Find the Exact Value arctan(tan((4pi)/5))

Problem

arctan(tan((4*π)/5))

Solution

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

2. Determine if the input angle (4*π)/5 lies within this range. Since (4*π)/5≈0.8*π and the upper bound is 0.5*π the angle is outside the principal range.

3. Find a coterminal or related angle θ such that tan(θ)=tan((4*π)/5) and θ is within (−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 principal range.

(4*π)/5−π=(4*π)/5−(5*π)/5=−π/5

6. Verify that −π/5 is within the interval (−π/2,π/2) Since −0.2*π is between −0.5*π and 0.5*π the condition is met.

Final Answer

arctan(tan((4*π)/5))=−π/5

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