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

The general solution for sin(3x)-sin(x)=sin(3x-x) is x=2pin,x=pi+2pin,x=(4pin)/3 ,x=(2pi)/3+(4pin)/3

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

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

x = 2 πn, x = π + 2 πn, x = \frac{4 πn}{3}, x = \frac{2 π}{3} + \frac{4 πn}{3}

Solution steps

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

Subtract

sin (3 x - x)

from both sides

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

Rewrite using trig identities

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

Factor

2 cos (2 x) sin (x) - 2 cos (x) sin (x) : 2 sin (x) (cos (2 x) - cos (x))

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

Solving each part separately

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

sin (x) = 0 : x = 2 πn, x = π + 2 πn

cos (2 x) - cos (x) = 0 : x = \frac{4 πn}{3}, x = \frac{2 π}{3} + \frac{4 πn}{3}

Combine all the solutions

x = 2 πn, x = π + 2 πn, x = \frac{4 πn}{3}, x = \frac{2 π}{3} + \frac{4 πn}{3}

sin (a) = - \frac{3}{2}

t, v = 311 cos (30 t + 3 0^{\circ}) v

cos^{2} (x) = 2 sin (x)

sin^{4} (θ) + cos^{4} (θ) = 2 sin^{2} (θ) - 1

x, \frac{- y}{2} + sin (x) = 0

What is the general solution for sin(3x)-sin(x)=sin(3x-x) ?
The general solution for sin(3x)-sin(x)=sin(3x-x) is x=2pin,x=pi+2pin,x=(4pin)/3 ,x=(2pi)/3+(4pin)/3