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

The general solution for solvefor x,sin(2x)=sin(x) is x=2pin,x=pi+2pin,x= pi/3+2pin,x=(5pi)/3+2pin

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solvefor x,sin(2x)=sin(x)

solve for

x, sin (2 x) = sin (x)

x = 2 πn, x = π + 2 πn, x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

Solution steps

sin (2 x) = sin (x)

Subtract

sin (x)

from both 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{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

Combine all the solutions

x = 2 πn, x = π + 2 πn, x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

- 3 cos (2 θ) + 7 cos (θ) - 10 = - 5

3 cos (2 θ) = - 9 cos (θ) - 6

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

1 + cos (θ) = 2 sin^{2} (θ)

cos (2 a) = - \frac{4}{7}

What is the general solution for solvefor x,sin(2x)=sin(x) ?
The general solution for solvefor x,sin(2x)=sin(x) is x=2pin,x=pi+2pin,x= pi/3+2pin,x=(5pi)/3+2pin