Find the Exact Value arctan(tan(pi/4))

Problem

arctan(tan(π/4))

Solution

1. Identify the range of the principal value for the inverse tangent function, arctan(x) which is (−π/2,π/2)

2. Determine if the input angle π/4 falls within this principal range.

3. Compare the values: since −π/2<π/4<π/2 the composition arctan(tan(θ)) simplifies directly to θ

4. Evaluate the expression using the property arctan(tan(x))=x for x in the interval (−π/2,π/2)

Final Answer

arctan(tan(π/4))=π/4

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