Inverse Trigonometric Functions
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Summary
The domains and ranges (principal value branches) of inverse trigonometric functions are given in the following table:
| Functions | Domain | Range (Principal Value Branches) |
|---|---|---|
| y = \(sin^{-1}x\) | [-1, 1] | \([-\frac{\pi}{2}, \frac{\pi}{2}]\) |
| y = \(cos^{-1}x\) | [-1, 1] | \([0, \pi]\) |
| y = \(cosec^{-1}x\) | R – (–1, 1) | \([-\frac{\pi}{2}, \frac{\pi}{2}]\) – {0} |
| y = \(sec^{-1}x\) | R – (–1, 1) | \([0, \pi]\) – {<span class="math inline">\(\frac{\pi}{2}\)} |
| y = \(tan^{-1}x\) | R | \((-\frac{\pi}{2}, \frac{\pi}{2})\) |
| y = \(cot^{-1}x\) | R | \((0, \pi)\) |
\(sin^{-1}x\) should not be confused with \((sin x)^{-1}\). In fact \((sin x)^{-1} = \frac{1}{sin x}\) and similarly for other trigonometric functions.
The value of an inverse trigonometric functions which lies in its principal value branch is called the principal value of that inverse trigonometric functions.
For suitable values of domain, we have \(y = sin^{-1}x \Rightarrow x = sin y\) \(x = sin y \Rightarrow y = sin^{-1}x\) \(sin(sin^{-1}x) = x\) \(sin^{-1}(sin x) = x\)