Question 4 of 42beginner🔧 ApplyShort Answer2 marks

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

Correct Answer

The principal value of tan⁻¹(-3\sqrt{3}) is -π/3.

Exercise: EXERCISE 2.1 | Q: 4 | (Page 9)
For More Understanding

Explanation

This question follows the same methodology as Example 2 in the textbook for finding principal values of inverse trigonometric functions. The key is to express the given value in terms of a known trigonometric value and then find an angle within the principal branch range.

Solution Steps

  1. Step 1: Let tan⁻¹(-3\sqrt{3}) = y. Then, tan y = -3\sqrt{3}

  2. Step 2: We know that tan(π/3) = 3\sqrt{3}, so tan y = -3\sqrt{3} = -tan(π/3) = tan(-π/3)

  3. Step 3: The range of principal value branch of tan⁻¹ is (-π/2, π/2)

  4. Step 4: Since -π/3 lies in the range (-π/2, π/2), the principal value of tan⁻¹(-3\sqrt{3}) is -π/3

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