Find the Exact Value arctan(tan(100))

Problem

arctan(tan(100))

Solution

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

2. Determine the value of the input 100 in terms of radians. Since 100 is not within the interval (−π/2,π/2) we must find an angle θ such that tan(θ)=tan(100) and θ falls within the principal range.

3. Apply the periodicity of the tangent function, which states tan(x)=tan(x−k*π) for any integer k

4. Calculate the integer k that shifts 100 into the interval (−π/2,π/2) by solving −π/2<100−k*π<π/2

5. Solve for k by approximating π≈3.14159

k≈100/π≈31.83

6. Select the integer k=32 to bring the value closest to 0 and within the required range.

7. Substitute the value back into the expression to find the exact principal value.

Final Answer

arctan(tan(100))=100−32*π

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