What is the general solution for sin(x)+sin(2x)=sin(3x) ?

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

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sin(x)+sin(2x)=sin(3x)

sin (x) + sin (2 x) = sin (3 x)

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

Solution steps

sin (x) + sin (2 x) = sin (3 x)

Subtract

sin (3 x)

from both sides

sin (x) + sin (2 x) - sin (3 x) = 0

Rewrite using trig identities

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

Factor

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

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

Solving each part separately

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

sin (x) = 0 : x = 2 πn, x = π + 2 πn

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

Combine all the solutions

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

2 cos^{2} (x) - 9 cos (x) + 4 = 0

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

x, - 2 sin (2 x) = 0

sin (θ) + cos (2 θ) = 1

- 6 sec (θ) - 3 = - 15

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