{"id":3198,"date":"2026-01-30T14:34:58","date_gmt":"2026-01-30T09:04:58","guid":{"rendered":"https:\/\/www.manabadi.co.in\/boards\/?p=3198"},"modified":"2026-01-30T14:35:02","modified_gmt":"2026-01-30T09:05:02","slug":"ts-inter-1st-year-maths-1a-trigonometric-ratios-upto-transformations-solutions-exercise-6b","status":"publish","type":"post","link":"https:\/\/www.manabadi.co.in\/boards\/ts-inter-1st-year-maths-1a-trigonometric-ratios-upto-transformations-solutions-exercise-6b\/","title":{"rendered":"TS Inter 1st Year Maths 1A Trigonometric Ratios upto Transformations Solutions Exercise 6(b)"},"content":{"rendered":"\n<h3>I.<br> Question 1.<br> Find the periods for the given 1 \u2013 5 functions.<br> (i) cos (3x + 5) + 7<\/h3>\n\n<p>Answer:<br> Let f(x) = cos (3x + 5) + 7<br> We have period of cos x is 2\u03c0 \u2200 x \u2208 R<br> \u2234 f (x) is periodic and period of f is = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-1-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-1\" style=\"width: 4.289em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 3.684em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.307em, 1003.68em, 3.078em, -999.998em); top: -2.328em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-2\"><span class=\"mfrac\" id=\"MathJax-Span-3\"><span style=\"display: inline-block; position: relative; width: 0.888em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.75em, 4.15em, -999.998em); top: -4.425em; left: 50%; margin-left: -0.37em;\"><span class=\"mrow\" id=\"MathJax-Span-4\"><span class=\"mn\" id=\"MathJax-Span-5\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span class=\"mi\" id=\"MathJax-Span-6\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic;\">\u03c0<span style=\"display: inline-block; overflow: hidden; height: 1px; width: 0.002em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.357em, 1000.65em, 4.336em, -999.998em); top: -3.586em; left: 50%; margin-left: -0.37em;\"><span class=\"mrow\" id=\"MathJax-Span-7\"><span class=\"texatom\" id=\"MathJax-Span-8\"><span class=\"mrow\" id=\"MathJax-Span-9\"><span class=\"mo\" id=\"MathJax-Span-10\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">|<\/span><\/span><\/span><span class=\"mn\" id=\"MathJax-Span-11\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">3<\/span><span class=\"texatom\" id=\"MathJax-Span-12\"><span class=\"mrow\" id=\"MathJax-Span-13\"><span class=\"mo\" id=\"MathJax-Span-14\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">|<\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1000.89em, 1.214em, -999.998em); top: -1.302em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.888em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-15\" style=\"font-family: MathJax_Main; padding-left: 0.282em;\">=<\/span><span class=\"mfrac\" id=\"MathJax-Span-16\" style=\"padding-left: 0.282em;\"><span style=\"display: inline-block; position: relative; width: 0.888em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.75em, 4.15em, -999.998em); top: -4.425em; left: 50%; margin-left: -0.37em;\"><span class=\"mrow\" id=\"MathJax-Span-17\"><span class=\"mn\" id=\"MathJax-Span-18\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span class=\"mi\" id=\"MathJax-Span-19\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic;\">\u03c0<span style=\"display: inline-block; overflow: hidden; height: 1px; width: 0.002em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1000.33em, 4.15em, -999.998em); top: -3.632em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-20\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">3<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1000.89em, 1.214em, -999.998em); top: -1.302em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.888em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 2.332em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.754em; border-left: 0px solid; width: 0px; height: 1.841em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-1\">\\frac{2 \\pi}{|3|}=\\frac{2 \\pi}{3}<\/script><br> (or) f (x + p) = f(x)<br> \u21d2 cos (3x + 3p + 5) + 7 = cos (2\u03c0+ 3x + 5) + 7<br> 3x + 3p + 5 = 2\u03c0 + 3x + 5<br> \u21d2 3x = 2\u03c0<br> \u21d2 x = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-2-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-21\" style=\"width: 1.354em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 1.167em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.26em, 1001.17em, 2.798em, -999.998em); top: -2.281em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-22\"><span class=\"mfrac\" id=\"MathJax-Span-23\"><span style=\"display: inline-block; position: relative; width: 0.888em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.75em, 4.15em, -999.998em); top: -4.425em; left: 50%; margin-left: -0.37em;\"><span class=\"mrow\" id=\"MathJax-Span-24\"><span class=\"mn\" id=\"MathJax-Span-25\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span class=\"mi\" id=\"MathJax-Span-26\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic;\">\u03c0<span style=\"display: inline-block; overflow: hidden; height: 1px; width: 0.002em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1000.33em, 4.15em, -999.998em); top: -3.632em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-27\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">3<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1000.89em, 1.214em, -999.998em); top: -1.302em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.888em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 2.286em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.484em; border-left: 0px solid; width: 0px; height: 1.57em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-2\">\\frac{2 \\pi}{3}<\/script><\/p>\n\n<h3>Question 2.