Here, you will learn domain and range of inverse trigonometric function class 12.<\/p>\n
Let’s begin – <\/p>\n
The inverse trigonometric functions, denoted by \\(sin^{-1}x\\) or (arc sinx), \\(cos^{-1}x\\) etc., denote the angles whose sine, cosine etc, is equal to x. The angles are usually smallest angles, except in case of \\(cot^{-1}x\\) and if the positive & negative angles have same numerical value, the positive angle has been chosen.<\/p>\n
It is worthwhile noting that the function sinx, cosx, tanx, cotx, cosecx, secx are in general not invertible. Their inverse is defined by choosing an appropriate domain & co-domain so that they become invertible. For this reason the chosen value is usually the simplest and easy way to remember.<\/p>\n
| S.No<\/td>\n | f(x)<\/td>\n | Domain<\/td>\n | Range<\/td>\n <\/tr>\n |
| 1<\/td>\n | \\(sin^{-1}x\\)<\/td>\n | |x| \\(\\le\\) 1<\/td>\n | [-\\(\\pi\\over 2\\), \\(\\pi\\over 2\\)]<\/td>\n <\/tr>\n |
| 2<\/td>\n | \\(cos^{-1}x\\)<\/td>\n | |x| \\(\\le\\) 1<\/td>\n | [0, \\(\\pi\\)]<\/td>\n <\/tr>\n |
| 3<\/td>\n | \\(tan^{-1}x\\)<\/td>\n | x \\(\\in\\) R<\/td>\n | (-\\(\\pi\\over 2\\), \\(\\pi\\over 2\\))<\/td>\n <\/tr>\n |
| 4<\/td>\n | \\(sec^{-1}x\\)<\/td>\n | |x| \\(\\ge\\) 1<\/td>\n | [0, \\(\\pi\\)] – {\\(\\pi\\over 2\\)} or [0, \\(\\pi\\over 2\\)) \\(\\cup\\) (\\(\\pi\\over 2\\), \\(\\pi\\)]<\/td>\n <\/tr>\n |
| 5<\/td>\n | \\(cosec^{-1}x\\)<\/td>\n | |x| \\(\\ge\\) 1<\/td>\n | [-\\(\\pi\\over 2\\), \\(\\pi\\over 2\\)] – {0}<\/td>\n <\/tr>\n |
| 6<\/td>\n | \\(cot^{-1}x\\)<\/td>\n | x \\(\\in\\) R<\/td>\n | (0, \\(\\pi\\))<\/td>\n <\/tr>\n <\/tbody><\/table> \n\n\n Note :<\/strong><\/p>\n (i) All the inverse trigonometric functions represent an angle.<\/p>\n (ii) If x > 0, then all six inverse trigonometric functions viz \\(sin^{-1}x\\), \\(cos^{-1}x\\), \\(tan^{-1}x\\), \\(sec^{-1}x\\), \\(cosec^{-1}x\\), \\(cot^{-1}x\\) represent an acute angle.<\/p>\n (iii) If x < 0, then \\(sin^{-1}x\\), \\(tan^{-1}x\\) & \\(cosec^{-1}x\\) represent an angle from -\\(\\pi\\over 2\\) to 0 (IV quadrant)<\/p>\n (iv) If x < 0, then \\(cos^{-1}x\\), \\(cot^{-1}x\\), & \\(sec^{-1}x\\) represent an obtuse angle. (II quadrant)<\/p>\n (v) Third (III) quadrant is never used in range of inverse trigonometric function.<\/p>\n\n\n Example : <\/span> The value of \\(tan^{-1}(1)\\) + \\(cos^{-1}({-1\\over 2})\\) + \\(sin^{-1}({-1\\over 2})\\) is equal to – <\/p>\n Solution :<\/span> We have, \\(tan^{-1}(1)\\) + \\(cos^{-1}({-1\\over 2})\\) + \\(sin^{-1}({-1\\over 2})\\) Hope, you learnt domain and range of inverse trigonometric function class 12, learn more concepts of inverse trigonometric function<\/a><\/span> and practice more questions to get ahead in competition. Good Luck!\u00a0\u00a0<\/p>\n\n\n |