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

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

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sin(4x)=sin(2x)

sin (4 x) = sin (2 x)

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

Solution steps

sin (4 x) = sin (2 x)

Subtract

sin (2 x)

from both sides

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

Rewrite using trig identities

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

Solving each part separately

cos (3 x) = 0 or sin (x) = 0

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

sin (x) = 0 : x = 2 πn, x = π + 2 πn

Combine all the solutions

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

cos^{2} (x) = sin^{2} (x) - \frac{2}{2}

cos (2 θ) - sin (θ) = 0

tan (θ) = - \frac{2 3}{3} sin (θ)

3 cos (x) tan (x) - cos (x) = 0

4 sin^{2} (θ) = 3

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