<br> tan x<\/h3>\n\n<p>Answer:<br> The function tan x is periodic with period \u03c0<br> \u2234 f(x) = tan 5x is periodic and its period is<br> <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-3-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-28\" style=\"width: 3.87em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 3.311em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.493em, 1003.31em, 3.078em, -999.998em); top: -2.328em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-29\"><span class=\"mfrac\" id=\"MathJax-Span-30\"><span style=\"display: inline-block; position: relative; width: 0.888em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.544em, 1000.42em, 4.15em, -999.998em); top: -4.425em; left: 50%; margin-left: -0.184em;\"><span class=\"mi\" id=\"MathJax-Span-31\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic;\">\u03c0<span style=\"display: inline-block; overflow: hidden; height: 1px; width: 0.002em;\"><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.357em, 1000.65em, 4.336em, -999.998em); top: -3.586em; left: 50%; margin-left: -0.37em;\"><span class=\"mrow\" id=\"MathJax-Span-32\"><span class=\"texatom\" id=\"MathJax-Span-33\"><span class=\"mrow\" id=\"MathJax-Span-34\"><span class=\"mo\" id=\"MathJax-Span-35\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">|<\/span><\/span><\/span><span class=\"mn\" id=\"MathJax-Span-36\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">5<\/span><span class=\"texatom\" id=\"MathJax-Span-37\"><span class=\"mrow\" id=\"MathJax-Span-38\"><span class=\"mo\" id=\"MathJax-Span-39\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">|<\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1000.89em, 1.214em, -999.998em); top: -1.302em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.888em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-40\" style=\"font-family: MathJax_Main; padding-left: 0.282em;\">=<\/span><span class=\"mfrac\" id=\"MathJax-Span-41\" style=\"padding-left: 0.282em;\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.544em, 1000.42em, 4.15em, -999.998em); top: -4.425em; left: 50%; margin-left: -0.184em;\"><span class=\"mi\" id=\"MathJax-Span-42\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic;\">\u03c0<span style=\"display: inline-block; overflow: hidden; height: 1px; width: 0.002em;\"><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1000.33em, 4.15em, -999.998em); top: -3.632em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-43\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">5<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1000.51em, 1.214em, -999.998em); top: -1.302em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.515em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 2.332em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.754em; border-left: 0px solid; width: 0px; height: 1.624em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-3\">\\frac{\\pi}{|5|}=\\frac{\\pi}{5}<\/script><\/p>\n\n<h3>Question 3.<br> cos<span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-4-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-44\" style=\"width: 3.777em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 3.264em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.866em, 1003.08em, 3.963em, -999.998em); top: -3.166em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-45\"><span class=\"mrow\" id=\"MathJax-Span-46\"><span class=\"mo\" id=\"MathJax-Span-47\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Size2;\">(<\/span><\/span><span class=\"mfrac\" id=\"MathJax-Span-48\"><span style=\"display: inline-block; position: relative; width: 1.773em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.404em, 1001.63em, 4.196em, -999.998em); top: -4.471em; left: 50%; margin-left: -0.836em;\"><span class=\"mrow\" id=\"MathJax-Span-49\"><span class=\"mn\" id=\"MathJax-Span-50\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">4<\/span><span class=\"mi\" id=\"MathJax-Span-51\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic;\">x<\/span><span class=\"mo\" id=\"MathJax-Span-52\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">+<\/span><span class=\"mn\" id=\"MathJax-Span-53\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">9<\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1000.33em, 4.15em, -999.998em); top: -3.632em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-54\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">5<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1001.77em, 1.214em, -999.998em); top: -1.302em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 1.773em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-55\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Size2;\">)<\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 3.171em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.808em; border-left: 0px solid; width: 0px; height: 2.219em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-4\">\\left(\\frac{4 x+9}{5}\\right)<\/script> (Mar. \u201914)<\/h3>\n\n\n\n<p>Answer:<br> The function f(x) = cos x \u2200 x \u2208 R has the period 2\u03c0<br> \u2234 f(x) = cos<span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-5-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-56\" style=\"width: 3.777em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 3.264em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.866em, 1003.08em, 3.963em, -999.998em); top: -3.166em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-57\"><span class=\"mrow\" id=\"MathJax-Span-58\"><span class=\"mo\" id=\"MathJax-Span-59\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Size2;\">(<\/span><\/span><span class=\"mfrac\" id=\"MathJax-Span-60\"><span style=\"display: inline-block; position: relative; width: 1.773em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.404em, 1001.63em, 4.196em, -999.998em); top: -4.471em; left: 50%; margin-left: -0.836em;\"><span class=\"mrow\" id=\"MathJax-Span-61\"><span class=\"mn\" id=\"MathJax-Span-62\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">4<\/span><span class=\"mi\" id=\"MathJax-Span-63\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic;\">x<\/span><span class=\"mo\" id=\"MathJax-Span-64\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">+<\/span><span class=\"mn\" id=\"MathJax-Span-65\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">9<\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1000.33em, 4.15em, -999.998em); top: -3.632em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-66\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">5<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1001.77em, 1.214em, -999.998em); top: -1.302em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 1.773em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-67\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Size2;\">)<\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 3.171em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.808em; border-left: 0px solid; width: 0px; height: 2.219em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-5\">\\left(\\frac{4 x+9}{5}\\right)<\/script> is periodic and period of f is <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-6-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-68\" style=\"width: 4.289em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 3.684em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.307em, 1003.68em, 3.357em, -999.998em); top: -2.328em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-69\"><span class=\"mfrac\" id=\"MathJax-Span-70\"><span style=\"display: inline-block; position: relative; width: 0.888em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.75em, 4.15em, -999.998em); top: -4.425em; left: 50%; margin-left: -0.37em;\"><span class=\"mrow\" id=\"MathJax-Span-71\"><span class=\"mn\" id=\"MathJax-Span-72\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span class=\"mi\" id=\"MathJax-Span-73\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic;\">\u03c0<span style=\"display: inline-block; overflow: hidden; height: 1px; width: 0.002em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.61em, 4.476em, -999.998em); top: -3.446em; left: 50%; margin-left: -0.324em;\"><span class=\"mfrac\" id=\"MathJax-Span-74\"><span style=\"display: inline-block; position: relative; width: 0.375em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.544em, 1000.24em, 4.15em, -999.998em); top: -4.331em; left: 50%; margin-left: -0.137em;\"><span class=\"mn\" id=\"MathJax-Span-75\" style=\"font-size: 50%; font-family: MathJax_Main;\">4<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.544em, 1000.24em, 4.15em, -999.998em); top: -3.679em; left: 50%; margin-left: -0.137em;\"><span class=\"mn\" id=\"MathJax-Span-76\" style=\"font-size: 50%; font-family: MathJax_Main;\">5<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1000.38em, 1.214em, -999.998em); top: -1.209em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.375em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1000.89em, 1.214em, -999.998em); top: -1.302em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.888em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-77\" style=\"font-family: MathJax_Main; padding-left: 0.282em;\">=<\/span><span class=\"mfrac\" id=\"MathJax-Span-78\" style=\"padding-left: 0.282em;\"><span style=\"display: inline-block; position: relative; width: 0.888em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.75em, 4.15em, -999.998em); top: -4.425em; left: 50%; margin-left: -0.37em;\"><span class=\"mrow\" id=\"MathJax-Span-79\"><span class=\"mn\" id=\"MathJax-Span-80\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">5<\/span><span