What is the general solution for 3sin(2x)*cos(x)=2sin^3(x) ?

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

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3sin(2x)*cos(x)=2sin^3(x)

3 sin (2 x) \cdot cos (x) = 2 sin^{3} (x)

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

Solution steps

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

Subtract

2 sin^{3} (x)

from both sides

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

Rewrite using trig identities

- 2 sin^{3} (x) + 6 cos^{2} (x) sin (x) = 0

Factor

- 2 sin^{3} (x) + 6 cos^{2} (x) sin (x) : 2 sin (x) (3 cos (x) + sin (x)) (3 cos (x) - sin (x))

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

Solving each part separately

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

sin (x) = 0 : x = 2 πn, x = π + 2 πn

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

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

Combine all the solutions

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

x, sin (x) \cdot cos (y) = 0

2 cos (3 x - 1) = - 1

\frac{0.866}{sin ( x )} = 1.53

3 sin (2 x) + 1 = 2

2 cos^{4} (x) - 2 cos^{2} (x) = 4 cos^{2} (x) - 1

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