What is the general solution for -sin(x/2)+sin(x)=0 ?

The general solution for -sin(x/2)+sin(x)=0 is x=(2pi)/3+(8pin)/3 ,x=2pi+(8pin)/3 ,x=8pin,x=4pi+8pin

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

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

x = \frac{2 π}{3} + \frac{8 πn}{3}, x = 2 π + \frac{8 πn}{3}, x = 8 πn, x = 4 π + 8 πn

Solution steps

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

Rewrite using trig identities

2 cos (\frac{3 x}{4}) sin (\frac{x}{4}) = 0

Solving each part separately

cos (\frac{3 x}{4}) = 0 or sin (\frac{x}{4}) = 0

cos (\frac{3 x}{4}) = 0 : x = \frac{2 π}{3} + \frac{8 πn}{3}, x = 2 π + \frac{8 πn}{3}

sin (\frac{x}{4}) = 0 : x = 8 πn, x = 4 π + 8 πn

Combine all the solutions

x = \frac{2 π}{3} + \frac{8 πn}{3}, x = 2 π + \frac{8 πn}{3}, x = 8 πn, x = 4 π + 8 πn

3 sin (2 θ) = 2 cos (2 θ)

cos (x) + cos^{2} (x) = 0

cos (θ) = cos (1)

tan (\frac{x}{2}) = \pm \frac{1 - cos ( x )}{1 + cos ( x )}

sin^{2} (x) - cos^{2} (x) = 0.5

What is the general solution for -sin(x/2)+sin(x)=0 ?
The general solution for -sin(x/2)+sin(x)=0 is x=(2pi)/3+(8pin)/3 ,x=2pi+(8pin)/3 ,x=8pin,x=4pi+8pin