class=\"mi\" id=\"MathJax-Span-81\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic;\">\u03c0<span style=\"display: inline-block; overflow: hidden; height: 1px; width: 0.002em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1000.33em, 4.15em, -999.998em); top: -3.632em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-82\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1000.89em, 1.214em, -999.998em); top: -1.302em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.888em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 2.332em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -1.078em; border-left: 0px solid; width: 0px; height: 2.165em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-6\">\\frac{2 \\pi}{\\frac{4}{5}}=\\frac{5 \\pi}{2}<\/script><\/p>\n\n<h3>Question 4.<br> |sin x|<\/h3>\n\n<p>Answer:<br> The function sin x has period 2\u03c0 \u2200 x \u2208 R<br> But f(x) = |sin x| is periodic and its period is \u03c0<br> [\u2235 f(x + \u03c0) = |sin(x + \u03c0)| = |-sin x| = sinx]<\/p>\n\n<h3>Question 5.<br> tan (x + 4x + 9x + \u2026. + n<sup>2<\/sup>x) (n any positive integer) (March 2015-A.P&amp;T.S)<\/h3>\n\n<p>Answer:<br> tan [1 + 2<sup>2<\/sup> + 3<sup>2<\/sup> + \u2026\u2026\u2026 + n<sup>2<\/sup>) x<br> = tan<span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-7-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-83\" style=\"width: 6.712em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 5.781em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.866em, 1005.55em, 3.963em, -999.998em); top: -3.166em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-84\"><span class=\"mrow\" id=\"MathJax-Span-85\"><span class=\"mo\" id=\"MathJax-Span-86\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Size2;\">[<\/span><\/span><span class=\"mfrac\" id=\"MathJax-Span-87\"><span style=\"display: inline-block; position: relative; width: 4.569em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.357em, 1004.38em, 4.336em, -999.998em); top: -4.564em; left: 50%; margin-left: -2.234em;\"><span class=\"mrow\" id=\"MathJax-Span-88\"><span class=\"texatom\" id=\"MathJax-Span-89\"><span class=\"mrow\" id=\"MathJax-Span-90\"><span class=\"mi\" id=\"MathJax-Span-91\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">n<\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-92\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">(<\/span><span class=\"texatom\" id=\"MathJax-Span-93\"><span class=\"mrow\" id=\"MathJax-Span-94\"><span class=\"mi\" id=\"MathJax-Span-95\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">n<\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-96\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">+<\/span><span class=\"mn\" id=\"MathJax-Span-97\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">1<\/span><span class=\"mo\" id=\"MathJax-Span-98\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">)<\/span><span class=\"mo\" id=\"MathJax-Span-99\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">(<\/span><span class=\"mn\" id=\"MathJax-Span-100\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span class=\"texatom\" id=\"MathJax-Span-101\"><span class=\"mrow\" id=\"MathJax-Span-102\"><span class=\"mi\" id=\"MathJax-Span-103\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">n<\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-104\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">+<\/span><span class=\"mn\" id=\"MathJax-Span-105\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">1<\/span><span class=\"mo\" id=\"MathJax-Span-106\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">)<\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1000.33em, 4.15em, -999.998em); top: -3.632em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-107\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">6<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1004.57em, 1.214em, -999.998em); top: -1.302em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 4.569em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-108\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Size2;\">]<\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 3.171em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.808em; border-left: 0px solid; width: 0px; height: 2.219em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-7\">\\left[\\frac{\\mathrm{n}(\\mathrm{n}+1)(2 \\mathrm{n}+1)}{6}\\right]<\/script>x<br> Period = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-8-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-109\" style=\"width: 5.734em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 4.942em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.26em, 1004.94em, 3.031em, -999.998em); top: -2.281em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-110\"><span class=\"mfrac\" id=\"MathJax-Span-111\"><span style=\"display: inline-block; position: relative; width: 4.662em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.75em, 4.15em, -999.998em); top: -4.425em; left: 50%; margin-left: -0.37em;\"><span class=\"mrow\" id=\"MathJax-Span-112\"><span