Respuesta :

Answer:   {0, pi/3, pi, 5pi/3, 2pi}

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Work Shown:

sin(2x) = sin(x)

sin(2x) - sin(x) = 0

2*sin(x)*cos(x) - sin(x) = 0

sin(x)*(2cos(x) - 1) = 0

sin(x) = 0 or 2cos(x)-1 = 0

sin(x) = 0 or cos(x) = 1/2

Using the unit circle, the solutions to sin(x) = 0 are x = 0, x = pi, x = 2pi

Also, the solutions to cos(x) = 1/2 are x = pi/3, x = 5pi/3

So the entire solution set on the interval [0,2pi] is {0, pi/3, pi, 5pi/3, 2pi} when written in increasing order.

All solutions to the equation in the interval [0, 2[tex]\pi[/tex]]  is  {0, [tex]\frac{\pi }{3}, \pi , \frac{5\pi }{3} ,2\pi[/tex]}.

What is solution to an equation?

A solution is an assignment of values to the unknown variables that makes the equality in the equation true.

As sin2x = 2 sinx cosx,

sin2x = sinx is equivalent to

2 sinx cosx = sinx or 2 sinx cosx − sinx = 0 or

sinx (2cosx − 1) = 0 i.e.

either sinx = 0 and x = 0, π, 2π

or 2cosx − 1 = 0 i.e. cosx = [tex]\frac{1}{2}[/tex] i.e. x = [tex]\frac{\pi }{3} , \frac{5\pi }{3}[/tex]

Hence, solution to an equation is x = {0, [tex]\frac{\pi }{3}, \pi , \frac{5\pi }{3} ,2\pi[/tex]}

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