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

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

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

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

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

Solution steps

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

Rewrite using trig identities

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

Factor

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

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

Solving each part separately

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

sin (x) = 0 : x = 2 πn, x = π + 2 πn

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

Combine all the solutions

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

Show solutions in decimal form

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

2 sin^{2} (x) - 1 = - sin (x)

cosh (x) cosh (1) - sinh (x) sinh (1) = 2

cos^{2} (x) = \frac{3 ( 1 - sin ( x ) )}{2}

2 cos (θ) = 2 cos (3 θ)

cos (t) + cos (2 t) = - 0.75

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