class=\"mn\" id=\"MathJax-Span-113\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">6<\/span><span class=\"mi\" id=\"MathJax-Span-114\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic;\">\u03c0<span style=\"display: inline-block; overflow: hidden; height: 1px; width: 0.002em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.357em, 1004.48em, 4.336em, -999.998em); top: -3.586em; left: 50%; margin-left: -2.281em;\"><span class=\"mrow\" id=\"MathJax-Span-115\"><span class=\"mi\" id=\"MathJax-Span-116\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic;\">n<\/span><span class=\"mo\" id=\"MathJax-Span-117\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">(<\/span><span class=\"mi\" id=\"MathJax-Span-118\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic;\">n<\/span><span class=\"mo\" id=\"MathJax-Span-119\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">+<\/span><span class=\"mn\" id=\"MathJax-Span-120\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">1<\/span><span class=\"mo\" id=\"MathJax-Span-121\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">)<\/span><span class=\"mo\" id=\"MathJax-Span-122\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">(<\/span><span class=\"mn\" id=\"MathJax-Span-123\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span class=\"mi\" id=\"MathJax-Span-124\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic;\">n<\/span><span class=\"mo\" id=\"MathJax-Span-125\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">+<\/span><span class=\"mn\" id=\"MathJax-Span-126\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">1<\/span><span class=\"mo\" id=\"MathJax-Span-127\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">)<\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1004.66em, 1.214em, -999.998em); top: -1.302em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 4.662em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 2.286em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.754em; border-left: 0px solid; width: 0px; height: 1.841em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-8\">\\frac{6 \\pi}{n(n+1)(2 n+1)}<\/script><\/p>\n\n\n\n<h3>Question 6.<br> Find a sine function whose period is <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-9-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-128\" style=\"width: 0.888em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 0.748em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.26em, 1000.75em, 2.798em, -999.998em); top: -2.281em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-129\"><span class=\"mfrac\" id=\"MathJax-Span-130\"><span style=\"display: inline-block; position: relative; width: 0.468em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.33em, 4.15em, -999.998em); top: -4.425em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-131\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1000.33em, 4.15em, -999.998em); top: -3.632em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-132\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">3<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1000.47em, 1.214em, -999.998em); top: -1.302em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.468em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 2.286em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.484em; border-left: 0px solid; width: 0px; height: 1.57em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-9\">\\frac{2}{3}<\/script>.<\/h3>\n\n<p>Answer:<br> <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-10-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-133\" style=\"width: 3.777em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 3.264em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.307em, 1003.26em, 2.845em, -999.998em); top: -2.328em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-134\"><span class=\"mfrac\" id=\"MathJax-Span-135\"><span style=\"display: inline-block; position: relative; width: 0.888em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.75em, 4.15em, -999.998em); top: -4.425em; left: 50%; margin-left: -0.37em;\"><span class=\"mrow\" id=\"MathJax-Span-136\"><span class=\"mn\" id=\"MathJax-Span-137\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span class=\"mi\" id=\"MathJax-Span-138\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic;\">\u03c0<span style=\"display: inline-block; overflow: hidden; height: 1px; width: 0.002em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.357em, 1000.38em, 4.15em, -999.998em); top: -3.632em; left: 50%; margin-left: -0.184em;\"><span class=\"texatom\" id=\"MathJax-Span-139\"><span class=\"mrow\" id=\"MathJax-Span-140\"><span class=\"mi\" id=\"MathJax-Span-141\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">k<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1000.89em, 1.214em, -999.998em); top: -1.302em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.888em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-142\" style=\"font-family: MathJax_Main; padding-left: 0.282em;\">=<\/span><span class=\"mfrac\" id=\"MathJax-Span-143\" style=\"padding-left: 0.282em;\"><span