Question 29 of 42intermediate🔧 ApplyShort Answer2 marks

Find the value of the following: cos⁻¹(cos(13π/6))

Correct Answer

The principal value branch of cos⁻¹ is [0, π]. The given angle 13π/6 can be written as 2π + π/6. Since cosine has period 2π, cos(13π/6) = cos(π/6). If the angle is not within the principal branch, we must find an equivalent angle within [0, π] that has the same cosine. Here, π/6 lies in [0, π], so cos⁻¹(cos(13π/6)) = cos⁻¹(cos(π/6)) = π/6.

Therefore, the required value is π/6.

Exercise: Miscellaneous Exercise on Chapter 2 | Q: 1 | (Page 14)
For More Understanding

Explanation

This question is from the Miscellaneous Exercise on Chapter 2. The method follows the same principle as Example 6, which adjusts the angle to lie within the principal branch. For cos⁻¹, the principal branch is [0, π]. The angle 13π/6 exceeds this interval, but using periodicity we reduce it to π/6, which is within [0, π], giving the result π/6.

Solution Steps

  1. Step 1: Recall that the principal value branch of cos⁻¹ is [0, π].

  2. Step 2: Express 13π/6 as 2π + π/6.

  3. Step 3: Use the periodicity of cosine: cos(2π + θ) = cos θ, so cos(13π/6) = cos(π/6).

  4. Step 4: Since π/6 ∈ [0, π], cos⁻¹(cos(π/6)) = π/6.

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