What is the value of \(\tan^{-1}\left(\frac{1}{\sqrt{3}}\right)\)?

Correct answer: \(\frac{\pi}{6}\)

Reason: \(\tan\frac{\pi}{6}=\frac{1}{\sqrt{3}}\), and the principal value of arctan lies in \((-\frac{\pi}{2},\frac{\pi}{2})\). Hence \(\tan^{-1}\left(\frac{1}{\sqrt{3}}\right)=\frac{\pi}{6}\). Checking close distractors: \(\tan\frac{\pi}{4}=1\), so \(\frac{\pi}{4}\) is incorrect; \(\tan\frac{\pi}{3}=\sqrt{3}\) and \(\tan\frac{\pi}{2}\) is undefined. Exam tip: Memorise standard angle values and remember the principal range for inverse trig functions when answering quickly.