Find the Exact Value arccos(cos((17pi)/6))

Problem

arccos(cos((17*π)/6))

Solution

1. Identify the domain and range of the inverse cosine function. The range of arccos(x) is [0,π]

2. Simplify the angle (17*π)/6 by finding its coterminal equivalent within the standard 2*π interval.

(17*π)/6=(12*π+5*π)/6

(17*π)/6=2*π+(5*π)/6

3. Apply the periodicity of the cosine function, where cos(θ+2*π)=cos(θ)

cos((17*π)/6)=cos((5*π)/6)

4. Evaluate the inverse cosine. Since (5*π)/6 is within the interval [0,π] the property arccos(cos(θ))=θ applies directly.

arccos(cos((5*π)/6))=(5*π)/6

Final Answer

arccos(cos((17*π)/6))=(5*π)/6

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