03b8<\/strong><\/p>\n<\/td>\n<td style=\"text-align: center;\" width=\"85\"><strong>0\u00b0<\/strong><\/td>\n<td style=\"text-align: center;\" width=\"85\"><strong>30\u00b0<\/strong><\/td>\n<td style=\"text-align: center;\" width=\"85\"><strong>45\u00b0<\/strong><\/td>\n<td style=\"text-align: center;\" width=\"85\"><strong>60\u00b0<\/strong><\/td>\n<td style=\"text-align: center;\" width=\"85\"><strong>90\u00b0<\/strong><\/td>\n<\/tr>\n<tr>\n<td width=\"85\">\n<p style=\"text-align: center;\"><strong>Sin<\/strong><\/p>\n<\/td>\n<td style=\"text-align: center;\" width=\"85\">0<\/td>\n<td style=\"text-align: center;\" width=\"85\">1\/2<\/td>\n<td style=\"text-align: center;\" width=\"85\">1\/\u221a2<\/td>\n<td style=\"text-align: center;\" width=\"85\">\u00a0\u221a3\/2<\/td>\n<td width=\"85\">\n<p style=\"text-align: center;\">1<\/p>\n<\/td>\n<\/tr>\n<tr>\n<td width=\"85\">\n<p style=\"text-align: center;\"><strong>Cos<\/strong><\/p>\n<\/td>\n<td style=\"text-align: center;\" width=\"85\">1<\/td>\n<td style=\"text-align: center;\" width=\"85\">\u00a0\u221a3\/2<\/td>\n<td style=\"text-align: center;\" width=\"85\">\u00a01\/\u221a2<\/td>\n<td style=\"text-align: center;\" width=\"85\">1\/2<\/td>\n<td style=\"text-align: center;\" width=\"85\">0<\/td>\n<\/tr>\n<tr>\n<td width=\"85\">\n<p style=\"text-align: center;\"><strong>Tan<\/strong><\/p>\n<\/td>\n<td style=\"text-align: center;\" width=\"85\">0<\/td>\n<td style=\"text-align: center;\" width=\"85\">\u00a01\/\u221a3<\/td>\n<td style=\"text-align: center;\" width=\"85\">1<\/td>\n<td style=\"text-align: center;\" width=\"85\">\u221a3<\/td>\n<td style=\"text-align: center;\" width=\"85\">\u221e<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\" width=\"85\"><strong>Cosec<\/strong><\/td>\n<td style=\"text-align: center;\" width=\"85\">\u221e<\/td>\n<td style=\"text-align: center;\" width=\"85\">2<\/td>\n<td style=\"text-align: center;\" width=\"85\">\u221a2<\/td>\n<td style=\"text-align: center;\" width=\"85\">2\/\u221a3<\/td>\n<td width=\"85\">\n<p style=\"text-align: center;\">1<\/p>\n<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\" width=\"85\"><strong>Sec<\/strong><\/td>\n<td style=\"text-align: center;\" width=\"85\">1<\/td>\n<td style=\"text-align: center;\" width=\"85\">2\/\u221a3<\/td>\n<td style=\"text-align: center;\" width=\"85\">\u00a0\u221a2<\/td>\n<td style=\"text-align: center;\" width=\"85\">2<\/td>\n<td width=\"85\">\n<p style=\"text-align: center;\">\u221e<\/p>\n<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\" width=\"85\"><strong>Cot<\/strong><\/td>\n<td style=\"text-align: center;\" width=\"85\">\u221e<\/td>\n<td style=\"text-align: center;\" width=\"85\">\u221a3<\/td>\n<td style=\"text-align: center;\" width=\"85\">1<\/td>\n<td style=\"text-align: center;\" width=\"85\">1\/\u221a3<\/td>\n<td width=\"85\">\n<p style=\"text-align: center;\">0<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h2><strong>Trigonometric Ratios Of Some Specific Angles With Examples<\/strong><\/h2>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-12883\" src=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2016\/08\/Trigonometric-Ratios.png\" alt=\"\" width=\"722\" height=\"466\" srcset=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2016\/08\/Trigonometric-Ratios.png 722w, https:\/\/www.aplustopper.com\/wp-content\/uploads\/2016\/08\/Trigonometric-Ratios-300x194.png 300w\" sizes=\"(max-width: 722px) 100vw, 722px\" \/><\/p>\n<p><strong>Example 1:<\/strong> \u00a0 \u00a0Evaluate each of the following in the simplest form:<br \/>\n(i) sin 60\u00ba cos 30\u00ba + cos 60\u00ba sin 30\u00ba<br \/>\n(ii) sin 60\u00ba cos 45\u00ba + cos 60\u00ba sin 45\u00ba<br \/>\n<strong>Sol.\u00a0\u00a0\u00a0\u00a0\u00a0 (i)\u00a0<\/strong>\u00a0sin 60\u00ba cos 30\u00ba + cos 60\u00ba sin 30\u00ba<br \/>\n\\(=\\frac{\\sqrt{3}}{2}\\times \\frac{\\sqrt{3}}{2}+\\frac{1}{2}\\times \\frac{1}{2}=\\frac{3}{4}+\\frac{1}{4}=1\\)<br \/>\n<strong>(ii)\u00a0<\/strong> sin 60\u00ba cos 45\u00ba + cos 60\u00ba sin 45\u00ba<br \/>\n\\( =\\frac{\\sqrt{3}}{2}\\times \\frac{1}{\\sqrt{2}}+\\frac{1}{2}\\times \\frac{1}{\\sqrt{2}} \\)<br \/>\n\\( =\\frac{\\sqrt{3}}{2\\sqrt{2}}+\\frac{1}{2\\sqrt{2}}=\\frac{\\sqrt{3}+1}{2\\sqrt{2}} \\)<\/p>\n<p><strong>Example 2:<\/strong>\u00a0 \u00a0 \u00a0Evaluate the following expression:<br \/>\n(i) tan 60\u00ba cosec<sup>2<\/sup> 45\u00ba + sec<sup>2<\/sup> 60\u00ba tan 45\u00ba<br \/>\n(ii) 4cot<sup>2<\/sup> 45\u00ba \u2013 sec<sup>2<\/sup> 60\u00ba + sin<sup>2<\/sup> 60\u00ba + cos<sup>2<\/sup> 90\u00ba.<br \/>\n<strong>Sol. \u00a0 \u00a0\u00a0\u00a0(i)<\/strong>\u00a0 tan 60\u00ba cosec<sup>2<\/sup> 45\u00ba + sec<sup>2<\/sup> 60\u00ba tan 45\u00ba<br \/>\n= tan 60\u00ba (cosec 45\u00ba)<sup>2<\/sup> + (sec 60\u00ba)<sup>2<\/sup> tan 45\u00ba<br \/>\n=\u00a0\u221a3\u00a0\u00d7 (\u221a2)<sup>2<\/sup> + (2)<sup>2<\/sup>\u00a0\u00d7 1<br \/>\n= \u221a3\u00a0\u00a0\u00d7 2 + 4 = 4 + 2\u00a0\u221a3<br \/>\n<strong>(ii)<\/strong>\u00a0 4cot<sup>2<\/sup> 45\u00ba \u2013 sec<sup>2<\/sup> 60\u00ba + sin<sup>2<\/sup> 60\u00ba + cos<sup>2<\/sup> 90\u00ba<br \/>\n= 4(cot 45\u00ba)<sup>2<\/sup> \u2013 (sec 60\u00ba)<sup>2<\/sup> + (sin 60\u00ba)<sup>2<\/sup> + (cos 90\u00ba)<sup>2<\/sup><br \/>\n= 4 \u00d7 (1)<sup>2<\/sup> \u2013 (2)<sup>2<\/sup> + (\u221a3\/2)<sup>2\u00a0<\/sup>+ 0<br \/>\n= 4 \u2013 4 + 3\/4 + 0 = 3\/4<\/p>\n<p><iframe loading=\"lazy\" width=\"500\" height=\"375\" src=\"https:\/\/www.youtube.com\/embed\/jNrizjTD4F8?feature=oembed\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen><\/iframe><\/p>\n<p><strong>Example 3:<\/strong>\u00a0 \u00a0 \u00a0Show that:<br \/>\n(i) 2(cos<sup>2<\/sup>45\u00ba + tan<sup>2<\/sup>60\u00ba) \u2013 6(sin<sup>2<\/sup>45\u00ba \u2013 tan<sup>2<\/sup>30\u00ba) = 6<br \/>\n(ii) 2(cos<sup>4<\/sup>60\u00ba + sin<sup>4<\/sup>30\u00ba) \u2013 (tan<sup>2<\/sup>60\u00ba + cot<sup>2<\/sup>45\u00ba) + 3 sec<sup>2<\/sup>30\u00ba = 1\/4<br \/>\n<strong>Sol. \u00a0 \u00a0(i)<\/strong>\u00a0\u00a02(cos<sup>2<\/sup>45\u00ba + tan<sup>2<\/sup>60\u00ba) \u2013 6(sin<sup>2<\/sup>45\u00ba \u2013 tan<sup>2<\/sup>30\u00ba)<br \/>\n\\( =2\\left( {{\\left( \\frac{1}{\\sqrt{2}} \\right)}^{2}}+{{(\\sqrt{3})}^{2}} \\right)-6\\left( {{\\left( \\frac{1}{\\sqrt{2}} \\right)}^{2}}-{{\\left( \\frac{1}{\\sqrt{3}} \\right)}^{2}} \\right) \\)<br \/>\n\\( =2\\left( \\frac{1}{2}+3 \\right)-6\\left( \\frac{1}{2}-\\frac{1}{3} \\right)=2\\left( \\frac{1+6}{2} \\right)-6\\left( \\frac{3-2}{6} \\right) \\)<br \/>\n\\( =2\\times \\frac{7}{2}-6\\times \\frac{1}{6}=7-1=6 \\)<br \/>\n<strong>(ii)\u00a0<\/strong> 2(cos<sup>4<\/sup>60\u00ba + sin<sup>4<\/sup>30\u00ba) \u2013 (tan<sup>2<\/sup>60\u00ba + cot<sup>2<\/sup>45\u00ba) + 3 sec<sup>2<\/sup>30\u00ba<br \/>\n\\( =2\\left( {{\\left( \\frac{1}{2} \\right)}^{4}}+{{\\left( \\frac{1}{2} \\right)}^{4}} \\right)-\\left( {{(\\sqrt{3})}^{2}}+{{(1)}^{2}} \\right)+3{{\\left( \\frac{2}{\\sqrt{3}} \\right)}^{2}} \\)<br \/>\n\\( =2\\left( \\frac{1}{16}+\\frac{1}{16} \\right)\\left( 3+1 \\right)+3\\times \\frac{4}{3} \\)<br \/>\n\\( =2\\times ~\\frac{1}{8}-4+4=\\frac{1}{4} \\)<\/p>\n<p><strong>Example 4:<\/strong>\u00a0 \u00a0 \u00a0Find the value of x in each of the following :<br \/>\n(i) tan 3x = sin 45\u00ba cos 45\u00ba + sin 30\u00ba<br \/>\n(ii) cos x = cos 60\u00ba cos 30\u00ba + sin 60\u00ba sin 30\u00ba<br \/>\n<strong>Sol. \u00a0 \u00a0(i)<\/strong>\u00a0 tan 3x = sin 45\u00ba cos 45\u00ba + sin 30\u00ba<br \/>\n\\( \\Rightarrow tan\\text{ }3x=\\frac{1}{\\sqrt{2}}\\times \\frac{1}{\\sqrt{2}}+\\frac{1}{2} \\)<br \/>\n\\( \\Rightarrow tan\\text{ }3x=\\frac{1}{2}+\\frac{1}{2} \u00a0\\)<br \/>\n\u21d2\u00a0tan 3x = 1<br \/>\n\u21d2\u00a0tan 3x = tan 45\u00ba<br \/>\n\u21d2\u00a03x = 45\u00ba \u00a0 \u00a0\u21d2\u00a0 \u00a0 \u00a0 x = 15\u00ba<br \/>\n<strong>(ii)\u00a0<\/strong> cos x = cos 60\u00ba cos 30\u00ba + sin 60\u00ba sin 30\u00ba<br \/>\n\\( \\Rightarrow cos\\text{ }x=\\frac{1}{2}\\times \\frac{\\sqrt{3}}{2}+\\frac{\\sqrt{3}}{2}\\times \\frac{1}{2} \\)<br \/>\n\\( \\Rightarrow cos\\text{ }x=\\frac{\\sqrt{3}}{4}+\\frac{\\sqrt{3}}{4} \\)<br \/>\n\u21d2\u00a0cos x = \u221a3\/2<br \/>\n\u21d2\u00a0cos x = cos 30\u00ba<br \/>\n\u21d2\u00a0x = 30\u00ba<\/p>\n<p><strong>Example 5:<\/strong>\u00a0 \u00a0 \u00a0If x = 30\u00b0, verify that<br \/>\n\\((i)\\tan 2x=\\frac{2\\tan x}{1-{{\\tan }^{2}}x}\\text{ \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 }(ii)\\sin x=\\sqrt{\\frac{1-\\cos 2x}{2}}\\)<br \/>\n<strong>Sol.(i)\u00a0\u00a0 <\/strong>When x = 30\u00b0, we have 2x = 60\u00b0 .<br \/>\n\u2234 \u00a0tan 2x = tan 60\u00b0 = \u221a3<br \/>\n\\( \\text{And, }\\frac{2\\tan x}{1-{{\\tan }^{2}}x}=\\frac{2\\tan 30{}^\\circ }{1-{{\\tan }^{2}}30{}^\\circ } \\)<br \/>\n\\( =\\frac{2\\times \\frac{1}{\\sqrt{3}}}{1-{{\\left( \\frac{1}{\\sqrt{3}} \\right)}^{2}}} \\)<br \/>\n\\( =\\frac{2\/\\sqrt{3}}{1-\\frac{1}{3}}=\\frac{2\/\\sqrt{3}}{2\/3}=\\frac{2}{\\sqrt{3}}\\times \\frac{3}{2}=\\sqrt{3} \\)<br \/>\n\\( \\tan 2x=\\frac{2\\tan x}{1-{{\\tan }^{2}}x} \\)<br \/>\n<strong>(ii)\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <\/strong>When x = 30\u00b0, we have 2x = 60\u00b0.<br \/>\n\\( \\sqrt{\\frac{1-\\cos 2x}{2}}=\\sqrt{\\frac{1-\\cos 60{}^\\circ }{2}} \\)<br \/>\n\\( \\sqrt{\\frac{1-\\frac{1}{2}}{2}}=\\sqrt{\\frac{1}{4}}=\\frac{1}{2} \\)<br \/>\nAnd, sinx = sin30\u00b0 = 1\/2<br \/>\n\\( \\sin x=\\frac{\\sqrt{1-\\cos 2x}}{2} \\)<\/p>\n<p><strong>Example 6:<\/strong>\u00a0 \u00a0 \u00a0Find the value of \u03b8\u00a0in each of the following :<br \/>\n(i) 2 sin 2\u03b8 = \u221a3 \u00a0 \u00a0 \u00a0(ii) 2 cos 3\u03b8 = 1<br \/>\n<strong>(i) \u00a0<\/strong>2 sin 2\u03b8= \u221a3<br \/>\n\u21d2\u00a0sin 2\u03b8= \u221a3\/2<br \/>\n\u21d2\u00a0sin 2\u03b8= sin 60\u00b0<br \/>\n\u21d2\u00a02\u03b8= 60\u00b0 \u00a0 \u21d2 \u00a0 \u00a0\u03b8 = 30\u00b0<br \/>\n<strong>(ii) \u00a0<\/strong>2 cos 3\u03b8 = 1<br \/>\n\u21d2\u00a0cos 3\u03b8 = 1\/2<br \/>\n\u21d2\u00a0cos 3\u03b8 = cos 60\u00b0<br \/>\n\u21d2\u00a03\u03b8 = 60\u00b0 \u00a0 \u21d2 \u00a0 \u03b8 = 20\u00b0.<\/p>\n<p><strong>Example 7: \u00a0<\/strong> \u00a0If \u03b8\u00a0is an acute angle and sin \u03b8\u00a0= cos \u03b8, find the value of 2 tan<sup>2<\/sup>\u03b8 + sin<sup>2<\/sup>\u03b8 \u2013 1.<br \/>\nSol. sin \u03b8\u00a0= cos\u00a0\u03b8<br \/>\n\\( \\Rightarrow \\frac{\\sin \\theta }{\\cos \\theta }=\\frac{\\cos \\theta }{\\cos \\theta } \\)<br \/>\n[Dividing both sides by cos \u03b8]<br \/>\n\u21d2\u00a0tan\u03b8 = 1<br \/>\n\u21d2\u00a0tan\u03b8 = tan45\u00b0 \u00a0\u21d2 \u00a0\u03b8= 45\u00b0<br \/>\n\u2234\u00a02 tan<sup>2<\/sup>\u03b8 + sin<sup>2<\/sup>\u03b8 \u2013 1<br \/>\n= 2tan<sup>2<\/sup>45\u00b0 + sin<sup>2<\/sup>45\u00b0 \u2013 1<br \/>\n\\( =2{{(2)}^{2}}+{{\\left( \\frac{1}{\\sqrt{2}} \\right)}^{2}}-1 \u00a0\\)<br \/>\n\\( =2+\\frac{1}{2}-1=\\frac{5}{2}-1=\\frac{3}{2} \\)<\/p>\n<p><strong>Example 8:<\/strong>\u00a0 \u00a0 \u00a0An equilateral triangle is inscribed in a circle of radius 6 cm. Find its side.<br \/>\n<strong>Sol.<\/strong>\u00a0 \u00a0 Let ABC be an equilateral triangle inscribed in a circle of radius 6 cm. Let O be the centre of the circle.<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-135644\" src=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2017\/04\/Trigonometric-Ratios-Of-Some-Specific-Angles-1.png\" alt=\"Trigonometric Ratios Of Some Specific Angles 1\" width=\"189\" height=\"194\" \/><br \/>\nThen, OA = OB = OC = 6 cm.<br \/>\nLet OD be perpendicular from O on side BC. Then, D is mid-point of BC and OB and OC are bisectors of \u2220B and \u2220C respectively.<br \/>\n\u2234\u00a0\u2220OBD = 30\u00b0<br \/>\nIn \u2206OBD, right angled at D, we have<br \/>\n\u2220OBD = 30\u00b0 and OB = 6 cm.<br \/>\n\\( \\cos \\angle OBD=\\frac{BD}{OB}\\Rightarrow \\cos {{60}^{0}}=\\frac{BD}{6} \\)<br \/>\n\\( \\Rightarrow BD=6\\cos {{60}^{0}}=6\\times \\frac{\\sqrt{3}}{2}=3\\sqrt{3}\\text{ } \\)<br \/>\n\u21d2\u00a0BC = 2 BD = 2(3\u221a3 )cm = 6\u00a0\u221a3 cm.<\/p>\n<p><strong>Example 9:<\/strong>\u00a0 \u00a0 \u00a0Using the formula, sin(A \u2013 B) = sinA cosB \u2013 cosA sinB, find the value of sin 15\u00ba.<br \/>\n<strong>Sol.<\/strong>\u00a0 \u00a0 Let A = 45\u00ba and B = 30\u00ba. Then A \u2013 B = 15\u00ba. Putting A = 45\u00ba and B = 30\u00ba in the given formula, we get<br \/>\nsin(45\u00ba \u2013 30\u00ba) = sin45\u00ba cos30\u00ba \u2013 cos45\u00ba sin30\u00ba<br \/>\n\\( or,\\sin (45{}^\\text{o}-30{}^\\text{o})=\\frac{1}{\\sqrt{2}}\\times \\frac{\\sqrt{3}}{2}-\\frac{1}{\\sqrt{2}}\\times \\frac{1}{2} \u00a0\\)<br \/>\n\\( =\\frac{\\sqrt{3}-1}{2\\sqrt{2}}\\Rightarrow \\sin 15{}^\\text{o}=\\frac{\\sqrt{3}-1}{2\\sqrt{2}} \\)<\/p>\n<p><strong>Example 10:<\/strong>\u00a0 \u00a0 \u00a0If tan (A + B) = \u221a3 and tan (A \u2013 B) = 1\/\u221a3 ; 0\u00b0 &lt; A + B \u2264 90\u00b0 ; A &gt; B, find A and B.<br \/>\ntan (A + B) = \u221a3 = tan 60\u00b0 &amp; tan (A \u2013 B) = 1\/\u221a3 = tan 30\u00b0<br \/>\nA + B = 60\u00b0 \u2026\u2026.(1)<br \/>\nA \u2013 B = 30\u00b0 \u2026\u2026.(2)<br \/>\n2A = 90\u00b0 \u00a0\u21d2 \u00a0A = 45\u00b0<br \/>\nadding (1) &amp; (2)<br \/>\nA + B = 60<br \/>\nA \u2013 B = 30<br \/>\nSubtract equation (2) from (1)<br \/>\nA + B = 60<br \/>\nA \u2013 B = 30<br \/>\n2B = 30\u00b0<br \/>\n\u21d2\u00a0B = 15\u00b0. Ans.