Inverse Trigonometric Functions

Let us now discuss \(cosec^{-1}x\) and \(sec^{-1}x\) as follows: Since, \(cosec~x = \frac{1}{sin~x}\), the domain of the cosec function is the set \(\{x : x \in R \text{ and } x \neq n\pi, n \in Z\}\) and the range is the set \(\{y : y \in R, y \ge 1 \text{ or } y \le -1\}\) i.e., the set \(R - (-1, 1)\). It means that \(y = cosec~x\) assumes all real values except \(-1 < y < 1\) and is not defined for integral multiple of \(\pi\). If we restrict the domain of cosec function to \([-\frac{\pi}{2}, \frac{\pi}{2}] - \{0\}\), then it is one to one and onto with its range as the set \(R - (-1, 1)\). Actually, cosec function restricted to any of the intervals \([\frac{-3\pi}{2}, \frac{-\pi}{2}] - \{-\pi\}, [\frac{-\pi}{2}, \frac{\pi}{2}] - \{0\}, [\frac{\pi}{2}, \frac{3\pi}{2}] - \{\pi\}\) etc., is bijective and its range is the set of all real numbers \(R - (-1, 1)\). Thus \(cosec^{-1}\) can be defined as a function whose domain is \(R - (-1, 1)\) and range could be any of the intervals \([\frac{-3\pi}{2}, \frac{-\pi}{2}] - \{-\pi\}\), \([\frac{-\pi}{2}, \frac{\pi}{2}] - \{0\}\), \([\frac{\pi}{2}, \frac{3\pi}{2}] - \{\pi\}\) etc. The function corresponding to the range \([\frac{-\pi}{2}, \frac{\pi}{2}] - \{0\}\) is called the principal value branch of \(cosec^{-1}\). We thus have principal branch as \(cosec^{-1}: R - (-1, 1) \rightarrow [-\frac{\pi}{2}, \frac{\pi}{2}] - \{0\}\).