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

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

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sin(x)=sin(x/2)

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

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

Subtract

sin (\frac{x}{2})

from both sides

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

sec (x) - 2 = - tan (x) - 3

- 9.81 + 5 (32.17) (\frac{12}{2}) sin (θ) = 0

2 sin^{3} (x) - sin^{2} (x) - 2 sin (x) + 1 = 0

2 sin (x) = - 2

3 sin (x) = sin (x) + 1

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