Question 1101623
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3*sin(2*a) = -3/sqrt(2) 
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        Due to "typography" issues,  I will replace  {{{theta}}}  in my post by simple  "a".



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{{{3*sin(2*a)}}} = {{{-3/sqrt(2)}}}      (1)       ====>  (divide both sides by 3)  ====>


{{{sin(2*a)}}} = {{{-1/sqrt(2)}}},   or, which is the same,

{{{sin(2*a)}}} = {{{-sqrt(2)/2}}}.


It implies  {{{2*a)}}} = {{{5pi/4}}}   or   {{{2*a}}} = {{{7pi/4}}}.



    Everything was simple to this point. 

    But in reality, accurate analysis only  <U>STARTS</U>  from this point.


1)  It is obvious that  {{{2*a)}}} = {{{5pi/4}}}  implies  {{{a}}} = {{{5pi/8}}}. 

    But if you stop here, you will loose another existing solution of the same family.

    It is  {{{a[2]}}} = {{{5pi/8 + pi}}} = {{{13pi/8}}}.

    Indeed,  {{{2*a[2]}}} = {{{5pi/4 + 2pi}}} = {{{13pi/4}}} is GEOMETRICALLY the same angle as {{{5pi/4}}}  and has the same value of sine,

    so {{{a[2]}}} is the solution to the original equation  (1), too.


    Thus the relation  {{{2*a)}}} = {{{5pi/4}}}  creates and generates not one solution {{{5pi/8}}}, but TWO solutions  {{{5pi/8}}}  and  {{{13pi/8}}}  

    of the same family.     Notice, that they BOTH belong to the interval  [0,{{{2pi}}}).



2)  The same or the similar story is with the solution  {{{2*a}}} = {{{7pi/4}}}.


    It is obvious that  {{{2*a)}}} = {{{7pi/4}}}  implies  {{{a}}} = {{{7pi/8}}}. 

    But if you stop here, you will loose another existing solution of the same family.

    It is  {{{a[4]}}} = {{{7pi/8 + pi}}} = {{{15pi/8}}}.

    Indeed,  {{{2*a[4]}}} = {{{7pi/4 + 2pi}}} = {{{15pi/4}}}  is GEOMETRICALLY the same angle as {{{7pi/4}}}  and has the same value of sine,

    so {{{a[4]}}} is the solution to the original equation  (1), too.


    Thus the relation  {{{2*a)}}} = {{{7pi/4}}}  creates and generates not one solution {{{7pi/8}}}, but TWO solutions  {{{7pi/8}}}  and  {{{15pi/8}}}  

    of the same family.     Notice, that they BOTH belong to the interval  [0,{{{2pi}}}).



3.  Thus the original equation (1) has 4 (four, FOUR) solutions in the interval  [0,{{{2pi}}}):

    {{{5pi/8}}},  {{{13pi/8}}},  {{{7pi/8}}}  and  {{{15pi/8}}}.



4.  The plot below visually confirms existing of 4 solutions to the given equality:


{{{graph( 330, 330, -0.5, 6.5, -4.5, 4.5,
          3*sin(2*x),  -3/sqrt(2)
)}}}


Plot y = {{{3*sin(2*teta)}}}  (red)  and y = {{{-3/sqrt(2)}}} (green)
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