Inverse Trigonometric Functions

Also, since \(sec~x = \frac{1}{cos~x}\), the domain of \(y=sec~x\) is the set \(R - \{x : x = (2n+1)\frac{\pi}{2}, n \in Z\}\) and range is the set \(R - (-1, 1)\). It means that sec (secant function) assumes all real values except \(-1 < y < 1\) and is not defined for odd multiples of \(\frac{\pi}{2}\). If we restrict the domain of secant function to \([0, \pi] - \{\frac{\pi}{2}\}\), then it is one-one and onto with its range as the set \(R - (-1, 1)\). Actually, secant function restricted to any of the intervals \([-\pi, 0] - \{-\frac{\pi}{2}\}\), \([0, \pi] - \{\frac{\pi}{2}\}\), \([\pi, 2\pi] - \{\frac{3\pi}{2}\}\) etc., is bijective and its range is \(R - \{-1, 1\}\). Thus \(sec^{-1}\) can be defined as a function whose domain is \(R - (-1, 1)\) and range could be any of the intervals \([-\pi, 0] - \{-\frac{\pi}{2}\}\), \([0, \pi] - \{\frac{\pi}{2}\}\), \([\pi, 2\pi] - \{\frac{3\pi}{2}\}\) etc. Corresponding to each of these intervals, we get different branches of the function \(sec^{-1}\). The branch with range \([0, \pi] - \{\frac{\pi}{2}\}\) is called the principal value branch of the function \(sec^{-1}\). We thus have \(sec^{-1}: R - (-1, 1) \rightarrow [0, \pi] - \{\frac{\pi}{2}\}\).