.
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, for example, the lesson Trigonometric functions of multiply argument 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
=
. (1)
In this way the equation (1) decomposes in two equations. One is
sin(x) = 0 with the solutions x = 0, +/-
, +/-
, +/-
, . . . , +/-
, . . . (2)
The other equation is
=
, or
sin(x) = +/-
.
It has the roots x = +/-
. (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 = 0, 1, 2, 3, . . .
Obviously, the last set overlay the previous one, so you may restrict yourself by the last formula.