Inverse Trigonometric Functions QRT – Quick Revision Tables bring the important concepts, formulas and results of the chapter together in one place. Use these tables for a quick recap while revising, practising questions or preparing for your exams.
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Table 1: Inverse Trigonometric Functions QRT – Basics
| Concept | Formula / Result |
|---|---|
| Inverse of a Function | If \displaystyle f:X\to Y is one-one and onto, then its inverse \displaystyle f^{-1} exists. |
| Domain of Inverse | The domain of \displaystyle f^{-1} is the range of \displaystyle f. |
| Range of Inverse | The range of \displaystyle f^{-1} is the domain of \displaystyle f. |
| Inverse of Inverse | If \displaystyle g=f^{-1}, then \displaystyle g^{-1}=f. |
| Composition with Inverse | \displaystyle f^{-1}\circ f=I_A and \displaystyle f\circ f^{-1}=I_B. |
| Why Restrict Trigonometric Functions? | Trigonometric functions are not one-one over their natural domains. Their domains are restricted so that the functions become one-one and onto, allowing inverses to be defined. |
| Inverse Trigonometric Functions | The inverses of the restricted trigonometric functions are denoted by \displaystyle \sin^{-1}x, \displaystyle \cos^{-1}x, \displaystyle \tan^{-1}x, etc. |
| Principal Value Branch | The branch of an inverse trigonometric function with the standard specified range is called its principal value branch. |
| Principal Value | The value of an inverse trigonometric function lying in the range of its principal value branch is called its principal value. |
Maths Better Tip… Remember: an inverse trigonometric function is defined by first restricting the corresponding trigonometric function to a suitable domain where it becomes one-one and onto. The principal value branch is the standard branch used unless another branch is specified.
The following standard domains, ranges and principal value branches form the foundation for working with inverse trigonometric functions.
Table 2: Inverse Trigonometric Functions QRT – Domains, Ranges & Principal Value Branches
| T | Dom | R=D | PVB | Inv |
|---|---|---|---|---|
| \sin | \mathbb{R} | [-1,1] | [-\frac{\pi}{2},\frac{\pi}{2}] | \sin^{-1} |
| \cos | \mathbb{R} | [-1,1] | [0,\pi] | \cos^{-1} |
| \tan | \mathbb{R}\setminus\{\frac{(2n+1)\pi}{2}\} | \mathbb{R} | (-\frac{\pi}{2},\frac{\pi}{2}) | \tan^{-1} |
| \cot | \mathbb{R}\setminus\{n\pi\} | \mathbb{R} | (0,\pi) | \cot^{-1} |
| \sec | \mathbb{R}\setminus\{\frac{(2n+1)\pi}{2}\} | \mathbb{R}\setminus(-1,1) | [0,\pi]\setminus\{\frac{\pi}{2}\} | \sec^{-1} |
| \csc | \mathbb{R}\setminus\{n\pi\} | \mathbb{R}\setminus(-1,1) | [-\frac{\pi}{2},\frac{\pi}{2}]\setminus\{0\} | \csc^{-1} |
Here, n\in\mathbb{Z} and PVB denotes Principal Value Branch.
Maths Better Tip… Memorise the principal value branches carefully. They determine the unique values returned by inverse trigonometric functions.
The following standard values are important to memorise. They are frequently used while solving trigonometric and inverse trigonometric problems.
