Question:
If \( 2\cos^{-1}x = y \), then
If \( 2\cos^{-1}x = y \), then
Show Hint
The range of \(\cos^{-1}x\) is \([0, \pi]\). Multiply this inequality by 2 to get the range of \(y\).
Updated On: Oct 22, 2026
- \( 0 \leq y \leq \pi \)
- \( -\pi \leq y \leq \pi \)
- \( 0 \leq y \leq 2\pi \)
- \( -\pi \leq y \leq 0 \)
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The Correct Option is C
Solution and Explanation
Step 1: Recall the range of cos inverse:
The principal value branch of \(\cos^{-1}x\) is \([0, \pi]\) for \(x \in [-1, 1]\).
So \(\cos^{-1}x\) always lies between 0 and \(\pi\), both included.
Step 2: Double the inequality:
We have \(0 \leq \cos^{-1}x \leq \pi\).
Multiply every part by 2. This keeps the direction of the inequality, since 2 is positive.
\[ 0 \leq 2\cos^{-1}x \leq 2\pi \]
Since \(y = 2\cos^{-1}x\), we get \(0 \leq y \leq 2\pi\).
Step 3: Check the options:
Option (A) gives only half of the range. The value \(y = 2\pi\) occurs at \(x = -1\), so \(y\) goes beyond \(\pi\). Option (A) is wrong.
Options (B) and (D) allow negative values of \(y\). But \(\cos^{-1}x\) is never negative, so \(y\) is never negative. Both are wrong.
Option (C) matches the interval we found.
Final Answer:
The value of \(y\) lies in \([0, 2\pi]\), which is option (C). \[ \boxed{0 \leq y \leq 2\pi} \]
The principal value branch of \(\cos^{-1}x\) is \([0, \pi]\) for \(x \in [-1, 1]\).
So \(\cos^{-1}x\) always lies between 0 and \(\pi\), both included.
Step 2: Double the inequality:
We have \(0 \leq \cos^{-1}x \leq \pi\).
Multiply every part by 2. This keeps the direction of the inequality, since 2 is positive.
\[ 0 \leq 2\cos^{-1}x \leq 2\pi \]
Since \(y = 2\cos^{-1}x\), we get \(0 \leq y \leq 2\pi\).
Step 3: Check the options:
Option (A) gives only half of the range. The value \(y = 2\pi\) occurs at \(x = -1\), so \(y\) goes beyond \(\pi\). Option (A) is wrong.
Options (B) and (D) allow negative values of \(y\). But \(\cos^{-1}x\) is never negative, so \(y\) is never negative. Both are wrong.
Option (C) matches the interval we found.
Final Answer:
The value of \(y\) lies in \([0, 2\pi]\), which is option (C). \[ \boxed{0 \leq y \leq 2\pi} \]
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