Solve the equation sin (5x) - sin (3x) = sin (x), giving all the solutions in the range of 0 < = x < = 180

Write the left side in terms of the sum and difference of 4x and x.

sin(4x +x) - sin(4x -x) = sin(x)

(sin(4x)cos(x) +cos(4x)sin(x)) - (sin(4x)cos(x) -cos(4x)sin(x)) = sin(x)

2cos(4x)sin(x) = sin(x)

sin(x)·(2cos(4x) -1) = 0 . . . . . subtract sin(x) and factor

Now, the zero product rule comes into play. Solutions are found where the factors are zero.

... sin(x) = 0 at x ∈ {0, 180°}

... 2cos(4x) -1 = 0 at x = arccos(1/2)/4 = (±60° +k·360°)/4 . . . . k an integer

... ... x ∈ {15°, 75°, 105°, 165°}

step-by-step explanation: