Question 1102804
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<pre>
{{{2*cos(2theta)}}} + 1 = 0  ====>


{{{cos(2theta)}}} = {{{-1/2}}}  ====>


{{{2*theta}}}  may have one of the two values:  {{{2*theta}}} = {{{2pi/3}}}  or  {{{2*theta}}} = {{{4pi/3}}}.


<U>Case 1</U>.  If  {{{2*theta}}} = {{{2pi/3}}},  then the solutions for {{{theta}}} in the interval  [0,{{{2pi}}})  are  these TWO values:

         {{{theta}}} = {{{pi/3}}}   and/or   {{{theta}}} = {{{pi/3 + pi}}} = {{{4pi/3}}}.



<U>Case 2</U>.  If  {{{2*theta}}} = {{{4pi/3}}},  then the solutions for {{{theta}}} in the interval  [0,{{{2pi}}})  are  these TWO values:

         {{{theta}}} = {{{2pi/3}}}   and/or   {{{theta}}} = {{{2pi/3 + pi}}} = {{{5pi/3}}}.


<U>Answer</U>.  The given equation has 4 (four, FOUR) solutions in the interval  [0,{{{2pi}}}):

                {{{pi/3}}},  {{{4pi/3}}},  {{{2pi/3}}},  and  {{{5pi/3}}}.
</pre>


The plot below visually confirms existing of 4 solutions:


{{{graph( 330, 330, -0.5, 6.5, -3.5, 4.5,
          2*cos(2*x) + 1
)}}}


Plot y = {{{2*cos(2*theta) + 1}}} 



&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;SOLVED.



----------------
<U>N O T E</U>. All the part of the condition &nbsp;IN PARENTHESES&nbsp; <U>makes no sense</U> &nbsp;and &nbsp;<U>IS NOT RELEVANT</U>&nbsp; to the problem at all.



&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;When I see such a &nbsp;NONSENSE&nbsp; in the condition, &nbsp;I become sad,
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;because I clearly see that the person who prepared this post &nbsp;<U>does not understand WHAT HE IS WRITING</U>.