Question 1092909
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{{{(3*tan(3theta))^2}}} - 27 = 0  ====>

{{{(3*tan(3theta))^2}}} = 27 ====>

{{{9*tan^2(3theta)}}} = 27  ====>  divide by 9 each side  ====>

{{{tan^2(3theta)}}} = 3  ====>

{{{tan(3theta)}}} = +/- {{{sqrt(3)}}}.


1)  {{{tan(3theta)}}} = {{{sqrt(3)}}}  ====>  {{{3theta}}} = {{{pi/3}}}  or  {{{4pi/3}}}  or  {{{7pi/3}}}  or  {{{10pi/3}}}  or  {{{13pi/3}}}  or  {{{16pi/3}}}.

    It gives 6 different solutions for {{{theta}}} in the interval [0,{{{2pi}}}):

    {{{pi/9}}},  {{{4pi/9}}},  {{{7pi/9}}},  {{{10pi/9}}},  {{{13pi/9}}},   {{{16pi/9}}}.



2)  {{{tan(3theta)}}} = {{{sqrt(3)}}}  ====>  {{{3theta}}} = {{{2pi/3}}}  or  {{{5pi/3}}}  or  {{{8pi/3}}}  or  {{{11pi/3}}}  or  {{{14pi/3}}}  or  {{{17pi/3}}}.

    It gives 6 different solutions for {{{theta}}} in the interval [0,{{{2pi}}}):

    {{{2pi/9}}},  {{{5pi/9}}},  {{{8pi/9}}},  {{{11pi/9}}},  {{{14pi/9}}},   {{{17pi/9}}}.


The final answer is the union of values of {{{theta}}} from the lists 1) and 2).
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