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=1.40334…+2pin,x=2pi-1.40334…+2pin

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3sin(2x)=sin(x)

3 sin (2 x) = sin (x)

x = 2 πn, x = π + 2 πn, x = 1.40334 \dots + 2 πn, x = 2 π - 1.40334 \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 = 2 π - 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 = 2 π - arccos (\frac{1}{6}) + 2 πn

Show solutions in decimal form

x = 2 πn, x = π + 2 πn, x = 1.40334 \dots + 2 πn, x = 2 π - 1.40334 \dots + 2 πn

cos (b) = \frac{12}{37}

27 (sin (2 t) cos (t) - (sin (t))^{3}) = 0

2 cos^{2} (x) + cos^{2} (x) = 0

cos (t) = \frac{1}{3}

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

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=1.40334…+2pin,x=2pi-1.40334…+2pin