Question 26 of 42intermediate🔧 ApplyMCQ1 mark

cos⁻¹(cos(7π/6)) is equal to

Exercise: EXERCISE 2.2 | Q: 13 | (Page 13)
For More Understanding

Explanation

The principal value branch of cos⁻¹ has range [0, π]. Since 7π/6 > π, it lies outside this range. Using the identity cos(θ) = cos(2π - θ), we get cos(7π/6) = cos(2π - 7π/6) = cos(5π/6). Since 5π/6 ∈ [0, π], this is the principal value. Therefore, cos⁻¹(cos(7π/6)) = 5π/6.

Solution Steps

  1. Step 1: Identify that the principal value branch of cos⁻¹ has range [0, π].

  2. Step 2: Check if 7π/6 lies in [0, π]. Since 7π/6 > π, it is outside the principal range.

  3. Step 3: Use the identity cos(θ) = cos(2π - θ) to find an equivalent angle in the principal range.

  4. Step 4: cos(7π/6) = cos(2π - 7π/6) = cos(12π/6 - 7π/6) = cos(5π/6).

  5. Step 5: Since 5π/6 ∈ [0, π], we have cos⁻¹(cos(7π/6)) = 5π/6.

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