Class CBSE Class 12 Mathematics Inverse Trigonometric Functions Q #587
KNOWLEDGE BASED
REMEMBER
1 Marks 2025 AISSCE(Board Exam) MCQ SINGLE
The principal value of \(\sin^{-1}(\sin(-\frac{10\pi}{3}))\) is:
(A) \(-\frac{2\pi}{3}\)
(B) \(-\frac{\pi}{3}\)
(C) \(\frac{\pi}{3}\)
(D) \(\frac{2\pi}{3}\)
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Correct Answer: C

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Detailed Solution

Step 1: Understanding the Principal Value Range

The principal value range for $\sin^{-1}(x)$ is $[-\frac{\pi}{2}, \frac{\pi}{2}]$. We need to find an angle within this range that has the same sine value as $\sin(-\frac{10\pi}{3})$.

Step 2: Simplifying the Angle

We can simplify $-\frac{10\pi}{3}$ by adding multiples of $2\pi$ to bring it within a more manageable range. $$-\frac{10\pi}{3} + 2\pi = -\frac{10\pi}{3} + \frac{6\pi}{3} = -\frac{4\pi}{3}$$ $$-\frac{4\pi}{3} + 2\pi = -\frac{4\pi}{3} + \frac{6\pi}{3} = \frac{2\pi}{3}$$ So, $-\frac{10\pi}{3}$ is coterminal with $\frac{2\pi}{3}$.

Step 3: Evaluating the Sine Function

Now we have $\sin(-\frac{10\pi}{3}) = \sin(\frac{2\pi}{3})$. We know that $\sin(\frac{2\pi}{3}) = \sin(\pi - \frac{\pi}{3}) = \sin(\frac{\pi}{3}) = \frac{\sqrt{3}}{2}$.

Step 4: Finding the Principal Value

We need to find an angle $\theta$ in the range $[-\frac{\pi}{2}, \frac{\pi}{2}]$ such that $\sin(\theta) = \frac{\sqrt{3}}{2}$. The angle $\theta = \frac{\pi}{3}$ satisfies this condition, since $\sin(\frac{\pi}{3}) = \frac{\sqrt{3}}{2}$ and $-\frac{\pi}{2} \le \frac{\pi}{3} \le \frac{\pi}{2}$.

Step 5: Final Answer

Therefore, the principal value of $\sin^{-1}(\sin(-\frac{10\pi}{3}))$ is $\frac{\pi}{3}$.

Final Answer: \(\frac{\pi}{3}\)

AI Suggestion: Option C

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Pedagogical Audit
Bloom's Analysis: This is an REMEMBER question because it requires recalling the definition of the principal value range of the inverse sine function and applying trigonometric identities to simplify the expression.
Knowledge Dimension: PROCEDURAL
Justification: The question requires applying a series of steps, including simplifying the angle, evaluating the sine function, and finding the principal value within the defined range. These are all procedural steps.
Syllabus Audit: In the context of CBSE Class 12, this is classified as KNOWLEDGE. The question directly tests the student's understanding of inverse trigonometric functions and their principal values, which is a core concept in the textbook.
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