style=\"display: inline-block; position: relative; width: 0.468em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.33em, 4.15em, -999.998em); top: -4.425em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-144\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1000.33em, 4.15em, -999.998em); top: -3.632em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-145\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">3<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1000.47em, 1.214em, -999.998em); top: -1.302em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.468em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 2.332em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.484em; border-left: 0px solid; width: 0px; height: 1.57em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-10\">\\frac{2 \\pi}{\\mathrm{k}}=\\frac{2}{3}<\/script> \u21d2 3\u03c0 = |k| \u2234 sin kx = sin (3n x)<\/p>\n\n<h3>Question 7.<br> Find a cosine function whose period is 7. (March 2013)<\/h3>\n\n<p>Answer:<br> f(x) = cos[<span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-11-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-146\" style=\"width: 1.354em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 1.167em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.26em, 1001.17em, 2.798em, -999.998em); top: -2.281em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-147\"><span class=\"mfrac\" id=\"MathJax-Span-148\"><span style=\"display: inline-block; position: relative; width: 0.888em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.75em, 4.15em, -999.998em); top: -4.425em; left: 50%; margin-left: -0.37em;\"><span class=\"mrow\" id=\"MathJax-Span-149\"><span class=\"mn\" id=\"MathJax-Span-150\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span class=\"mi\" id=\"MathJax-Span-151\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic;\">\u03c0<span style=\"display: inline-block; overflow: hidden; height: 1px; width: 0.002em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1000.33em, 4.15em, -999.998em); top: -3.632em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-152\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">7<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1000.89em, 1.214em, -999.998em); top: -1.302em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.888em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 2.286em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.484em; border-left: 0px solid; width: 0px; height: 1.57em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-11\">\\frac{2 \\pi}{7}<\/script> .x] (<span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-12-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-153\" style=\"width: 1.354em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 1.167em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.26em, 1001.17em, 2.798em, -999.998em); top: -2.281em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-154\"><span class=\"mfrac\" id=\"MathJax-Span-155\"><span style=\"display: inline-block; position: relative; width: 0.888em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.75em, 4.15em, -999.998em); top: -4.425em; left: 50%; margin-left: -0.37em;\"><span class=\"mrow\" id=\"MathJax-Span-156\"><span class=\"mn\" id=\"MathJax-Span-157\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span class=\"mi\" id=\"MathJax-Span-158\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic;\">\u03c0<span style=\"display: inline-block; overflow: hidden; height: 1px; width: 0.002em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.357em, 1000.38em, 4.15em, -999.998em); top: -3.632em; left: 50%; margin-left: -0.184em;\"><span class=\"mi\" id=\"MathJax-Span-159\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic;\">k<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1000.89em, 1.214em, -999.998em); top: -1.302em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.888em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 2.286em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.484em; border-left: 0px solid; width: 0px; height: 1.57em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-12\">\\frac{2 \\pi}{k}<\/script> = 7 \u21d2 <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-13-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-160\" style=\"width: 1.354em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 1.167em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.26em, 1001.17em, 2.798em, -999.998em); top: -2.281em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-161\"><span class=\"mfrac\" id=\"MathJax-Span-162\"><span style=\"display: inline-block; position: relative; width: 0.888em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.75em, 4.15em, -999.998em); top: -4.425em; left: 50%; margin-left: -0.37em;\"><span class=\"mrow\" id=\"MathJax-Span-163\"><span class=\"mn\" id=\"MathJax-Span-164\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span class=\"mi\" id=\"MathJax-Span-165\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic;\">\u03c0<span style=\"display: inline-block; overflow: hidden; height: 1px; width: 0.002em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1000.33em, 4.15em, -999.998em); top: -3.632em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-166\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">7<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1000.89em, 1.214em, -999.998em); top: -1.302em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.888em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 2.286em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.484em; border-left: 0px solid; width: 0px; height: 1.57em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-13\">\\frac{2 \\pi}{7}<\/script> = k)<\/p>\n\n\n\n<h3>II. Sketch the graph of the following functions<br> Question 1.