Verified answer

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Without a calculator:

You might expect this to simplify to 7π/6.

But, arctan only returns angles between -π/2 and π/2.

i.e. arctan only returns 1st and 4th Quadrant angles.

Going back to 7π/6:

7π/6 is a 3rd Quadrant angle with π/6 as its reference angle.

And, since the tangent of 3rd Quadrant angles is positive,

it follows that we are looking for a 1st or 4th Quadrant angle with a reference

angle of π/6 for which the tangent is positive.

That puts our angle in the 1st Quadrant with a reference angle of π/6.

                     ANSWER

           arctan(tan(7π/6)) = π/6

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Look up (7π/6) on the Unit Circle.

sin(7π/6) = -1/2

cos(7π/6) = -(√3)/2

tan(7π/6) = 1/√3

tan(π/6) is also 1/√3.

arctan(tan(7π/6)) = π/6

arctan(tan(7π/6))

= arctan(1/sqrt3)

= pi/6