Find the Exact Value arccos(cos((4pi)/5))

Problem

arccos(cos((4*π)/5))

Solution

1. Identify the definition of the inverse cosine function. The expression arccos(cos(θ)) simplifies to θ if and only if θ falls within the restricted range of the principal value of arccos(x) which is [0,π]

2. Check if the given angle θ=(4*π)/5 is within the interval [0,π] Since 0≤4/5≤1 it follows that 0≤(4*π)/5≤π

3. Apply the property arccos(cos(θ))=θ for θ∈[0,π] Because (4*π)/5 is in the required range, the function and its inverse cancel each other out directly.

Final Answer

arccos(cos((4*π)/5))=(4*π)/5

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