<br> tan x between 0 and <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-14-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-167\" style=\"width: 0.934em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 0.795em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.447em, 1000.79em, 2.798em, -999.998em); top: -2.281em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-168\"><span class=\"mfrac\" id=\"MathJax-Span-169\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.544em, 1000.42em, 4.15em, -999.998em); top: -4.425em; left: 50%; margin-left: -0.184em;\"><span class=\"mi\" id=\"MathJax-Span-170\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic;\">\u03c0<span style=\"display: inline-block; overflow: hidden; height: 1px; width: 0.002em;\"><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1000.33em, 4.15em, -999.998em); top: -3.632em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-171\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">4<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1000.51em, 1.214em, -999.998em); top: -1.302em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.515em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 2.286em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.484em; border-left: 0px solid; width: 0px; height: 1.354em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-14\">\\frac{\\pi}{4}<\/script>.<\/h3>\n\n<p>Answer:<br> <img fetchpriority=\"high\" decoding=\"async\" class=\"alignnone size-full wp-image-7506\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-6-Trigonometric-Ratios-upto-Transformations-Ex-6b-1.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 6 Trigonometric Ratios upto Transformations Ex 6(b) 1\" width=\"311\" height=\"353\" sizes=\"(max-width: 311px) 100vw, 311px\" data-pin-description=\"TS Inter 1st Year Maths 1A Solutions Chapter 6 Trigonometric Ratios upto Transformations Ex 6(b) 1\" data-pin-title=\"TS Inter 1st Year Maths 1A Solutions Chapter 6 Trigonometric Ratios upto Transformations Ex 6(b)\"><\/p>\n\n<h3>Question 2.<br> cos 2x in the interval [0, \u03c0]<\/h3>\n\n<p>Answer:<br> <img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-7505\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-6-Trigonometric-Ratios-upto-Transformations-Ex-6b-2.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 6 Trigonometric Ratios upto Transformations Ex 6(b) 2\" width=\"367\" height=\"349\" sizes=\"auto, (max-width: 367px) 100vw, 367px\" data-pin-description=\"TS Inter 1st Year Maths 1A Solutions Chapter 6 Trigonometric Ratios upto Transformations Ex 6(b) 2\" data-pin-title=\"TS Inter 1st Year Maths 1A Solutions Chapter 6 Trigonometric Ratios upto Transformations Ex 6(b)\"><\/p>\n\n<h3>Question 3.<br> sin 2x in the interval (0, \u03c0).<\/h3>\n\n<p>Answer:<br> <img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-7504\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-6-Trigonometric-Ratios-upto-Transformations-Ex-6b-3.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 6 Trigonometric Ratios upto Transformations Ex 6(b) 3\" width=\"359\" height=\"371\" sizes=\"auto, (max-width: 359px) 100vw, 359px\" data-pin-description=\"TS Inter 1st Year Maths 1A Solutions Chapter 6 Trigonometric Ratios upto Transformations Ex 6(b) 3\" data-pin-title=\"TS Inter 1st Year Maths 1A Solutions Chapter 6 Trigonometric Ratios upto Transformations Ex 6(b)\"><\/p>\n\n<h3>Question 4.<br> sin x in the interval [-\u03c0, +\u03c0]. (May 2014)<\/h3>\n\n<p>Answer:<br> <img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-7503\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-6-Trigonometric-Ratios-upto-Transformations-Ex-6b-4.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 6 Trigonometric Ratios upto Transformations Ex 6(b) 4\" width=\"366\" height=\"395\" sizes=\"auto, (max-width: 366px) 100vw, 366px\" data-pin-description=\"TS Inter 1st Year Maths 1A Solutions Chapter 6 Trigonometric Ratios upto Transformations Ex 6(b) 4\" data-pin-title=\"TS Inter 1st Year Maths 1A Solutions Chapter 6 Trigonometric Ratios upto Transformations Ex 6(b)\"><\/p>\n\n<h3>Question 5.<br> cos<sup>2<\/sup>x in [0, \u03c0].<\/h3>\n\n<p>Answer:<br> <img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-7502\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-6-Trigonometric-Ratios-upto-Transformations-Ex-6b-5.