What is the general solution for sin(θ+pi/6)=2sin(θ-pi/6) ?

The general solution for sin(θ+pi/6)=2sin(θ-pi/6) is θ= pi/3+pin

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sin(θ+pi/6)=2sin(θ-pi/6)

sin (θ + \frac{π}{6}) = 2 sin (θ - \frac{π}{6})

θ = \frac{π}{3} + πn

Solution steps

sin (θ + \frac{π}{6}) = 2 sin (θ - \frac{π}{6})

Rewrite using trig identities

\frac{3}{2} sin (θ) + \frac{1}{2} cos (θ) = 3 sin (θ) - cos (θ)

Subtract

3 sin (θ) - cos (θ)

from both sides

\frac{3}{2} cos (θ) + \frac{3}{2} sin (θ) - 3 sin (θ) = 0

Simplify

\frac{3}{2} cos (θ) + \frac{3}{2} sin (θ) - 3 sin (θ) : \frac{3 cos ( θ ) - 3 sin ( θ )}{2}

\frac{3 cos ( θ ) - 3 sin ( θ )}{2} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

3 cos (θ) - 3 sin (θ) = 0

Rewrite using trig identities

3 - 3 tan (θ) = 0

Move

3

to the right side

- 3 tan (θ) = - 3

Divide both sides by

- 3

tan (θ) = 3

General solutions for

tan (θ) = 3

θ = \frac{π}{3} + πn

4 sin^{2} (x) + 2 cos^{2} (x) - 3 = 0

4 cos^{2} (θ) tan (θ) - tan (θ) = 0

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

cos (x) = \frac{- 8}{9}

cos (x) = \frac{6}{7 76}

What is the general solution for sin(θ+pi/6)=2sin(θ-pi/6) ?
The general solution for sin(θ+pi/6)=2sin(θ-pi/6) is θ= pi/3+pin