What is the general solution for 6sin(2t)+9sin(t)=0 ?

The general solution for 6sin(2t)+9sin(t)=0 is t=2pin,t=pi+2pin,t=2.41885…+2pin,t=-2.41885…+2pin

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6sin(2t)+9sin(t)=0

6 sin (2 t) + 9 sin (t) = 0

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

Solution steps

6 sin (2 t) + 9 sin (t) = 0

Rewrite using trig identities

9 sin (t) + 12 cos (t) sin (t) = 0

Factor

9 sin (t) + 12 cos (t) sin (t) : 3 sin (t) (4 cos (t) + 3)

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

Solving each part separately

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

sin (t) = 0 : t = 2 πn, t = π + 2 πn

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

Combine all the solutions

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

Show solutions in decimal form

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

sin (2 x + 60) + sin (x + 30) = 0

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

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

4 sec (x) + 8 = 0

x, ln (y) = e^{y} sin (x)

What is the general solution for 6sin(2t)+9sin(t)=0 ?
The general solution for 6sin(2t)+9sin(t)=0 is t=2pin,t=pi+2pin,t=2.41885…+2pin,t=-2.41885…+2pin