Find the Exact Value arccos(cos((5pi)/7))

Problem

arccos(cos((5*π)/7))

Solution

1. Identify the definition of the inverse cosine function. The expression arccos(cos(θ)) simplifies to θ if and only if θ is within the restricted range of the arccosine function.

2. Determine the range of the arccosine function. The range of y=arccos(x) is [0,π]

3. Compare the given angle to the range. The angle θ=(5*π)/7 must be checked against the interval [0,π]

4. Verify the inequality. Since 0≤5/7≤1 it follows that 0≤(5*π)/7≤π

5. Apply the property arccos(cos(θ))=θ for θ∈[0,π] Because (5*π)/7 is within the required interval, the functions cancel out directly.

Final Answer

arccos(cos((5*π)/7))=(5*π)/7

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