2.8 Summary- Inverse Trigonometric Functions - Class 12 Mathematics
2.8 Summary
Accessible NCERT Class 12 Mathematics Textbook for Blind and Visually Impaired Students facilitated by Professor T K Bansal.
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The domain and the range (principal value branches) for all the 6 inverse trigonometric functions are given in the following table:
| Function | Domain | Range (Principal Value Branches) |
| \(y\ =\ \sin^{−1}x\) | \([−1,\ 1]\) | \([\frac{−π}{2},\ \frac{π}{2}]\) |
| \(y\ =\ \cos^{−1} x\) | \([−1,\ 1]\) | \([0,\ π]\) |
| \(y\ =\ \csc^{−1} x\) | \(R\ −\ (−1,\ 1)\) | \([\frac{−π}{2},\ \frac{π}{2}]\ −\ \{0\}\) |
| \(y\ =\ \sec^{−1} x\) | \(R\ −\ (−1, 1)\) | \([0,\ π]\ −\ \{\frac{π}{2}\}\) |
| \(y\ =\ \tan^{−1}x\) | \(R\) | \((\frac{−π}{2},\ \frac{π}{2})\) |
| \(y\ =\ \cot^{−1}x\) | \(R\) | \((0,\ π)\) |
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Please note:
\[\sin^{−1} x\ ≠\\(sin x)^{−1}.\]
In fact
\[(\sin x)^{−1}\ =\ \frac{1}{\sin x}\]
and similarly for other trigonometric functions.
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The value of an inverse trigonometric functions which lies in its principal value branch is called the principal value of that inverse trigonometric function.
For suitable values of domain, we have
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\[y\ =\ \sin^{−1}x\ ⇒\ x\ =\ \sin y\]
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\[x\ =\ \sin y\ ⇒\ y\ =\ \sin^{−1} x\]
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\[\sin( \sin^{−1} x)\ =\ x\]
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\[\sin^{−1} (\sin x)\ =\ x\]
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\[\sin^{−1} \frac{1}{x}\ =\ \csc^{−1} x\]
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\[\cos^{−1} (−x)\ =\ π\ −\ \cos^{−1} x\]
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\[\cos^{−1} \frac{1}{x}\ =\ \sec^{−1} x\]
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\[\cot^{−1} (−x)\ =\ π\ −\ \cot^{−1} x\]
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\[\tan^{−1} \frac{1}{x},\ =\ \cot^{−1} x\]
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\[\sec^{−1} −x\ =\ π\ −\ \sec^{−1} x\]
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\[\sin^{−1} −x\ =\ −\ \sin^{−1} x\]
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\[\tan^{−1} −x\ =\ −\ \tan^{−1} x\]
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\[\tan^{−1} x\ +\ \cot^{−1} x\ =\ \frac{π}{2}\]
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\[\csc^{−1} −x\ =\ −\ \csc^{−1} x\]
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\[\tan^{−1} x\ +\ \tan^{−1} y\ =\ \tan^{−1} \frac{x+y}{1−xy}\]
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\[\sin^{−1}x\ +\ \cos^{−1}x\ =\ \frac{π}{2}\]
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\[\tan^{−1}x\ +\ \tan^{−1}y\ =\ \tan^{−1}\frac{x+y}{1−xy}\]
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\[\csc^{−1} x\ +\ \sec^{−1} x\ =\ \frac{π}{2}\]
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\[2 \tan^{−1} x\ =\ \sin^{−1} \frac{2x}{1+x^2}\ =\ \cos^{−1} \frac{1−x^2}{1+x^2}\]
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\[2 \tan^{−1} x\ =\ \tan^{−1}\frac{2x}{1−x^2}\]
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\[\tan^{−1} x\ +\ \tan^{−1}y\ =\ π\ +\ \tan^{−1} \frac{x+y}{1−xy},\ xy\ >\ 1;\ x,\ y\ >\ 0\]