Table 3: Standard Trigonometric and Inverse Values
| Function | 0 | \frac{\pi}{6} | \frac{\pi}{4} | \frac{\pi}{3} | \frac{\pi}{2} |
|---|---|---|---|---|---|
| \sin\theta | 0 | \frac{1}{2} | \frac{1}{\sqrt2} | \frac{\sqrt3}{2} | 1 |
| \cos\theta | 1 | \frac{\sqrt3}{2} | \frac{1}{\sqrt2} | \frac{1}{2} | 0 |
| \tan\theta | 0 | \frac{1}{\sqrt3} | 1 | \sqrt3 | \text{N.D.} |
| \cot\theta | \text{N.D.} | \sqrt3 | 1 | \frac{1}{\sqrt3} | 0 |
| \sec\theta | 1 | \frac{2}{\sqrt3} | \sqrt2 | 2 | \text{N.D.} |
| \cosec\theta | \text{N.D.} | 2 | \sqrt2 | \frac{2}{\sqrt3} | 1 |
| Function | 0 | \frac{1}{2} | \frac{1}{\sqrt2} | \frac{\sqrt3}{2} | 1 |
|---|---|---|---|---|---|
| \sin^{-1}x | 0 | \frac{\pi}{6} | \frac{\pi}{4} | \frac{\pi}{3} | \frac{\pi}{2} |
| \cos^{-1}x | \frac{\pi}{2} | \frac{\pi}{3} | \frac{\pi}{4} | \frac{\pi}{6} | 0 |
| Function | -\sqrt3 | -1 | -\frac{1}{\sqrt3} | 0 | \frac{1}{\sqrt3} | 1 | \sqrt3 |
|---|---|---|---|---|---|---|---|
| \tan^{-1}x | -\frac{\pi}{3} | -\frac{\pi}{4} | -\frac{\pi}{6} | 0 | \frac{\pi}{6} | \frac{\pi}{4} | \frac{\pi}{3} |
| \cot^{-1}x | \frac{5\pi}{6} | \frac{3\pi}{4} | \frac{2\pi}{3} | \frac{\pi}{2} | \frac{\pi}{3} | \frac{\pi}{4} | \frac{\pi}{6} |
| Function | -2 | -1 | 1 | 2 |
|---|---|---|---|---|
| \sec^{-1}x | \frac{2\pi}{3} | \pi | 0 | \frac{\pi}{3} |
| \cosec^{-1}x | -\frac{\pi}{6} | -\frac{\pi}{2} | \frac{\pi}{2} | \frac{\pi}{6} |
Maths Better Tip… Learn the standard trigonometric values first. The inverse values then become much easier to recall. For inverse functions, always choose the principal value from the prescribed range i.e. PVB.
Also note that, some books write ∞ for values such as \tan\frac{\pi}{2} or \sec\frac{\pi}{2}. Strictly speaking, these values are not defined (N.D.). The symbol ∞ describes the unbounded behaviour near the angle; it is not the value of the trigonometric function at that angle.
The following fundamental identities are among the most frequently used results for simplifying inverse trigonometric expressions.
Table 4: Inverse Trigonometric Functions QRT – Fundamental Identities
| Identity | Result |
|---|---|
| \sin(\sin^{-1}x) | x,\quad x\in[-1,1] |
| \cos(\cos^{-1}x) | x,\quad x\in[-1,1] |
| \tan(\tan^{-1}x) | x,\quad x\in\mathbb{R} |
| \cot(\cot^{-1}x) | x,\quad x\in\mathbb{R} |
| \sec(\sec^{-1}x) | x,\quad x\in\mathbb{R}\setminus(-1,1) |
| \cosec(\cosec^{-1}x) | x,\quad x\in\mathbb{R}\setminus(-1,1) |
| \sin^{-1}(\sin x) | x,\quad x\in\left[-\frac{\pi}{2},\frac{\pi}{2}\right] |
| \cos^{-1}(\cos x) | x,\quad x\in[0,\pi] |
| \tan^{-1}(\tan x) | x,\quad x\in\left(-\frac{\pi}{2},\frac{\pi}{2}\right) |
| \cot^{-1}(\cot x) | x,\quad x\in(0,\pi) |
| \sec^{-1}(\sec x) | x,\quad x\in[0,\pi]\setminus\left\{\frac{\pi}{2}\right\} |
| \cosec^{-1}(\cosec x) | x,\quad x\in\left[-\frac{\pi}{2},\frac{\pi}{2}\right]\setminus\{0\} |
Maths Better Tip… Remember: \sin(\sin^{-1}x)=x is true for every allowed x, but \sin^{-1}(\sin x)=x is true only when x lies in the principal value branch.
