Find all solutions in the interval [0, 2pi) : sin5x+sinx=sin3x

This is the concept of trigonometry, to solve for the value of x we proceed as follows;

sin5x+sinx=sin3x

this can be written as:

(sinx) ^5+sinx = (sinx) ^3

dividing through by sin x we get:

(sinx) ^4+1 = (sinx) ^2

let y = (sinx) ^2

thus our expression will be given by:

y^2+1=y

y^2-y+1=0

solving the quadratic equation above we get:

y = (-1) ^ (1/3) or - (-1) ^ (2/3)

the above is a complex equation, therefore our expression has no defined answer