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

The general solution for sin(t)+2sin(2t)=0 is t=2pin,t=pi+2pin,t=1.82347…+2pin,t=-1.82347…+2pin

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

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

t = 2 πn, t = π + 2 πn, t = 1.82347 \dots + 2 πn, t = - 1.82347 \dots + 2 πn

Solution steps

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

Rewrite using trig identities

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

Factor

sin (t) + 4 cos (t) sin (t) : sin (t) (4 cos (t) + 1)

sin (t) (4 cos (t) + 1) = 0

Solving each part separately

sin (t) = 0 or 4 cos (t) + 1 = 0

sin (t) = 0 : t = 2 πn, t = π + 2 πn

4 cos (t) + 1 = 0 : t = arccos (- \frac{1}{4}) + 2 πn, t = - arccos (- \frac{1}{4}) + 2 πn

Combine all the solutions

t = 2 πn, t = π + 2 πn, t = arccos (- \frac{1}{4}) + 2 πn, t = - arccos (- \frac{1}{4}) + 2 πn

Show solutions in decimal form

t = 2 πn, t = π + 2 πn, t = 1.82347 \dots + 2 πn, t = - 1.82347 \dots + 2 πn

1 + cos (θ) = 0

x, (D^{2} - 3 D + 2) y = cos (e^{- x})

2 + cos (2 x) = - 5 sin (x)

cos^{2} (θ) - \frac{1}{4} = 0

2 cot (x) + sec^{2} (x) = 0

What is the general solution for sin(t)+2sin(2t)=0 ?
The general solution for sin(t)+2sin(2t)=0 is t=2pin,t=pi+2pin,t=1.82347…+2pin,t=-1.82347…+2pin