{"id":1852,"date":"2026-05-21T09:09:53","date_gmt":"2026-05-21T09:09:53","guid":{"rendered":"https:\/\/mymockmate.com\/notes\/?p=1852"},"modified":"2026-05-21T09:09:58","modified_gmt":"2026-05-21T09:09:58","slug":"inverse-trigonometric-functions-exercise-2-1-solutions","status":"publish","type":"post","link":"https:\/\/mymockmate.com\/notes\/inverse-trigonometric-functions-exercise-2-1-solutions\/","title":{"rendered":"Inverse Trigonometric Functions Exercise 2.1 Solutions"},"content":{"rendered":"\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h1 class=\"wp-block-heading\">Short Intro<\/h1>\n\n\n\n<p class=\"wp-block-paragraph\">In this post, students can find complete step-by-step solutions for Class 12 Maths Chapter 2 Exercise 2.1 \u2013 Inverse Trigonometric Functions based on the latest NCERT syllabus. This exercise covers principal values, domains and ranges of inverse trigonometric functions, and evaluation of inverse trigonometric expressions in a simple and exam-oriented format.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h1 class=\"wp-block-heading\">Quick Information Box<\/h1>\n\n\n\n<figure class=\"wp-block-table\"><table class=\"has-fixed-layout\"><thead><tr><th>Item<\/th><th>Details<\/th><\/tr><\/thead><tbody><tr><td>Board<\/td><td>NCERT \/ CBSE<\/td><\/tr><tr><td>Class<\/td><td>12<\/td><\/tr><tr><td>Subject<\/td><td>Mathematics<\/td><\/tr><tr><td>Chapter<\/td><td>Inverse Trigonometric Functions<\/td><\/tr><tr><td>Exercise<\/td><td>2.1<\/td><\/tr><tr><td>Main Topics<\/td><td>Principal Values<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h1 class=\"wp-block-heading\">Concepts Used (Topics Covered)<\/h1>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Principal Value Branch<\/li>\n\n\n\n<li>Inverse Sine Function<\/li>\n\n\n\n<li>Inverse Cosine Function<\/li>\n\n\n\n<li>Inverse Tangent Function<\/li>\n\n\n\n<li>Inverse Cotangent Function<\/li>\n\n\n\n<li>Inverse Secant Function<\/li>\n\n\n\n<li>Inverse Cosecant Function<\/li>\n\n\n\n<li>Domain &amp; Range of Inverse Trigonometric Functions<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">The chapter explains principal value branches of inverse trigonometric functions.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h1 class=\"wp-block-heading\">Important Formulas<\/h1>\n\n\n\n<h2 class=\"wp-block-heading\">Principal Value Range of sin\u207b\u00b9x<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mrow><mi>sin<\/mi><mo>\u2061<\/mo><\/mrow><mrow><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><mi>x<\/mi><mo>\u2208<\/mo><mrow><mo fence=\"true\">[<\/mo><mo>\u2212<\/mo><mfrac><mi>\u03c0<\/mi><mn>2<\/mn><\/mfrac><mo separator=\"true\">,<\/mo><mfrac><mi>\u03c0<\/mi><mn>2<\/mn><\/mfrac><mo fence=\"true\">]<\/mo><\/mrow><\/mrow><annotation encoding=\"application\/x-tex\">\\sin^{-1}x\\in\\left[-\\frac{\\pi}{2},\\frac{\\pi}{2}\\right]<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\">Principal Value Range of cos\u207b\u00b9x<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mrow><mi>cos<\/mi><mo>\u2061<\/mo><\/mrow><mrow><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><mi>x<\/mi><mo>\u2208<\/mo><mo stretchy=\"false\">[<\/mo><mn>0<\/mn><mo separator=\"true\">,<\/mo><mi>\u03c0<\/mi><mo stretchy=\"false\">]<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">\\cos^{-1}x\\in[0,\\pi]<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\">Principal Value Range of tan\u207b\u00b9x<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mrow><mi>tan<\/mi><mo>\u2061<\/mo><\/mrow><mrow><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><mi>x<\/mi><mo>\u2208<\/mo><mrow><mo fence=\"true\">(<\/mo><mo>\u2212<\/mo><mfrac><mi>\u03c0<\/mi><mn>2<\/mn><\/mfrac><mo separator=\"true\">,<\/mo><mfrac><mi>\u03c0<\/mi><mn>2<\/mn><\/mfrac><mo fence=\"true\">)<\/mo><\/mrow><\/mrow><annotation encoding=\"application\/x-tex\">\\tan^{-1}x\\in\\left(-\\frac{\\pi}{2},\\frac{\\pi}{2}\\right)<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\">Principal