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

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

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sin(3x)=sin(x)

sin (3 x) = sin (x)

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

Solution steps

sin (3 x) = sin (x)

Subtract

sin (x)

from both sides

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

Rewrite using trig identities

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

Solving each part separately

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

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

sin (x) = 0 : x = 2 πn, x = π + 2 πn

Combine all the solutions

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

cos (\frac{x}{2}) - sin (x) = 0

2 tan^{2} (θ) + tan (θ) = 0

2 cos^{2} (θ) + sin (θ) = 1

tan (2 x) - 3 = 0

0 = cos (2 x)

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