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

The general solution for 2sin(2x)+3sin(x)=0 is x=2pin,x=pi+2pin,x=2.41885…+2pin,x=-2.41885…+2pin

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

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

x = 2 πn, x = π + 2 πn, x = 2.41885 \dots + 2 πn, x = - 2.41885 \dots + 2 πn

Solution steps

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

Rewrite using trig identities

3 sin (x) + 4 cos (x) sin (x) = 0

Factor

3 sin (x) + 4 cos (x) sin (x) : sin (x) (4 cos (x) + 3)

sin (x) (4 cos (x) + 3) = 0

Solving each part separately

sin (x) = 0 or 4 cos (x) + 3 = 0

sin (x) = 0 : x = 2 πn, x = π + 2 πn

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

Combine all the solutions

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

Show solutions in decimal form

x = 2 πn, x = π + 2 πn, x = 2.41885 \dots + 2 πn, x = - 2.41885 \dots + 2 πn

sin (x) = - \frac{2}{5}

tan (2 x) = - 1.8

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

5 cot (x) + 3 = - 2

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

What is the general solution for 2sin(2x)+3sin(x)=0 ?
The general solution for 2sin(2x)+3sin(x)=0 is x=2pin,x=pi+2pin,x=2.41885…+2pin,x=-2.41885…+2pin