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

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

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2sin(x)=sin(2x)

2 sin (x) = sin (2 x)

x = 2 πn, x = π + 2 πn

Solution steps

2 sin (x) = sin (2 x)

Subtract

sin (2 x)

from both sides

2 sin (x) - sin (2 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 (2 x) cos (x) = 0, 0 \leq x \leq 2 π

tan (x + π) + 2 sin (x + π) = 0

sin (x) = \frac{3}{2}

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

sin (θ) - 1 = cos (θ)

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