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

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

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

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

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

Solution steps

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

Switch sides

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

Rewrite using trig identities

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

Factor

- sin (x) - 2 cos (x) sin (x) : - sin (x) (2 cos (x) + 1)

- sin (x) (2 cos (x) + 1) = 0

Solving each part separately

sin (x) = 0 or 2 cos (x) + 1 = 0

sin (x) = 0 : x = 2 πn, x = π + 2 πn

2 cos (x) + 1 = 0 : x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn

Combine all the solutions

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

x, 4 sin (x) cos (x) = 1

sin (θ) = - 3

\frac{sin ( 4 0 ^{\circ} )}{8} = \frac{sin ( x )}{12}

y, \frac{3}{2} = sin (y)

tan (β) = 2, 18 0^{\circ} \leq β \leq 27 0^{\circ}

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