<br \/>\n<strong>Note:<\/strong> sin(A + B) = sin A cos B + cos A sin B<br \/>\nsin(A + B) \u2260 sin A + sin B.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Trigonometric Ratios Of Some Specific Angles The angles 0\u00b0, 30\u00b0, 45\u00b0, 60\u00b0, 90\u00b0 are angles for which we have values of T.R. \u03b8 0\u00b0 30\u00b0 45\u00b0 60\u00b0 90\u00b0 Sin 0 1\/2 1\/\u221a2 \u00a0\u221a3\/2 1 Cos 1 \u00a0\u221a3\/2 \u00a01\/\u221a2 1\/2 0 Tan 0 \u00a01\/\u221a3 1 \u221a3 \u221e Cosec \u221e 2 \u221a2 2\/\u221a3 1 Sec 1 2\/\u221a3 [&hellip;]<\/p>\n","protected":false},"author":3,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_genesis_hide_title":false,"_genesis_hide_breadcrumbs":false,"_genesis_hide_singular_image":false,"_genesis_hide_footer_widgets":false,"_genesis_custom_body_class":"","_genesis_custom_post_class":"","_genesis_layout":"","footnotes":""},"categories":[5],"tags":[147,148,146],"yoast_head":"<!-- This site is optimized with the Yoast SEO Premium plugin v17.8 (Yoast SEO v17.8) - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Trigonometric Ratios Of Some Specific Angles - A Plus Topper<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/www.aplustopper.com\/trigonometry-for-specific-angles\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Trigonometric Ratios Of Some Specific Angles\" \/>\n<meta property=\"og:description\" content=\"Trigonometric Ratios Of Some Specific Angles The angles 0\u00b0, 30\u00b0, 45\u00b0, 60\u00b0, 90\u00b0 are angles for which we have values of T.R. \u03b8 0\u00b0 30\u00b0 45\u00b0 60\u00b0 90\u00b0 Sin 0 1\/2 1\/\u221a2 \u00a0\u221a3\/2 1 Cos 1 \u00a0\u221a3\/2 \u00a01\/\u221a2 1\/2 0 Tan 0 \u00a01\/\u221a3 1 \u221a3 \u221e Cosec \u221e 2 \u221a2 2\/\u221a3 1 Sec 1 2\/\u221a3 [&hellip;]\" \/>\n<meta property=\"og:url\" content=\"https:\/\/www.aplustopper.com\/trigonometry-for-specific-angles\/\" \/>\n<meta property=\"og:site_name\" content=\"A Plus Topper\" \/>\n<meta property=\"article:publisher\" content=\"https:\/\/www.facebook.com\/aplustopper\/\" \/>\n<meta property=\"article:published_time\" content=\"2023-05-05T04:30:42+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2023-05-06T03:49:01+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2016\/08\/Trigonometric-Ratios.png\" \/>\n<meta name=\"twitter:card\" content=\"summary\" \/>\n<meta name=\"twitter:label1\" content=\"Written by\" \/>\n\t<meta name=\"twitter:data1\" content=\"Veerendra\" \/>\n\t<meta name=\"twitter:label2\" content=\"Est. reading time\" \/>\n\t<meta name=\"twitter:data2\" content=\"4 minutes\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"Organization\",\"@id\":\"https:\/\/www.aplustopper.com\/#organization\",\"name\":\"Aplus Topper\",\"url\":\"https:\/\/www.aplustopper.com\/\",\"sameAs\":[\"https:\/\/www.facebook.com\/aplustopper\/\"],\"logo\":{\"@type\":\"ImageObject\",\"@id\":\"https:\/\/www.aplustopper.com\/#logo\",\"inLanguage\":\"en-US\",\"url\":\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2018\/12\/Aplus_380x90-logo.jpg\",\"contentUrl\":\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2018\/12\/Aplus_380x90-logo.jpg\",\"width\":1585,\"height\":375,\"caption\":\"Aplus Topper\"},\"image\":{\"@id\":\"https:\/\/www.aplustopper.com\/#logo\"}},{\"@type\":\"WebSite\",\"@id\":\"https:\/\/www.aplustopper.com\/#website\",\"url\":\"https:\/\/www.aplustopper.com\/\",\"name\":\"A