Question 8 of 42beginner🔧 ApplyShort Answer2 marks

Find the principal values of the following: cot⁻¹(3\sqrt{3})

Correct Answer

Let cot⁡−1(3)=y\cot^{-1}(\sqrt{3}) = y. Then cot⁡y=3\cot y = \sqrt{3}.

We know that cot⁡(π6)=3\cot\left(\frac{\pi}{6}\right) = \sqrt{3}.

From the principal value branch, the range of cot⁡−1\cot^{-1} is (0,π)(0, \pi). Since π6∈(0,π)\frac{\pi}{6} \in (0, \pi), the principal value of cot⁡−1(3)\cot^{-1}(\sqrt{3}) is π6\frac{\pi}{6}.

Exercise: EXERCISE 2.1 | Q: 8 | (Page 10)
For More Understanding

Explanation

This question tests understanding of inverse cotangent function and its principal value branch. The student needs to: (1) Set up the equation by letting the inverse function equal y, (2) Convert to a cotangent equation, (3) Identify the angle where cotangent equals 3\sqrt{3}, and (4) Verify the angle lies within the principal branch range (0, π). The context clearly states that cot⁻¹ : R → (0, π), confirming π/6 is valid since it lies in this interval.

Solution Steps

  1. Step 1: Let cot⁻¹(3\sqrt{3}) = y, which means cot y = 3\sqrt{3}

  2. Step 2: Identify that cot(π/6) = 3\sqrt{3}

  3. Step 3: Check that π/6 lies in the principal branch range (0, π)

  4. Step 4: Conclude the principal value is π/6

PreviousNext