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

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

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2sin(x/2)=sin(x)

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

x = 4 πn, x = 2 π + 4 πn

Solution steps

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

Subtract

sin (x)

from both sides

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

Let:

u = \frac{x}{2}

2 sin (u) - sin (2 u) = 0

Rewrite using trig identities

2 sin (u) - 2 cos (u) sin (u) = 0

Factor

2 sin (u) - 2 cos (u) sin (u) : - 2 sin (u) (cos (u) - 1)

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

Solving each part separately

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

sin (u) = 0 : u = 2 πn, u = π + 2 πn

cos (u) - 1 = 0 : u = 2 πn

Combine all the solutions

u = 2 πn, u = π + 2 πn

Substitute back

u = \frac{x}{2}

\frac{x}{2} = 2 πn : x = 4 πn

\frac{x}{2} = π + 2 πn : x = 2 π + 4 πn

x = 4 πn, x = 2 π + 4 πn

cos (5 x) + cos (7 x) = 0

3 tan (4 x) - 1 = 0

2 sin (θ) = 3, 0 \leq θ < 2 π

6 arccos (x) = π

2 tan (x) - 2 = 0

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