Question 1169221
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The given equation

    8*sin(6x) + 8*sin(2x) = 0     (1)


is equivalent to this one

    sin(6x) + sin(2x) = 0         (2)


(after canceling the factor 8).


In the plot below I show the left side function of equation (2) on the interval [0,2pi).



    {{{graph( 400, 400, -1, 7, -3, 3,
          sin(6x) + sin(2x)         
)}}}


                 Plot  y = sin(6x) + sin(2x)



From the plot, you see that there are 8 roots (solutions) on the interval [0,2pi).


To find all these solutions, you need to solve these equations

    6x = 2x,          (3)

    6x = 2x + {{{pi}}}     (4)

    6x = 2x + {{{2pi}}}    (5)

    6x = 2x + {{{3pi}}}    (6)

    6x = 2x + {{{4pi}}}    (7)

    6x = 2x + {{{5pi}}}    (8)

    6x = 2x + {{{6pi}}}    (9)

    6x = 2x + {{{7pi}}}    (10)


Equations (3)- (10) give the solutions x such that 6x and 2x differ in {{{k*pi}}}:  then sin(6x) and sin(2x) have the opposite signs.



The solution of these equations is ELEMENTARY, and I leave it to you.


When you complete your calculations, COMPARE the found roots with that you see in the plot.
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You are guided, explained, directed and instructed.