Here, you will learn domain and range of inverse trigonometric function class 12.
Let’s begin –
Inverse Trigonometric Function
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.
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.
Domain and Range
| S.No | f(x) | Domain | Range |
| 1 | \(sin^{-1}x\) | |x| \(\le\) 1 | [-\(\pi\over 2\), \(\pi\over 2\)] |
| 2 | \(cos^{-1}x\) | |x| \(\le\) 1 | [0, \(\pi\)] |
| 3 | \(tan^{-1}x\) | x \(\in\) R | (-\(\pi\over 2\), \(\pi\over 2\)) |
| 4 | \(sec^{-1}x\) | |x| \(\ge\) 1 | [0, \(\pi\)] – {\(\pi\over 2\)} or [0, \(\pi\over 2\)) \(\cup\) (\(\pi\over 2\), \(\pi\)] |
| 5 | \(cosec^{-1}x\) | |x| \(\ge\) 1 | [-\(\pi\over 2\), \(\pi\over 2\)] – {0} |
| 6 | \(cot^{-1}x\) | x \(\in\) R | (0, \(\pi\)) |
Note :
(i) All the inverse trigonometric functions represent an angle.
(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.
(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)
(iv) If x < 0, then \(cos^{-1}x\), \(cot^{-1}x\), & \(sec^{-1}x\) represent an obtuse angle. (II quadrant)
(v) Third (III) quadrant is never used in range of inverse trigonometric function.
Example : The value of \(tan^{-1}(1)\) + \(cos^{-1}({-1\over 2})\) + \(sin^{-1}({-1\over 2})\) is equal to –
Solution : We have, \(tan^{-1}(1)\) + \(cos^{-1}({-1\over 2})\) + \(sin^{-1}({-1\over 2})\)
= \(\pi\over 4\) + \(2\pi\over 3\) – \(\pi\over 6\) = \(3\pi\over 4\)
Hope, you learnt domain and range of inverse trigonometric function class 12, learn more concepts of inverse trigonometric function and practice more questions to get ahead in competition. Good Luck!