What is the general solution for sin(4x)-sin(2x)=cos(3x) ?

The general solution for sin(4x)-sin(2x)=cos(3x) is x= pi/6+(2pin)/3 ,x= pi/2+(2pin)/3

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sin(4x)-sin(2x)=cos(3x)

sin (4 x) - sin (2 x) = cos (3 x)

x = \frac{π}{6} + \frac{2 πn}{3}, x = \frac{π}{2} + \frac{2 πn}{3}

Solution steps

sin (4 x) - sin (2 x) = cos (3 x)

Subtract

cos (3 x)

from both sides

sin (4 x) - sin (2 x) - cos (3 x) = 0

Rewrite using trig identities

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

Factor

- cos (3 x) + 2 cos (3 x) sin (x) : cos (3 x) (2 sin (x) - 1)

cos (3 x) (2 sin (x) - 1) = 0

Solving each part separately

cos (3 x) = 0 or 2 sin (x) - 1 = 0

cos (3 x) = 0 : x = \frac{π}{6} + \frac{2 πn}{3}, x = \frac{π}{2} + \frac{2 πn}{3}

2 sin (x) - 1 = 0 : x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

Combine all the solutions

x = \frac{π}{6} + \frac{2 πn}{3}, x = \frac{π}{2} + \frac{2 πn}{3}, x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

Merge Overlapping Intervals

x = \frac{π}{6} + \frac{2 πn}{3}, x = \frac{π}{2} + \frac{2 πn}{3}

sin (x) = 0.7392

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

3 + 2 sin (x) = 0

tan (x) = \frac{6}{4}

cot (x) sin^{2} (x) = 4 cot (x)

What is the general solution for sin(4x)-sin(2x)=cos(3x) ?
The general solution for sin(4x)-sin(2x)=cos(3x) is x= pi/6+(2pin)/3 ,x= pi/2+(2pin)/3