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

The general solution for 3sin(2x)=-sin(x) is x=2pin,x=pi+2pin,x=1.73824…+2pin,x=-1.73824…+2pin

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

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

x = 2 πn, x = π + 2 πn, x = 1.73824 \dots + 2 πn, x = - 1.73824 \dots + 2 πn

Solution steps

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

Subtract

- sin (x)

from both sides

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

Rewrite using trig identities

sin (x) + 6 cos (x) sin (x) = 0

Factor

sin (x) + 6 cos (x) sin (x) : sin (x) (6 cos (x) + 1)

sin (x) (6 cos (x) + 1) = 0

Solving each part separately

sin (x) = 0 or 6 cos (x) + 1 = 0

sin (x) = 0 : x = 2 πn, x = π + 2 πn

6 cos (x) + 1 = 0 : x = arccos (- \frac{1}{6}) + 2 πn, x = - arccos (- \frac{1}{6}) + 2 πn

Combine all the solutions

x = 2 πn, x = π + 2 πn, x = arccos (- \frac{1}{6}) + 2 πn, x = - arccos (- \frac{1}{6}) + 2 πn

Show solutions in decimal form

x = 2 πn, x = π + 2 πn, x = 1.73824 \dots + 2 πn, x = - 1.73824 \dots + 2 πn

10 sin (A) + 16 = 3 sin (A) + 9

y, x = cos (y)

tan^{2} (x) - 2 tan (x) - 5 = 0

tan (x) = \frac{10}{11}

sin (θ) = \frac{9}{13}

What is the general solution for 3sin(2x)=-sin(x) ?
The general solution for 3sin(2x)=-sin(x) is x=2pin,x=pi+2pin,x=1.73824…+2pin,x=-1.73824…+2pin