The following identities are useful for reducing inverse trigonometric expressions and converting negative arguments or complementary angles into simpler forms.
Table 5: Inverse Trigonometric Functions QRT – Negative Arguments & Complementary Relations
| Form | Result |
|---|---|
| \sin^{-1}(-x) | -\sin^{-1}x |
| \tan^{-1}(-x) | -\tan^{-1}x |
| \cosec^{-1}(-x) | -\cosec^{-1}x |
| \cos^{-1}(-x) | \pi-\cos^{-1}x |
| \cot^{-1}(-x) | \pi-\cot^{-1}x |
| \sec^{-1}(-x) | \pi-\sec^{-1}x |
| \sin^{-1}x+\cos^{-1}x | \frac{\pi}{2} |
| \tan^{-1}x+\cot^{-1}x | \frac{\pi}{2} |
| \sec^{-1}x+\cosec^{-1}x | \frac{\pi}{2} |
| \cos^{-1}x | \frac{\pi}{2}-\sin^{-1}x |
| \cot^{-1}x | \frac{\pi}{2}-\tan^{-1}x |
| \cosec^{-1}x | \frac{\pi}{2}-\sec^{-1}x |
The identities involving \cos^{-1},\cot^{-1},\sec^{-1} and their complementary forms are understood with the principal value branches specified in Table 2.
Maths Better Tip… For negative arguments, remember the pattern: \sin^{-1},\tan^{-1},\cosec^{-1} change sign, while \cos^{-1},\cot^{-1},\sec^{-1} become \pi-\text{(positive-angle value)}.
View or Download the free Inverse Trigonometric Functions QRT – Class 12 Maths PDF for quick revision anytime.
QRT 6: Inverse Trigonometry – Addition & Subtraction Results
These results are especially useful for combining or simplifying inverse trigonometric expressions.
| Form | Result |
|---|---|
| \tan^{-1}x+\tan^{-1}y | \tan^{-1}\left(\frac{x+y}{1-xy}\right) |
| \tan^{-1}x-\tan^{-1}y | \tan^{-1}\left(\frac{x-y}{1+xy}\right) |
| \sin^{-1}x+\sin^{-1}y | \sin^{-1}\left(x\sqrt{1-y^2}+y\sqrt{1-x^2}\right) |
| \sin^{-1}x-\sin^{-1}y | \sin^{-1}\left(x\sqrt{1-y^2}-y\sqrt{1-x^2}\right) |
| \cos^{-1}x+\cos^{-1}y | \cos^{-1}\left(xy-\sqrt{1-x^2}\sqrt{1-y^2}\right) |
| \cos^{-1}x-\cos^{-1}y | \cos^{-1}\left(xy+\sqrt{1-x^2}\sqrt{1-y^2}\right) |
These identities are subject to the appropriate domain and principal-value conditions.
Maths Better Tip… Do not apply an addition or subtraction formula blindly. After simplification, always check that the resulting angle lies in the principal value branch.
Table 7: Inverse Trigonometric Functions QRT – Conversion & Simplification Forms
These forms are useful for converting inverse trigonometric expressions into simpler forms while solving questions.
| Expression | Simplest Form |
|---|---|
| \sin^{-1}x | \cos^{-1}\sqrt{1-x^2} |
| \cos^{-1}x | \sin^{-1}\sqrt{1-x^2} |
| \tan^{-1}x | \sin^{-1}\frac{x}{\sqrt{1+x^2}} |
| \tan^{-1}x | \cos^{-1}\frac{1}{\sqrt{1+x^2}} |
| \cot^{-1}x | \tan^{-1}\frac{1}{x} for x\gt0 |
| \sec^{-1}x | \cos^{-1}\frac{1}{x} |
| \cosec^{-1}x | \sin^{-1}\frac{1}{x} |
| \sin(\tan^{-1}x) | \frac{x}{\sqrt{1+x^2}} |
| \cos(\tan^{-1}x) | \frac{1}{\sqrt{1+x^2}} |
| \tan(\sin^{-1}x) | \frac{x}{\sqrt{1-x^2}} |
| \tan(\cos^{-1}x) | \frac{\sqrt{1-x^2}}{x} |
The expressions are understood wherever the functions are defined and the corresponding principal-value conditions are satisfied.
