What is the general solution for sin(3x)-sin(x)=-2sin(2x)sin(x) ?

The general solution for sin(3x)-sin(x)=-2sin(2x)sin(x) is x=2pin,x=pi+2pin,x=(3pi)/8+(pin)/2

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sin(3x)-sin(x)=-2sin(2x)sin(x)

sin (3 x) - sin (x) = - 2 sin (2 x) sin (x)

x = 2 πn, x = π + 2 πn, x = \frac{3 π}{8} + \frac{πn}{2}

Solution steps

sin (3 x) - sin (x) = - 2 sin (2 x) sin (x)

Subtract

- 2 sin (2 x) sin (x)

from both sides

sin (3 x) - sin (x) + 2 sin (2 x) sin (x) = 0

Rewrite using trig identities

2 cos (2 x) sin (x) + 2 sin (2 x) sin (x) = 0

Factor

2 cos (2 x) sin (x) + 2 sin (2 x) sin (x) : 2 sin (x) (cos (2 x) + sin (2 x))

2 sin (x) (cos (2 x) + sin (2 x)) = 0

Solving each part separately

sin (x) = 0 or cos (2 x) + sin (2 x) = 0

sin (x) = 0 : x = 2 πn, x = π + 2 πn

cos (2 x) + sin (2 x) = 0 : x = \frac{3 π}{8} + \frac{πn}{2}

Combine all the solutions

x = 2 πn, x = π + 2 πn, x = \frac{3 π}{8} + \frac{πn}{2}

x, 0.05 cos (x) = - sin (x)

cos (x^{\circ}) = \frac{20.26}{26.4473}

cos (x) = \frac{- \frac{4}{3} 3}{\frac{8}{3} 3}

cos (θ) = \frac{3}{10}

27 tan^{2} (x) - 9 = 0

What is the general solution for sin(3x)-sin(x)=-2sin(2x)sin(x) ?
The general solution for sin(3x)-sin(x)=-2sin(2x)sin(x) is x=2pin,x=pi+2pin,x=(3pi)/8+(pin)/2