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 6 Trigonometric Ratios upto Transformations Ex 6(b) 5\" width=\"363\" height=\"389\" sizes=\"auto, (max-width: 363px) 100vw, 363px\" data-pin-description=\"TS Inter 1st Year Maths 1A Solutions Chapter 6 Trigonometric Ratios upto Transformations Ex 6(b) 5\" data-pin-title=\"TS Inter 1st Year Maths 1A Solutions Chapter 6 Trigonometric Ratios upto Transformations Ex 6(b)\"><\/p>\n\n<h3>Question 6.<br> Sketch the region enclosed by y = sin x, y = cos x and X \u2013 axis in the interval [0, \u03c0].<\/h3>\n\n<p>Answer:<br> y = sin x<br> <img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-7501\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-6-Trigonometric-Ratios-upto-Transformations-Ex-6b-6.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 6 Trigonometric Ratios upto Transformations Ex 6(b) 6\" width=\"365\" height=\"504\" sizes=\"auto, (max-width: 365px) 100vw, 365px\" data-pin-description=\"TS Inter 1st Year Maths 1A Solutions Chapter 6 Trigonometric Ratios upto Transformations Ex 6(b) 6\" data-pin-title=\"TS Inter 1st Year Maths 1A Solutions Chapter 6 Trigonometric Ratios upto Transformations Ex 6(b)\"><\/p>\n","protected":false},"excerpt":{"rendered":"<p>I. Question 1. Find the periods for the given 1 \u2013 5 functions. (i) cos (3x + 5) + 7 Answer: Let f(x) = cos (3x + 5) + 7 We have period of cos x is 2\u03c0 \u2200 x \u2208 R \u2234 f (x) is periodic and period of f is = 2\u03c0|3|=2\u03c03 (or) [&hellip;]<\/p>\n","protected":false},"author":2,"featured_media":3354,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[1544,15,38],"tags":[],"class_list":{"0":"post-3198","1":"post","2":"type-post","3":"status-publish","4":"format-standard","5":"has-post-thumbnail","7":"category-study-material-tg-inter","8":"category-telangana","9":"category-tg-inter"},"acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.4 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\r\n<title>TS Inter 1st Year Maths 1A Trigonometric Ratios upto Transformations Solutions Exercise 6(b)<\/title>\r\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\r\n<link rel=\"canonical\" href=\"https:\/\/www.manabadi.co.in\/boards\/ts-inter-1st-year-maths-1a-trigonometric-ratios-upto-transformations-solutions-exercise-6b\/\" \/>\r\n<meta property=\"og:locale\" content=\"en_US\" \/>\r\n<meta property=\"og:type\" content=\"article\" \/>\r\n<meta property=\"og:title\" content=\"TS Inter 1st Year Maths 1A Trigonometric Ratios upto Transformations Solutions Exercise 6(b)\" \/>\r\n<meta property=\"og:description\" content=\"I. Question 1. Find the periods for the given 1 \u2013 5 functions. (i) cos (3x + 5) + 7 Answer: Let f(x) = cos (3x + 5) + 7 We have period of cos x is 2\u03c0 \u2200 x \u2208 R \u2234 f (x) is periodic and period of f is = 2\u03c0|3|=2\u03c03 (or) [&hellip;]\" \/>\r\n<meta property=\"og:url\" content=\"https:\/\/www.manabadi.co.in\/boards\/ts-inter-1st-year-maths-1a-trigonometric-ratios-upto-transformations-solutions-exercise-6b\/\" \/>\r\n<meta property=\"og:site_name\" content=\"Manabadi Boards\" \/>\r\n<meta property=\"article:published_time\" content=\"2026-01-30T09:04:58+00:00\" \/>\r\n<meta property=\"article:modified_time\" content=\"2026-01-30T09:05:02+00:00\" \/>\r\n<meta property=\"og:image\" content=\"https:\/\/boardscdn.manabadi.co.in\/wp-content\/uploads\/2026\/01\/07114617\/ts-inter-1st-year-maths-1a-trigonometric-ratios-upto-transformations-solutions-exercise-6b.jpg\" \/>\r\n\t<meta property=\"og:image:width\" content=\"900\" \/>\r\n\t<meta property=\"og:image:height\" content=\"600\" \/>\r\n\t<meta property=\"og:image:type\" 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Question 1. Find the periods for the given 1 \u2013 5 functions. (i) cos (3x + 5) + 7 Answer: Let f(x) = cos (3x + 5) + 7 We have period of cos x is 2\u03c0 \u2200 x \u2208 R \u2234 f (x) is periodic and period of f is = 2\u03c0|3|=2\u03c03 (or) [&hellip;]","og_url":"https:\/\/www.manabadi.co.in\/boards\/ts-inter-1st-year-maths-1a-trigonometric-ratios-upto-transformations-solutions-exercise-6b\/","og_site_name":"Manabadi Boards","article_published_time":"2026-01-30T09:04:58+00:00","article_modified_time":"2026-01-30T09:05:02+00:00","og_image":[{"width":900,"height":600,"url":"https:\/\/boardscdn.manabadi.co.in\/wp-content\/uploads\/2026\/01\/07114617\/ts-inter-1st-year-maths-1a-trigonometric-ratios-upto-transformations-solutions-exercise-6b.jpg","type":"image\/jpeg"}],"author":"Junaid","twitter_card":"summary_large_image","twitter_misc":{"Written by":"Junaid","Est. reading time":"4 minutes"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"Article","@id":"https:\/\/www.manabadi.co.in\/boards\/ts-inter-1st-year-maths-1a-trigonometric-ratios-upto-transformations-solutions-exercise-6b\/#article","isPartOf":{"@id":"https:\/\/www.manabadi.co.in\/boards\/ts-inter-1st-year-maths-1a-trigonometric-ratios-upto-transformations-solutions-exercise-6b\/"},"author":{"name":"Junaid","@id":"https:\/\/www.manabadi.co.in\/boards\/#\/schema\/person\/a55695e135e9dc26536b290e31b4179d"},"headline":"TS Inter 1st Year Maths 1A Trigonometric Ratios upto Transformations Solutions Exercise 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