Value Range of cot\u207b\u00b9x<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mrow><mi>cot<\/mi><mo>\u2061<\/mo><\/mrow><mrow><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><mi>x<\/mi><mo>\u2208<\/mo><mo stretchy=\"false\">(<\/mo><mn>0<\/mn><mo separator=\"true\">,<\/mo><mi>\u03c0<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">\\cot^{-1}x\\in(0,\\pi)<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h1 class=\"wp-block-heading\">Questions &amp; Step-by-step Solutions<\/h1>\n\n\n\n<h1 class=\"wp-block-heading\">Question 1<\/h1>\n\n\n\n<p class=\"wp-block-paragraph\">Find:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mrow><mi>sin<\/mi><mo>\u2061<\/mo><\/mrow><mrow><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><mrow><mo fence=\"true\">(<\/mo><mo>\u2212<\/mo><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><mo fence=\"true\">)<\/mo><\/mrow><\/mrow><annotation encoding=\"application\/x-tex\">\\sin^{-1}\\left(-\\frac{1}{2}\\right)<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Solution<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">We know:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>sin(\u2212\u03c0\/6)=\u22121\/2<\/code><\/pre>\n\n\n\n<p class=\"wp-block-paragraph\">Principal value range of:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>sin\u207b\u00b9x<\/code><\/pre>\n\n\n\n<p class=\"wp-block-paragraph\">is:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>&#91;\u2212\u03c0\/2, \u03c0\/2]<\/code><\/pre>\n\n\n\n<p class=\"wp-block-paragraph\">Therefore:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>Answer = \u2212\u03c0\/6<\/code><\/pre>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h1 class=\"wp-block-heading\">Question 2<\/h1>\n\n\n\n<p class=\"wp-block-paragraph\">Find:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mrow><mi>cos<\/mi><mo>\u2061<\/mo><\/mrow><mrow><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><mrow><mo fence=\"true\">(<\/mo><mfrac><msqrt><mn>3<\/mn><\/msqrt><mn>2<\/mn><\/mfrac><mo fence=\"true\">)<\/mo><\/mrow><\/mrow><annotation encoding=\"application\/x-tex\">\\cos^{-1}\\left(\\frac{\\sqrt{3}}{2}\\right)<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Solution<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">We know:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>cos(\u03c0\/6)=\u221a3\/2<\/code><\/pre>\n\n\n\n<p class=\"wp-block-paragraph\">Principal value range of:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>cos\u207b\u00b9x<\/code><\/pre>\n\n\n\n<p class=\"wp-block-paragraph\">is:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>&#91;0, \u03c0]<\/code><\/pre>\n\n\n\n<p class=\"wp-block-paragraph\">Therefore:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>Answer = \u03c0\/6<\/code><\/pre>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h1 class=\"wp-block-heading\">Question 3<\/h1>\n\n\n\n<p class=\"wp-block-paragraph\">Find:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mrow><mi>cosec<\/mi><mo>\u2061<\/mo><\/mrow><mrow><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><mo stretchy=\"false\">(<\/mo><mn>2<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">\\cosec^{-1}(2)<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Solution<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">We know:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>cosec(\u03c0\/6)=2<\/code><\/pre>\n\n\n\n<p class=\"wp-block-paragraph\">Therefore:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>Answer = \u03c0\/6<\/code><\/pre>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h1 class=\"wp-block-heading\">Question 4<\/h1>\n\n\n\n<p class=\"wp-block-paragraph\">Find:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mrow><mi>tan<\/mi><mo>\u2061<\/mo><\/mrow><mrow><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><mo stretchy=\"false\">(<\/mo><mo>\u2212<\/mo><msqrt><mn>3<\/mn><\/msqrt><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">\\tan^{-1}(-\\sqrt{3})<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Solution<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">We know:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>tan(\u2212\u03c0\/3)=\u2212\u221a3<\/code><\/pre>\n\n\n\n<p class=\"wp-block-paragraph\">Therefore:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>Answer = \u2212\u03c0\/3<\/code><\/pre>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h1 class=\"wp-block-heading\">Question 5<\/h1>\n\n\n\n<p class=\"wp-block-paragraph\">Find:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mrow><mi>cos<\/mi><mo>\u2061<\/mo><\/mrow><mrow><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><mrow><mo fence=\"true\">(<\/mo><mo>\u2212<\/mo><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><mo fence=\"true\">)<\/mo><\/mrow><\/mrow><annotation encoding=\"application\/x-tex\">\\cos^{-1}\\left(-\\frac{1}{2}\\right)<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Solution<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">We know:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>cos(2\u03c0\/3)=\u22121\/2<\/code><\/pre>\n\n\n\n<p class=\"wp-block-paragraph\">Since principal value range of cosine inverse is:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>&#91;0, \u03c0]<\/code><\/pre>\n\n\n\n<p class=\"wp-block-paragraph\">Therefore:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>Answer = 2\u03c0\/3<\/code><\/pre>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h1 class=\"wp-block-heading\">Question 6<\/h1>\n\n\n\n<p class=\"wp-block-paragraph\">Find:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mrow><mi>tan<\/mi><mo>\u2061<\/mo><\/mrow><mrow><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><mo stretchy=\"false\">(<\/mo><mo>\u2212<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">\\tan^{-1}(-1)<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Solution<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">We know:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>tan(\u2212\u03c0\/4)=\u22121<\/code><\/pre>\n\n\n\n<p class=\"wp-block-paragraph\">Therefore:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>Answer = \u2212\u03c0\/4<\/code><\/pre>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h1 class=\"wp-block-heading\">Question 7<\/h1>\n\n\n\n<p class=\"wp-block-paragraph\">Find:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mrow><mi>sec<\/mi><mo>\u2061<\/mo><\/mrow><mrow><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><mrow><mo fence=\"true\">(<\/mo><mo>\u2212<\/mo><mfrac><mn>2<\/mn><msqrt><mn>3<\/mn><\/msqrt><\/mfrac><mo fence=\"true\">)<\/mo><\/mrow><\/mrow><annotation encoding=\"application\/x-tex\">\\sec^{-1}\\left(-\\frac{2}{\\sqrt{3}}\\right)<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Solution<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">We know:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>sec(5\u03c0\/6)=\u22122\/\u221a3<\/code><\/pre>\n\n\n\n<p class=\"wp-block-paragraph\">Therefore:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>Answer = 5\u03c0\/6<\/code><\/pre>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h1 class=\"wp-block-heading\">Question 8<\/h1>\n\n\n\n<p class=\"wp-block-paragraph\">Find:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mrow><mi>cot<\/mi><mo>\u2061<\/mo><\/mrow><mrow><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><mo stretchy=\"false\">(<\/mo><msqrt><mn>3<\/mn><\/msqrt><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">\\cot^{-1}(\\sqrt{3})<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Solution<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">We know:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>cot(\u03c0\/6)=\u221a3<\/code><\/pre>\n\n\n\n<p class=\"wp-block-paragraph\">Therefore:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>Answer = \u03c0\/6<\/code><\/pre>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h1 class=\"wp-block-heading\">Question 9<\/h1>\n\n\n\n<p