Plus Topper\",\"description\":\"Improve your Grades\",\"publisher\":{\"@id\":\"https:\/\/www.aplustopper.com\/#organization\"},\"potentialAction\":[{\"@type\":\"SearchAction\",\"target\":{\"@type\":\"EntryPoint\",\"urlTemplate\":\"https:\/\/www.aplustopper.com\/?s={search_term_string}\"},\"query-input\":\"required name=search_term_string\"}],\"inLanguage\":\"en-US\"},{\"@type\":\"ImageObject\",\"@id\":\"https:\/\/www.aplustopper.com\/trigonometry-for-specific-angles\/#primaryimage\",\"inLanguage\":\"en-US\",\"url\":\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2016\/08\/Trigonometric-Ratios.png\",\"contentUrl\":\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2016\/08\/Trigonometric-Ratios.png\",\"width\":722,\"height\":466},{\"@type\":\"WebPage\",\"@id\":\"https:\/\/www.aplustopper.com\/trigonometry-for-specific-angles\/#webpage\",\"url\":\"https:\/\/www.aplustopper.com\/trigonometry-for-specific-angles\/\",\"name\":\"Trigonometric Ratios Of Some Specific Angles - A Plus Topper\",\"isPartOf\":{\"@id\":\"https:\/\/www.aplustopper.com\/#website\"},\"primaryImageOfPage\":{\"@id\":\"https:\/\/www.aplustopper.com\/trigonometry-for-specific-angles\/#primaryimage\"},\"datePublished\":\"2023-05-05T04:30:42+00:00\",\"dateModified\":\"2023-05-06T03:49:01+00:00\",\"breadcrumb\":{\"@id\":\"https:\/\/www.aplustopper.com\/trigonometry-for-specific-angles\/#breadcrumb\"},\"inLanguage\":\"en-US\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"https:\/\/www.aplustopper.com\/trigonometry-for-specific-angles\/\"]}]},{\"@type\":\"BreadcrumbList\",\"@id\":\"https:\/\/www.aplustopper.com\/trigonometry-for-specific-angles\/#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Home\",\"item\":\"https:\/\/www.aplustopper.com\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"Trigonometric Ratios Of Some Specific Angles\"}]},{\"@type\":\"Article\",\"@id\":\"https:\/\/www.aplustopper.com\/trigonometry-for-specific-angles\/#article\",\"isPartOf\":{\"@id\":\"https:\/\/www.aplustopper.com\/trigonometry-for-specific-angles\/#webpage\"},\"author\":{\"@id\":\"https:\/\/www.aplustopper.com\/#\/schema\/person\/1363657f2432a17193a1f5d6404b8a0e\"},\"headline\":\"Trigonometric Ratios Of Some Specific Angles\",\"datePublished\":\"2023-05-05T04:30:42+00:00\",\"dateModified\":\"2023-05-06T03:49:01+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\/\/www.aplustopper.com\/trigonometry-for-specific-angles\/#webpage\"},\"wordCount\":827,\"commentCount\":1,\"publisher\":{\"@id\":\"https:\/\/www.aplustopper.com\/#organization\"},\"image\":{\"@id\":\"https:\/\/www.aplustopper.com\/trigonometry-for-specific-angles\/#primaryimage\"},\"thumbnailUrl\":\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2016\/08\/Trigonometric-Ratios.png\",\"keywords\":[\"Specific Angles\",\"Trigonometric Ratios Of Specific Angles\",\"Trigonometry\"],\"articleSection\":[\"Mathematics\"],\"inLanguage\":\"en-US\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"https:\/\/www.aplustopper.com\/trigonometry-for-specific-angles\/#respond\"]}]},{\"@type\":\"Person\",\"@id\":\"https:\/\/www.aplustopper.com\/#\/schema\/person\/1363657f2432a17193a1f5d6404b8a0e\",\"name\":\"Veerendra\",\"image\":{\"@type\":\"ImageObject\",\"@id\":\"https:\/\/www.aplustopper.com\/#personlogo\",\"inLanguage\":\"en-US\",\"url\":\"https:\/\/secure.gravatar.com\/avatar\/93ba5dedf76e89af65ae2e1d6ac09855?s=96&d=mm&r=g\",\"contentUrl\":\"https:\/\/secure.gravatar.com\/avatar\/93ba5dedf76e89af65ae2e1d6ac09855?s=96&d=mm&r=g\",\"caption\":\"Veerendra\"},\"url\":\"https:\/\/www.aplustopper.com\/author\/veerendra\/\"}]}<\/script>\n<!