Maths Better Tip… When simplifying, introduce a suitable angle such as x=\tan\theta or x=\sec\theta. Then use the principal value branch to identify the correct angle.
Table 8: Inverse Trigonometric Functions QRT – Double & Triple Angle Results
Useful double-angle and triple-angle results for quick recall and application.
| Expression | Result |
|---|---|
| \sin^{-1}x | \frac{1}{2}\sin^{-1}\left(2x\sqrt{1-x^2}\right) |
| \cos^{-1}x | \frac{1}{2}\cos^{-1}\left(2x^2-1\right) |
| \tan^{-1}x | \frac{1}{2}\sin^{-1}\frac{2x}{1+x^2} |
| \tan^{-1}x | \frac{1}{2}\cos^{-1}\frac{1-x^2}{1+x^2} |
| \sin^{-1}x | \frac{1}{3}\sin^{-1}(3x-4x^3) |
| \cos^{-1}x | \frac{1}{3}\cos^{-1}(4x^3-3x) |
| \tan^{-1}x | \frac{1}{3}\tan^{-1}\left(\frac{3x-x^3}{1-3x^2}\right) |
Important: These results are subject to the appropriate domain and principal-value conditions. They should not be applied directly for every value of x without checking these conditions.
Maths Better Tip… Never forget the principal-value range. In inverse trigonometry, getting the correct ratio is only half the job — the angle must also lie in the correct branch. So, always check the principal-value range before accepting a transformed result.
Table 9: Inverse Trigonometric Functions QRT – Useful Substitutions
These substitutions can make complicated inverse trigonometric expressions easier to simplify.
| Expression / Form | Try This Substitution |
|---|---|
| \sin^{-1}x,\ \cos^{-1}x | Put x=\sin\theta or x=\cos\theta, choosing \theta in the appropriate principal range. |
| \tan^{-1}x | Put x=\tan\theta, with \theta\in\left(-\frac{\pi}{2},\frac{\pi}{2}\right). |
| \sqrt{1-x^2} | Put x=\sin\theta or x=\cos\theta. |
| \sqrt{1+x^2} | Put x=\tan\theta, so that \sqrt{1+x^2}=\sec\theta. |
| \frac{2x}{1+x^2} | Recognise \frac{2x}{1+x^2}=\sin(2\tan^{-1}x). |
| \frac{1-x^2}{1+x^2} | Recognise \frac{1-x^2}{1+x^2}=\cos(2\tan^{-1}x). |
| \frac{2x}{1-x^2} | Recognise \frac{2x}{1-x^2}=\tan(2\tan^{-1}x), where defined. |
| 3x-4x^3 | Recognise 3x-4x^3=\sin(3\sin^{-1}x). |
| 4x^3-3x | Recognise 4x^3-3x=\cos(3\cos^{-1}x). |
| \frac{3x-x^3}{1-3x^2} | Recognise \frac{3x-x^3}{1-3x^2}=\tan(3\tan^{-1}x), where defined. |
Remember: After making a substitution, always check the principal-value range before simplifying an inverse trigonometric expression.
Maths Better Tip… When you see a square root such as \sqrt{1-x^2}, or a fraction involving 1+x^2, think of a trigonometric substitution before trying lengthy algebra. In inverse trigonometry, do not identify an angle from its trigonometric ratio alone. Always bring the angle into the required principal value branch.
Quick Questions — Test Yourself
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Continue Your Revision
Want to practise further? Explore the detailed NCERT Solutions for Inverse Trigonometric Functions & other chapters and also, continue learning with the Maths Better YouTube playlist on Inverse Trigonometric Functions. Use the Inverse Trigonometric Functions QRTs for a quick recap, and return to the detailed solutions whenever you need step-by-step practice.
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📘 Inverse Trigonometric Functions QRT – Class 12 Maths
9 Quick Revision Tables + Quick Q&A
