Inverse Trigonometric Functions

Finally, we now discuss \(tan^{-1}\) and \(cot^{-1}\). We know that the domain of the tan function (tangent function) is the set \(\{x: x \in R \text{ and } x \neq (2n+1)\frac{\pi}{2}, n \in Z\}\) and the range is R. It means that tan function is not defined for odd multiples of \(\frac{\pi}{2}\). If we restrict the domain of tangent function to \((\frac{-\pi}{2}, \frac{\pi}{2})\), then it is one-one and onto with its range as R. Actually, tangent function restricted to any of the intervals \((\frac{-3\pi}{2}, \frac{-\pi}{2}), (\frac{-\pi}{2}, \frac{\pi}{2}), (\frac{\pi}{2}, \frac{3\pi}{2})\) etc., is bijective and its range is R. Thus \(tan^{-1}\) can be defined as a function whose domain is R and range could be any of the intervals \((\frac{-3\pi}{2}, \frac{-\pi}{2}), (\frac{-\pi}{2}, \frac{\pi}{2}), (\frac{\pi}{2}, \frac{3\pi}{2})\) and so on. These intervals give different branches of the function \(tan^{-1}\). The branch with range \((\frac{-\pi}{2}, \frac{\pi}{2})\) is called the principal value branch of the function \(tan^{-1}\). We thus have \(tan^{-1}: R \rightarrow (\frac{-\pi}{2}, \frac{\pi}{2})\)