Question 9 of 42beginner🔧 ApplyShort Answer2 marks

Find the principal values of the following: cos⁻¹(-1/2\sqrt{2})

Correct Answer

Let cos⁻¹(-1/2\sqrt{2}) = y. Then, cos y = -1/2\sqrt{2}.

We know that cos(π/4) = 1/2\sqrt{2}. Therefore, cos(3π/4) = cos(π - π/4) = -cos(π/4) = -1/2\sqrt{2}.

Since the range of principal value branch of cos⁻¹ is [0, π] and 3π/4 lies in this interval, the principal value of cos⁻¹(-1/2\sqrt{2}) is 3π/4.

Exercise: EXERCISE 2.1 | Q: 9 | (Page 10)
For More Understanding

Explanation

This question follows the same pattern as Example 2 in the textbook. The student correctly identifies that the principal value branch of cos⁻¹ has range [0, π] as stated in the context. Using the identity cos(π - θ) = -cos(θ), they find that cos(3π/4) = -1/2\sqrt{2}. Since 3π/4 ∈ [0, π], this is the principal value.

Solution Steps

  1. Step 1: Let cos⁻¹(-1/2\sqrt{2}) = y, so cos y = -1/2\sqrt{2}

  2. Step 2: Use identity cos(π - θ) = -cos(θ), where cos(π/4) = 1/2\sqrt{2}

  3. Step 3: Thus cos(3π/4) = -1/2\sqrt{2}

  4. Step 4: Verify 3π/4 ∈ [0, π] (principal range of cos⁻¹)

  5. Step 5: Principal value is 3π/4

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