class=\"wp-block-paragraph\">Find:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mrow><mi>cos<\/mi><mo>\u2061<\/mo><\/mrow><mrow><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><mrow><mo fence=\"true\">(<\/mo><mo>\u2212<\/mo><mfrac><mn>1<\/mn><msqrt><mn>2<\/mn><\/msqrt><\/mfrac><mo fence=\"true\">)<\/mo><\/mrow><\/mrow><annotation encoding=\"application\/x-tex\">\\cos^{-1}\\left(-\\frac{1}{\\sqrt{2}}\\right)<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Solution<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">We know:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>cos(3\u03c0\/4)=\u22121\/\u221a2<\/code><\/pre>\n\n\n\n<p class=\"wp-block-paragraph\">Therefore:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>Answer = 3\u03c0\/4<\/code><\/pre>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h1 class=\"wp-block-heading\">Question 10<\/h1>\n\n\n\n<p class=\"wp-block-paragraph\">Find:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mrow><mi>cosec<\/mi><mo>\u2061<\/mo><\/mrow><mrow><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><mo stretchy=\"false\">(<\/mo><mo>\u2212<\/mo><msqrt><mn>2<\/mn><\/msqrt><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">\\cosec^{-1}(-\\sqrt{2})<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Solution<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">We know:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>cosec(\u2212\u03c0\/4)=\u2212\u221a2<\/code><\/pre>\n\n\n\n<p class=\"wp-block-paragraph\">Therefore:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>Answer = \u2212\u03c0\/4<\/code><\/pre>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h1 class=\"wp-block-heading\">Question 11<\/h1>\n\n\n\n<p class=\"wp-block-paragraph\">Find the value of:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mrow><mi>tan<\/mi><mo>\u2061<\/mo><\/mrow><mrow><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><mo stretchy=\"false\">(<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><msup><mrow><mi>cos<\/mi><mo>\u2061<\/mo><\/mrow><mrow><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><mrow><mo fence=\"true\">(<\/mo><mo>\u2212<\/mo><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><mo fence=\"true\">)<\/mo><\/mrow><mo>+<\/mo><msup><mrow><mi>sin<\/mi><mo>\u2061<\/mo><\/mrow><mrow><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><mrow><mo fence=\"true\">(<\/mo><mo>\u2212<\/mo><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><mo fence=\"true\">)<\/mo><\/mrow><\/mrow><annotation encoding=\"application\/x-tex\">\\tan^{-1}(1)+\\cos^{-1}\\left(-\\frac{1}{2}\\right)+\\sin^{-1}\\left(-\\frac{1}{2}\\right)<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Solution<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Using standard values:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>tan\u207b\u00b9(1)=\u03c0\/4<\/code><\/pre>\n\n\n\n<pre class=\"wp-block-code\"><code>cos\u207b\u00b9(\u22121\/2)=2\u03c0\/3<\/code><\/pre>\n\n\n\n<pre class=\"wp-block-code\"><code>sin\u207b\u00b9(\u22121\/2)=\u2212\u03c0\/6<\/code><\/pre>\n\n\n\n<p class=\"wp-block-paragraph\">Adding:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>\u03c0\/4 + 2\u03c0\/3 \u2212 \u03c0\/6<\/code><\/pre>\n\n\n\n<p class=\"wp-block-paragraph\">LCM = 12<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>= 3\u03c0\/12 + 8\u03c0\/12 \u2212 2\u03c0\/12<\/code><\/pre>\n\n\n\n<p class=\"wp-block-paragraph\">Therefore:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>Answer = 3\u03c0\/4<\/code><\/pre>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h1 class=\"wp-block-heading\">Question 12<\/h1>\n\n\n\n<p class=\"wp-block-paragraph\">Find:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mrow><mi>cos<\/mi><mo>\u2061<\/mo><\/mrow><mrow><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><mrow><mo fence=\"true\">(<\/mo><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><mo fence=\"true\">)<\/mo><\/mrow><mo>+<\/mo><mn>2<\/mn><msup><mrow><mi>sin<\/mi><mo>\u2061<\/mo><\/mrow><mrow><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><mrow><mo