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

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

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sin(2x)=2sin(x)

sin (2 x) = 2 sin (x)

x = 2 πn, x = π + 2 πn

Solution steps

sin (2 x) = 2 sin (x)

Subtract

2 sin (x)

from both sides

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

Rewrite using trig identities

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

Factor

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

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

Solving each part separately

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

sin (x) = 0 : x = 2 πn, x = π + 2 πn

cos (x) - 1 = 0 : x = 2 πn

Combine all the solutions

x = 2 πn, x = π + 2 πn

sin (x) - 1 = 0

cos (x) = \frac{1}{3}

tan (x) + 1 = 0

sin (x) = - cos (x)

tan (x) = \frac{1}{3}

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