Question 1005843
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SOLVE THE EQUATION SIN3X=SINX
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Start with the formula of triple argument for sine:

sin(3x) = -4*sin^3(x) + 3*sin(x)

(see, &nbsp;for example, &nbsp;the lesson &nbsp;<A HREF=http://www.algebra.com/algebra/homework/Trigonometry-basics/Trigonometric-functions-of-multiply-argument.lesson>Trigonometric functions of multiply argument</A>&nbsp; in this site).

It gives you an equation 

-4*sin^3(x) + 3*sin(x) = sin(x).

Move the term sin(x) from the right side to the left with the opposite sign and then simplify.

-4*sin^3(x) + 3*sin(x) - sin(x) = 0,

-4*sin^3(x) + 2*sin(x) = 0.

Now factor the left side

{{{-4*sin(x)*( sin^2(x) - (1/2) )}}} = {{{0}}}.   (1)

In this way the equation (1) decomposes in two equations. One is

sin(x) = 0 with the solutions x = 0, +/-{{{pi}}}, +/-{{{2pi}}}, +/-{{{3pi}}}, . . . , +/-{{{k*pi}}}, . . . (2)

The other equation is 

{{{ sin^2(x) - (1/2)}}} = {{{0}}},   or

sin(x) = +/- {{{sqrt(2)/2}}}.          

It has the roots x = +/- {{{k*(pi/2)}}}.   (3)

So, the answer is the combination (the union) of solutions (2) and (3).

x = +/- {k*pi}}}, k = 0, 1, 2, 3, . . .  and

x = +/- {{{k*(pi/2)}}}, k = 0, 1, 2, 3, . . . 

Obviously, the last set overlay the previous one, so you may restrict yourself by the last formula.
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