Q586: The principal value of $\cot^{-1}(-\frac{1}{\sqrt{3}})$ is...

Class CBSE Class 12 Mathematics Inverse Trigonometric Functions Q #586

KNOWLEDGE BASED

REMEMBER

1 Marks 2025 AISSCE(Board Exam) MCQ SINGLE

The principal value of $\cot^{-1}(-\frac{1}{\sqrt{3}})$ is:

(A) $-\frac{\pi}{3}$

(B) $-\frac{2\pi}{3}$

(D) $\frac{2\pi}{3}$

Prev Next

AI Tutor Explanation

Detailed Solution

Step 1: Understanding the Range of Principal Value

The principal value range of $\cot^{-1}(x)$ is $(0, \pi)$.

Step 2: Finding the Angle

We need to find an angle $\theta$ in the range $(0, \pi)$ such that $\cot(\theta) = -\frac{1}{\sqrt{3}}$. Since cotangent is negative, $\theta$ must be in the second quadrant.

Step 3: Using the Known Value

We know that $\cot(\frac{\pi}{3}) = \frac{1}{\sqrt{3}}$. Therefore, we need to find an angle in the second quadrant with a reference angle of $\frac{\pi}{3}$.

Step 4: Calculating the Angle in the Second Quadrant

The angle in the second quadrant is $\pi - \frac{\pi}{3} = \frac{2\pi}{3}$. Thus, $\cot(\frac{2\pi}{3}) = -\frac{1}{\sqrt{3}}$.

Step 5: Determining the Principal Value

Since $\frac{2\pi}{3}$ is in the range $(0, \pi)$, the principal value of $\cot^{-1}(-\frac{1}{\sqrt{3}})$ is $\frac{2\pi}{3}$.

Final Answer: $\frac{2\pi}{3}$

AI Suggestion: Option D

AI generated content. Review strictly for academic accuracy.

Pedagogical Audit

Bloom's Analysis: This is an REMEMBER question because it requires recalling the range of the principal value of the inverse cotangent function and applying a known trigonometric value.

Knowledge Dimension: FACTUAL

Justification: The question directly tests the knowledge of the principal value range of the inverse cotangent function and the values of cotangent at standard angles.

Syllabus Audit: In the context of CBSE Class 12, this is classified as KNOWLEDGE. It directly assesses the student's understanding of inverse trigonometric functions and their principal values, a core concept in the syllabus.