Question

If \( \cos^{-1} x = y \), then _____

• A. \( 0 \le y \le \pi \)
• B. \( 0<y<\pi \)
• C. \( -\frac{\pi}{2} \le y \le \frac{\pi}{2} \)
• D. \( -\frac{\pi}{2}<y<\frac{\pi}{2} \)

Hint:

Remember standard ranges: \(\sin^{-1}x \in [-\frac{\pi}{2}, \frac{\pi}{2}]\), \(\cos^{-1}x \in [0,\pi]\).

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
The "range" of an inverse trigonometric function is also known as its Principal Value Branch. For $\cos^{-1} x$, we restrict the domain of the original cosine function to make it bijective.
Step 2: Formula Application:
The cosine function is one-one and onto in the interval $[0, \pi]$. Therefore, the range of the inverse function $\cos^{-1} x$ is $[0, \pi]$.
Step 3: Explanation:
This means the value of $y$ (the angle) must fall between $0$ and $\pi$ radians ($0^\circ$ and $180^\circ$), inclusive.
Step 4: Final Answer:
The correct option is (a).