Class CBSE Class 12 Mathematics Inverse Trigonometric Functions Q #942
KNOWLEDGE BASED
APPLY
2 Marks 2024 VSA
Find the principal value of \(\tan^{-1}(1)+\cos^{-1}(-\frac{1}{2})+\sin^{-1}(-\frac{1}{\sqrt{2}}).\)
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Explanation
Let the expression be $E$.$$E = \tan^{-1}(1) + \cos^{-1}\left(-\frac{1}{2}\right) + \sin^{-1}\left(-\frac{1}{\sqrt{2}}\right)$$
Evaluate each term using principal value branches:
$\tan^{-1}(1) = \frac{\pi}{4}$(Since $\tan\frac{\pi}{4} = 1$)
$\cos^{-1}\left(-\frac{1}{2}\right) = \pi - \frac{\pi}{3} = \frac{2\pi}{3}$(Since $\cos^{-1}(-x) = \pi - \cos^{-1}x$)
$\sin^{-1}\left(-\frac{1}{\sqrt{2}}\right) = -\frac{\pi}{4}$(Since $\sin^{-1}(-x) = -\sin^{-1}x$)

Substitute these values back into $E$:$$E = \frac{\pi}{4} + \frac{2\pi}{3} - \frac{\pi}{4}$$Cancel the $\frac{\pi}{4}$ terms:$$E = \frac{2\pi}{3}$$

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Step-by-Step Solution

First, we find the principal value of each inverse trigonometric function:

  1. \(\tan^{-1}(1)\): The principal value is \(\frac{\pi}{4}\) because \(\tan(\frac{\pi}{4}) = 1\) and \(-\frac{\pi}{2} < \frac{\pi}{4} < \frac{\pi}{2}\).

  2. \(\cos^{-1}(-\frac{1}{2})\): The principal value is \(\frac{2\pi}{3}\) because \(\cos(\frac{2\pi}{3}) = -\frac{1}{2}\) and \(0 \le \frac{2\pi}{3} \le \pi\).

  3. \(\sin^{-1}(-\frac{1}{\sqrt{2}})\): The principal value is \(-\frac{\pi}{4}\) because \(\sin(-\frac{\pi}{4}) = -\frac{1}{\sqrt{2}}\) and \(-\frac{\pi}{2} \le -\frac{\pi}{4} \le \frac{\pi}{2}\).

Now, we add the principal values:

\(\tan^{-1}(1) + \cos^{-1}(-\frac{1}{2}) + \sin^{-1}(-\frac{1}{\sqrt{2}}) = \frac{\pi}{4} + \frac{2\pi}{3} - \frac{\pi}{4}\)

\(= \frac{2\pi}{3}\)

Correct Answer: 2π/3

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Pedagogical Audit
Bloom's Analysis: This is an APPLY question because it requires the student to apply their knowledge of inverse trigonometric functions and their principal values to solve the given expression.
Knowledge Dimension: PROCEDURAL
Justification: The question requires the student to follow a specific procedure to find the principal values of the inverse trigonometric functions and then sum them up.
Syllabus Audit: In the context of CBSE Class 12, this is classified as KNOWLEDGE. The question directly tests the student's understanding and application of formulas and concepts related to inverse trigonometric functions as covered in the textbook.

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