Question

The principal branch of $\cos^{-1} x$ is:

• A. $\left[ \frac{\pi}{2}, \frac{3\pi}{2} \right]$
• B. $[\pi, 2\pi]$
• C. $[0, \pi]$
• D. $[2\pi, 3\pi]$

Hint:

The range of the principal branch of the inverse cosine function $\cos^{-1} x$ is always $[0, \pi]$. Remember this when working with inverse trigonometric functions.

Answer 1

The Correct Option is C

The principal branch of the inverse trigonometric function $\cos^{-1} x$ is defined over a specific angle range, ensuring a real output within a restricted interval.
The domain of $\cos^{-1} x$ is $x \in [-1, 1]$. Its output is restricted to an interval where the cosine function is both decreasing and one-to-one. Consequently, the principal branch of $\cos^{-1} x$ is the interval $[0, \pi]$, aligning with the decreasing nature of the cosine function in this range.
Therefore, the principal branch is established as the range $[0, \pi]$.