fence=\"true\">(<\/mo><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><mo fence=\"true\">)<\/mo><\/mrow><\/mrow><annotation encoding=\"application\/x-tex\">\\cos^{-1}\\left(\\frac{1}{2}\\right)+2\\sin^{-1}\\left(\\frac{1}{2}\\right)<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Solution<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Using standard values:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>cos\u207b\u00b9(1\/2)=\u03c0\/3<\/code><\/pre>\n\n\n\n<pre class=\"wp-block-code\"><code>sin\u207b\u00b9(1\/2)=\u03c0\/6<\/code><\/pre>\n\n\n\n<p class=\"wp-block-paragraph\">Hence:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>\u03c0\/3 + 2(\u03c0\/6)<\/code><\/pre>\n\n\n\n<pre class=\"wp-block-code\"><code>= \u03c0\/3 + \u03c0\/3<\/code><\/pre>\n\n\n\n<p class=\"wp-block-paragraph\">Therefore:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>Answer = 2\u03c0\/3<\/code><\/pre>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h1 class=\"wp-block-heading\">Question 13<\/h1>\n\n\n\n<p class=\"wp-block-paragraph\">If:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mrow><mi>sin<\/mi><mo>\u2061<\/mo><\/mrow><mrow><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><mi>x<\/mi><mo>=<\/mo><mi>y<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\sin^{-1}x=y<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">then:<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Solution<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Range of:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>sin\u207b\u00b9x<\/code><\/pre>\n\n\n\n<p class=\"wp-block-paragraph\">is:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>&#91;\u2212\u03c0\/2, \u03c0\/2]<\/code><\/pre>\n\n\n\n<p class=\"wp-block-paragraph\">Therefore correct option is:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>(B)<\/code><\/pre>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h1 class=\"wp-block-heading\">Question 14<\/h1>\n\n\n\n<p class=\"wp-block-paragraph\">Evaluate:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mrow><mi>tan<\/mi><mo>\u2061<\/mo><\/mrow><mrow><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><mo stretchy=\"false\">(<\/mo><msqrt><mn>3<\/mn><\/msqrt><mo stretchy=\"false\">)<\/mo><mo>\u2212<\/mo><msup><mrow><mi>sec<\/mi><mo>\u2061<\/mo><\/mrow><mrow><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><mo stretchy=\"false\">(<\/mo><mo>\u2212<\/mo><mn>2<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">\\tan^{-1}(\\sqrt{3})-\\sec^{-1}(-2)<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Solution<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">We know:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>tan\u207b\u00b9(\u221a3)=\u03c0\/3<\/code><\/pre>\n\n\n\n<p class=\"wp-block-paragraph\">and:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>sec\u207b\u00b9(\u22122)=2\u03c0\/3<\/code><\/pre>\n\n\n\n<p class=\"wp-block-paragraph\">Therefore:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>\u03c0\/3 \u2212 2\u03c0\/3<\/code><\/pre>\n\n\n\n<p class=\"wp-block-paragraph\">Hence:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>Answer = \u2212\u03c0\/3<\/code><\/pre>\n\n\n\n<p class=\"wp-block-paragraph\">Correct option:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>(B)<\/code><\/pre>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h1 class=\"wp-block-heading\">Common Mistakes<\/h1>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Ignoring principal value ranges<\/li>\n\n\n\n<li>Confusing sin\u207b\u00b9x with 1\/sinx<\/li>\n\n\n\n<li>Using wrong quadrant values<\/li>\n\n\n\n<li>Forgetting domain restrictions<\/li>\n\n\n\n<li>Wrong sign in inverse functions<\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h1 class=\"wp-block-heading\">Exam Tips<\/h1>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Memorize standard inverse trigonometric values.<\/li>\n\n\n\n<li>Always check principal value branch.