Find the Exact Value arctan(tan((11pi)/12))

Problem

arctan(tan((11*π)/12))

Solution

1. Identify the range of the inverse tangent function. The range of y=arctan(x) is (−π/2,π/2)

2. Check if the input angle (11*π)/12 falls within this range. Since (11*π)/12>π/2 (which is (6*π)/12, the property arctan(tan(θ))=θ cannot be applied directly.

3. Find a coterminal or related angle within the interval (−π/2,π/2) that has the same tangent value. The tangent function has a period of π

4. Subtract π from the angle to find the equivalent value within the required range.

(11*π)/12−π=(11*π)/12−(12*π)/12

(11*π)/12−(12*π)/12=−π/12

5. Verify that −π/12 is within the interval (−π/2,π/2) Since −(6*π)/12<−π/12<(6*π)/12 the condition is met.

6. Evaluate the expression using the identity tan(θ)=tan(θ−π)

arctan(tan((11*π)/12))=arctan(tan(−π/12))

arctan(tan(−π/12))=−π/12

Final Answer

arctan(tan((11*π)/12))=−π/12

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