-- \/ Yoast SEO Premium plugin. -->","yoast_head_json":{"title":"Trigonometric Ratios Of Some Specific Angles - A Plus Topper","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"https:\/\/www.aplustopper.com\/trigonometry-for-specific-angles\/","og_locale":"en_US","og_type":"article","og_title":"Trigonometric Ratios Of Some Specific Angles","og_description":"Trigonometric Ratios Of Some Specific Angles The angles 0\u00b0, 30\u00b0, 45\u00b0, 60\u00b0, 90\u00b0 are angles for which we have values of T.R. \u03b8 0\u00b0 30\u00b0 45\u00b0 60\u00b0 90\u00b0 Sin 0 1\/2 1\/\u221a2 \u00a0\u221a3\/2 1 Cos 1 \u00a0\u221a3\/2 \u00a01\/\u221a2 1\/2 0 Tan 0 \u00a01\/\u221a3 1 \u221a3 \u221e Cosec \u221e 2 \u221a2 2\/\u221a3 1 Sec 1 2\/\u221a3 [&hellip;]","og_url":"https:\/\/www.aplustopper.com\/trigonometry-for-specific-angles\/","og_site_name":"A Plus Topper","article_publisher":"https:\/\/www.facebook.com\/aplustopper\/","article_published_time":"2023-05-05T04:30:42+00:00","article_modified_time":"2023-05-06T03:49:01+00:00","og_image":[{"url":"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2016\/08\/Trigonometric-Ratios.png"}],"twitter_card":"summary","twitter_misc":{"Written by":"Veerendra","Est. reading time":"4 minutes"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"Organization","@id":"https:\/\/www.aplustopper.com\/#organization","name":"Aplus Topper","url":"https:\/\/www.aplustopper.com\/","sameAs":["https:\/\/www.facebook.com\/aplustopper\/"],"logo":{"@type":"ImageObject","@id":"https:\/\/www.aplustopper.com\/#logo","inLanguage":"en-US","url":"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2018\/12\/Aplus_380x90-logo.jpg","contentUrl":"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2018\/12\/Aplus_380x90-logo.jpg","width":1585,"height":375,"caption":"Aplus Topper"},"image":{"@id":"https:\/\/www.aplustopper.com\/#logo"}},{"@type":"WebSite","@id":"https:\/\/www.aplustopper.com\/#website","url":"https:\/\/www.aplustopper.com\/","name":"A Plus Topper","description":"Improve your Grades","publisher":{"@id":"https:\/\/www.aplustopper.com\/#organization"},"potentialAction":[{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"https:\/\/www.aplustopper.com\/?s={search_term_string}"},"query-input":"required name=search_term_string"}],"inLanguage":"en-US"},{"@type":"ImageObject","@id":"https:\/\/www.aplustopper.com\/trigonometry-for-specific-angles\/#primaryimage","inLanguage":"en-US","url":"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2016\/08\/Trigonometric-Ratios.png","contentUrl":"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2016\/08\/Trigonometric-Ratios.png","width":722,"height":466},{"@type":"WebPage","@id":"https:\/\/www.aplustopper.com\/trigonometry-for-specific-angles\/#webpage","url":"https:\/\/www.aplustopper.com\/trigonometry-for-specific-angles\/","name":"Trigonometric Ratios Of Some Specific Angles - A Plus