<\/li>\n\n\n\n<li>Use unit circle for quick answers.<\/li>\n\n\n\n<li>Practice simplification regularly.<\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h1 class=\"wp-block-heading\">Practice MCQs<\/h1>\n\n\n\n<h2 class=\"wp-block-heading\">MCQ 1<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">Principal value range of:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mrow><mi>cos<\/mi><mo>\u2061<\/mo><\/mrow><mrow><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><mi>x<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\cos^{-1}x<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">is:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">A. [\u2212\u03c0\/2, \u03c0\/2]<br>B. [0, \u03c0]<br>C. R<br>D. (\u2212\u03c0, \u03c0)<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Answer:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>B. &#91;0, \u03c0]<\/code><\/pre>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\">MCQ 2<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">Value of:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mrow><mi>tan<\/mi><mo>\u2061<\/mo><\/mrow><mrow><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><mo stretchy=\"false\">(<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">\\tan^{-1}(1)<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">is:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">A. \u03c0<br>B. \u03c0\/2<br>C. \u03c0\/4<br>D. 0<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Answer:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>C. \u03c0\/4<\/code><\/pre>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\">MCQ 3<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">Value of:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mrow><mi>sin<\/mi><mo>\u2061<\/mo><\/mrow><mrow><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><mrow><mo fence=\"true\">(<\/mo><mo>\u2212<\/mo><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><mo fence=\"true\">)<\/mo><\/mrow><\/mrow><annotation encoding=\"application\/x-tex\">\\sin^{-1}\\left(-\\frac{1}{2}\\right)<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">is:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">A. \u03c0\/6<br>B. \u2212\u03c0\/6<br>C. 5\u03c0\/6<br>D. \u22125\u03c0\/6<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Answer:<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>B. \u2212\u03c0\/6<\/code><\/pre>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h1 class=\"wp-block-heading\">FAQ Section<\/h1>\n\n\n\n<h2 class=\"wp-block-heading\">What is principal value?<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">The value lying in the principal branch of inverse trigonometric function.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\">What is the range of sin\u207b\u00b9x?<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mo fence=\"true\">[<\/mo><mo>\u2212<\/mo><mfrac><mi>\u03c0<\/mi><mn>2<\/mn><\/mfrac><mo separator=\"true\">,<\/mo><mfrac><mi>\u03c0<\/mi><mn>2<\/mn><\/mfrac><mo fence=\"true\">]<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">\\left[-\\frac{\\pi}{2},\\frac{\\pi}{2}\\right]<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\">Is sin\u207b\u00b9x equal to 1\/sinx?<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">No.<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>sin\u207b\u00b9x means inverse sine function.<\/code><\/pre>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\">What is the range of tan\u207b\u00b9x?<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mo fence=\"true\">(<\/mo><mo>\u2212<\/mo><mfrac><mi>\u03c0<\/mi><mn>2<\/mn><\/mfrac><mo separator=\"true\">,<\/mo><mfrac><mi>\u03c0<\/mi><mn>2<\/mn><\/mfrac><mo fence=\"true\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">\\left(-\\frac{\\pi}{2},\\frac{\\pi}{2}\\right)<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h1 class=\"wp-block-heading\">CTA (Call To Action)<\/h1>\n\n\n\n<p class=\"wp-block-paragraph\">\ud83d\udcd8 Prepare Smarter with MyMockMate!