Question 13 of 42beginner💡 UnderstandMCQ1 mark

If sin⁻¹x = y, then

Exercise: EXERCISE 2.1 | Q: 13 | (Page 10)
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Explanation

The provided text explicitly states that from the definition of the inverse functions, sin⁡−1(sin⁡x)=x\sin^{-1}(\sin x) = x if −π2≤x≤π2-\frac{\pi}{2} \leq x \leq \frac{\pi}{2}. This interval represents the principal value branch of the sin⁡−1\sin^{-1} function. Therefore, if y=sin⁡−1xy = \sin^{-1} x, the value of yy must lie within this range, making Option (B) the correct choice. Option (A) corresponds to the principal value branch of the cosine function mentioned later in the text.

Solution Steps

  1. Step 1: Identify the relevant definition in the text: sin⁡−1(sin⁡x)=x\sin^{-1}(\sin x) = x if −π2≤x≤π2-\frac{\pi}{2} \leq x \leq \frac{\pi}{2}.

  2. Step 2: Recognize that this interval defines the range (principal value branch) of the sin⁡−1x\sin^{-1} x function.

  3. Step 3: Apply this to the given equation y=sin⁡−1xy = \sin^{-1} x to conclude that −π2≤y≤π2-\frac{\pi}{2} \leq y \leq \frac{\pi}{2}.

  4. Step 4: Match this conclusion with Option (B).

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