Topper","isPartOf":{"@id":"https:\/\/www.aplustopper.com\/#website"},"primaryImageOfPage":{"@id":"https:\/\/www.aplustopper.com\/trigonometry-for-specific-angles\/#primaryimage"},"datePublished":"2023-05-05T04:30:42+00:00","dateModified":"2023-05-06T03:49:01+00:00","breadcrumb":{"@id":"https:\/\/www.aplustopper.com\/trigonometry-for-specific-angles\/#breadcrumb"},"inLanguage":"en-US","potentialAction":[{"@type":"ReadAction","target":["https:\/\/www.aplustopper.com\/trigonometry-for-specific-angles\/"]}]},{"@type":"BreadcrumbList","@id":"https:\/\/www.aplustopper.com\/trigonometry-for-specific-angles\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https:\/\/www.aplustopper.com\/"},{"@type":"ListItem","position":2,"name":"Trigonometric Ratios Of Some Specific Angles"}]},{"@type":"Article","@id":"https:\/\/www.aplustopper.com\/trigonometry-for-specific-angles\/#article","isPartOf":{"@id":"https:\/\/www.aplustopper.com\/trigonometry-for-specific-angles\/#webpage"},"author":{"@id":"https:\/\/www.aplustopper.com\/#\/schema\/person\/1363657f2432a17193a1f5d6404b8a0e"},"headline":"Trigonometric Ratios Of Some Specific Angles","datePublished":"2023-05-05T04:30:42+00:00","dateModified":"2023-05-06T03:49:01+00:00","mainEntityOfPage":{"@id":"https:\/\/www.aplustopper.com\/trigonometry-for-specific-angles\/#webpage"},"wordCount":827,"commentCount":1,"publisher":{"@id":"https:\/\/www.aplustopper.com\/#organization"},"image":{"@id":"https:\/\/www.aplustopper.com\/trigonometry-for-specific-angles\/#primaryimage"},"thumbnailUrl":"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2016\/08\/Trigonometric-Ratios.png","keywords":["Specific Angles","Trigonometric Ratios Of Specific Angles","Trigonometry"],"articleSection":["Mathematics"],"inLanguage":"en-US","potentialAction":[{"@type":"CommentAction","name":"Comment","target":["https:\/\/www.aplustopper.com\/trigonometry-for-specific-angles\/#respond"]}]},{"@type":"Person","@id":"https:\/\/www.aplustopper.com\/#\/schema\/person\/1363657f2432a17193a1f5d6404b8a0e","name":"Veerendra","image":{"@type":"ImageObject","@id":"https:\/\/www.aplustopper.com\/#personlogo","inLanguage":"en-US","url":"https:\/\/secure.gravatar.com\/avatar\/93ba5dedf76e89af65ae2e1d6ac09855?s=96&d=mm&r=g","contentUrl":"https:\/\/secure.gravatar.com\/avatar\/93ba5dedf76e89af65ae2e1d6ac09855?s=96&d=mm&r=g","caption":"Veerendra"},"url":"https:\/\/www.aplustopper.com\/author\/veerendra\/"}]}},"jetpack_sharing_enabled":true,"jetpack_featured_media_url":"","_links":{"self":[{"href":"https:\/\/www.aplustopper.com\/wp-json\/wp\/v2\/posts\/899"}],"collection":[{"href":"https:\/\/www.aplustopper.com\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.aplustopper.com\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.aplustopper.com\/wp-json\/wp\/v2\/users\/3"}],"replies":[{"embeddable":true,"href":"https:\/\/www.aplustopper.com\/wp-json\/wp\/v2\/comments?post=899"}],"version-history":[{"count":1,"href":"https:\/\/www.aplustopper.com\/wp-json\/wp\/v2\/posts\/899\/revisions"}],"predecessor-version":[{"id":159341,"href":"https:\/\/www.aplustopper.com\/wp-json\/wp\/v2\/posts\/899\/revisions\/159341"}],"wp:attachment":[{"href":"https:\/\/www.aplustopper.com\/wp-json\/wp\/v2\/media?parent=899"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.aplustopper.com\/wp-json\/wp\/v2\/categories?post=899"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.aplustopper.com\/wp-json\/wp\/v2\/tags?post=899"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}