Find the principal values of the following: sec⁻¹(2/√3)

Question 7 of 42beginner🔧 ApplyShort Answer2 marks

Find the principal values of the following: sec⁻¹(2/ $\sqrt{3}$ )

Correct Answer

Let sec⁻¹(2/ $\sqrt{3}$ ) = y. Then, sec y = 2/ $\sqrt{3}$ , which means cos y = $\sqrt{3}$ /2. We know that cos(π/6) = $\sqrt{3}$ /2. The principal value branch of sec⁻¹ is [0, π] - {π/2}. Since π/6 lies in this interval and sec(π/6) = 2/ $\sqrt{3}$ , the principal value of sec⁻¹(2/ $\sqrt{3}$ ) is π/6.

Exercise: EXERCISE 2.1 | Q: 7 | (Page 10)

For More Understanding

Explanation

The question asks for the principal value of sec⁻¹(2/ $\sqrt{3}$ ). Following the method shown in Example 2 of the textbook, we let the inverse function equal y and solve for the angle. The key is to identify that sec y = 2/ $\sqrt{3}$ implies cos y = $\sqrt{3}$ /2, which corresponds to π/6. The principal value branch of sec⁻¹ is [0, π] - {π/2} as stated in the context, and π/6 falls within this range.

Solution Steps

Step 1: Let sec⁻¹(2/ $\sqrt{3}$ ) = y, then sec y = 2/ $\sqrt{3}$
Step 2: Convert to cosine: cos y = $\sqrt{3}$ /2
Step 3: Identify the angle: cos(π/6) = $\sqrt{3}$ /2
Step 4: Verify π/6 ∈ [0, π] - {π/2} (principal value branch of sec⁻¹)
Step 5: Principal value = π/6

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