Which of the following statements is correct about \(\tan^{-1}5\)?

Correct answer: It is defined — equal to a finite angle in \((-\tfrac{\pi}{2},\tfrac{\pi}{2})\)

The inverse tangent function \(\tan^{-1}x\) is defined for every real number \(x\in\mathbb{R}\) and its principal value lies in the interval \((-\tfrac{\pi}{2},\tfrac{\pi}{2})\). Hence \(\tan^{-1}5\) is defined and equals a finite angle inside that open interval. The choice \(\tfrac{\pi}{2}\) is incorrect because \(\tan\theta\) has a vertical asymptote at \(\theta=\tfrac{\pi}{2}\) and arctan never attains that endpoint; \(\pi\) is also impossible since arctan’s principal value cannot be \(\pi\). Exam tip: memorize domain = all real numbers and range = \((-\tfrac{\pi}{2},\tfrac{\pi}{2})\) for \(\tan^{-1}x\).