Simplify arctan(tan((2pi)/3))

Problem

arctan(tan((2*π)/3))

Solution

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

2. Determine if the input angle (2*π)/3 falls within this range. Since (2*π)/3≈2.09 and π/2≈1.57 the angle is outside the principal range.

3. Find a coterminal or related angle θ such that tan(θ)=tan((2*π)/3) and θ is within the interval (−π/2,π/2)

4. Apply the periodicity of the tangent function, tan(x)=tan(x−n*π)

5. Subtract π from the angle to bring it into the required range.

(2*π)/3−π=−π/3

6. Verify that −π/3 is within the interval (−π/2,π/2)

7. Conclude that arctan(tan((2*π)/3))=arctan(tan(−π/3))=−π/3

Final Answer

arctan(tan((2*π)/3))=−π/3

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