Question:
One branch of $\cos ^{-1}$ other than the principal value branch corresponds to
(a) $\left[\frac{\pi}{2}, \frac{3 \pi}{2}\right]$
(b) $[\pi, 2 \pi]-\left\{\frac{3 \pi}{2}\right\}$
(c) $(0, \pi)$
(d) $[2 \pi, 3 \pi]$
Solution:
The domain of the function $f(x)=\cos ^{-1} x$ is $[-1,1]$. The range of $\cos ^{-1} x$ in one of the intervals $\ldots,[-\pi, 0],[0, \pi],[\pi, 2 \pi],[2 \pi, 3 \pi], \ldots$ is one-one and onto with the range $[-1,1]$.
Thus, one branch of $\cos ^{-1} x$ other than the principal value branch corresponds to $[2 \pi, 3 \pi]$.
Hence, the correct answer is option (d).