How do I make 2sin(x)^2=sin(x)?

Question

Please help

Answer 1

The equation is obviously true for

sin(x)=0 <=> x = 0

For the other solutions divide both sides by sin(x)

2sin²(x) = sin(x)

<=>

2 sin(x) = 1

<=>

sin(x) = 1/2

=>

x = arcsin(1/2) = (1/6)∙π

because

sin((1/2)∙π - x) = sin((1/2)∙π + x)

There is another solution:

x' =  (1/2)∙π + (1/6)∙π = (5/6)∙π

So the equation has three solutions within the interval [0, 2π]:

x = 0

x = (1/6)∙π

x = (5/6)∙π

Adding a integer multiple of 2∙π to any these solutions gives another solution. So the entity of solutions is given by:

x = ± n∙2∙π

x = (1/6)∙π ± n∙2∙π

x = (5/6)∙π ± n∙2∙π

with n= 0,1,2,3,....