What is the general solution for sin(5x)-sin(x)+3cos(3x)=0 ?

The general solution for sin(5x)-sin(x)+3cos(3x)=0 is x= pi/6+(2pin)/3 ,x= pi/2+(2pin)/3

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sin(5x)-sin(x)+3cos(3x)=0

sin (5 x) - sin (x) + 3 cos (3 x) = 0

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

Solution steps

sin (5 x) - sin (x) + 3 cos (3 x) = 0

Rewrite using trig identities

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

Factor

3 cos (3 x) + 2 cos (3 x) sin (2 x) : cos (3 x) (2 sin (2 x) + 3)

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

Solving each part separately

cos (3 x) = 0 or 2 sin (2 x) + 3 = 0

cos (3 x) = 0 : x = \frac{π}{6} + \frac{2 πn}{3}, x = \frac{π}{2} + \frac{2 πn}{3}

2 sin (2 x) + 3 = 0 :

No Solution

Combine all the solutions

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

6 cos (2 x) = 3 + 5 sin^{2} (2 x)

2 + 2 sin (θ) = cos^{2} (θ)

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

- 19 sin (θ) + \frac{19}{2} = 0

y, e^{8 x} = sin (x + 3 y)

What is the general solution for sin(5x)-sin(x)+3cos(3x)=0 ?
The general solution for sin(5x)-sin(x)+3cos(3x)=0 is x= pi/6+(2pin)/3 ,x= pi/2+(2pin)/3