<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u2705 Chapter-wise NCERT Solutions<br>\u2705 Important Notes &amp; MCQs<br>\u2705 Online Mock Tests<br>\u2705 Instant Result &amp; Analysis<br>\u2705 CBSE Board Preparation<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Start learning now on <a target=\"_blank\" rel=\"noreferrer noopener\" href=\"https:\/\/www.mymockmate.com?utm_source=chatgpt.com\">MyMockMate<\/a><\/p>\n\n    <div class=\"xs_social_share_widget xs_share_url after_content \t\tmain_content  wslu-style-1 wslu-share-box-shaped wslu-fill-colored wslu-none wslu-share-horizontal wslu-theme-font-no wslu-main_content\">\n\n\t\t\n        <ul>\n\t\t\t        <\/ul>\n    <\/div> \n","protected":false},"excerpt":{"rendered":"<p>Short Intro In this post, students can find complete step-by-step solutions for Class 12 Maths Chapter 2 Exercise<\/p>\n","protected":false},"author":1,"featured_media":1853,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_angie_page":false,"_surecart_dashboard_logo_width":"180px","_surecart_dashboard_show_logo":true,"_surecart_dashboard_navigation_orders":true,"_surecart_dashboard_navigation_invoices":true,"_surecart_dashboard_navigation_subscriptions":true,"_surecart_dashboard_navigation_downloads":true,"_surecart_dashboard_navigation_billing":true,"_surecart_dashboard_navigation_account":true,"postBodyCss":"","postBodyMargin":[],"postBodyPadding":[],"postBodyBackground":{"backgroundType":"classic","gradient":""},"page_builder":"","footnotes":"","jetpack_publicize_message":"","jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":true,"jetpack_social_options":{"image_generator_settings":{"template":"highway","default_image_id":0,"font":"","enabled":false},"version":2}},"categories":[7,100,8],"tags":[187,175,226,174,205,85,78,221,217,219,220,216,218,227,213,214,225,223,76,53,47,215,222,224],"class_list":["post-1852","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-class-12","category-inverse-trigonometric-functions","category-maths","tag-cbse-board-maths-preparation","tag-cbse-class-12-maths","tag-chapter-2-maths-solutions","tag-class-12-maths","tag-class-12-maths-notes","tag-exercise-2-1-ncert-solutions","tag-exercise-2-1-solutions","tag-inverse-cosecant-function","tag-inverse-cosine-function","tag-inverse-cotangent-function","tag-inverse-secant-function","tag-inverse-sine-function","tag-inverse-tangent-function","tag-inverse-trigonometric-formulas","tag-inverse-trigonometric-functions","tag-inverse-trigonometric-functions-chapter-2","tag-inverse-trigonometric-functions-mcqs","tag-inverse-trigonometric-identities","tag-maths-important-questions","tag-ncert-maths-solutions","tag-ncert-solutions","tag-principal-value-branch","tag-principal-value-problems","tag-trigonometric-functions"],"jetpack_publicize_connections":[],"_links":{"self":[{"href":"https:\/\/mymockmate.com\/notes\/wp-json\/wp\/v2\/posts\/1852","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/mymockmate.com\/notes\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mymockmate.com\/notes\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mymockmate.com\/notes\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/mymockmate.com\/notes\/wp-json\/wp\/v2\/comments?post=1852"}],"version-history":[{"count":1,"href":"https:\/\/mymockmate.com\/notes\/wp-json\/wp\/v2\/posts\/1852\/revisions"}],"predecessor-version":[{"id":1854,"href":"https:\/\/mymockmate.com\/notes\/wp-json\/wp\/v2\/posts\/1852\/revisions\/1854"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/mymockmate.com\/notes\/wp-json\/wp\/v2\/media\/1853"}],"wp:attachment":[{"href":"https:\/\/mymockmate.com\/notes\/wp-json\/wp\/v2\/media?parent=1852"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mymockmate.com\/notes\/wp-json\/wp\/v2\/categories?post=1852"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mymockmate.com\/notes\/wp-json\/wp\/v2\/tags?post=1852"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}