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This problem can be solved very easily with calculus.     At the maximum value of θ ,  the derivative of the
expression will be zero.   
This derivative is    12 cos(θ) - 18 sin(θ) cos(θ)   setting this to zero results in
12 - 18 sin(θ) =0     (after dividing through by cos(θ)   )  So
  sin(θ) = 2/3    substituting back into the expression yields the value 4.
 
This can be checked using a graphing calculator.

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