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

The general solution for sin(x)+sin(x/2)=0 is x=2pi+8pin,x=6pi+8pin,x=(8pin)/3 ,x=(4pi)/3+(8pin)/3

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

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

x = 2 π + 8 πn, x = 6 π + 8 πn, x = \frac{8 πn}{3}, x = \frac{4 π}{3} + \frac{8 πn}{3}

Solution steps

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

Rewrite using trig identities

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

Solving each part separately

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

cos (\frac{x}{4}) = 0 : x = 2 π + 8 πn, x = 6 π + 8 πn

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

Combine all the solutions

x = 2 π + 8 πn, x = 6 π + 8 πn, x = \frac{8 πn}{3}, x = \frac{4 π}{3} + \frac{8 πn}{3}

sin^{3} (x) + sin (x) = 2 cos^{2} (x)

4 sin^{2} (x) + 2 cos^{2} (x) = 3

cos^{2} (θ) - 3 sin (2 θ) + sin^{2} (θ) + 2 sin^{2} (2 θ) = 0

tan^{2} (x) cos (x) + cos^{2} (x) = 1

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

What is the general solution for sin(x)+sin(x/2)=0 ?
The general solution for sin(x)+sin(x/2)=0 is x=2pi+8pin,x=6pi+8pin,x=(8pin)/3 ,x=(4pi)/3+(8pin)/3