What is the general solution for sin(2x)+3sin(x)=0 ?

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

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

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

x = 2 πn, x = π + 2 πn

Solution steps

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

Rewrite using trig identities

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

Factor

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

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

Solving each part separately

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

sin (x) = 0 : x = 2 πn, x = π + 2 πn

2 cos (x) + 3 = 0 :

No Solution

Combine all the solutions

x = 2 πn, x = π + 2 πn

\frac{cos ^{3} ( x )}{sin ( x )} = cot (x)

tan (2 x) = \frac{3}{4}

sin (x) = 1 + cos (x)

2 sin (x) = 1 + \frac{cos ( 2 x )}{sin ( x )}

sin (x) = \frac{21}{5}

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