What is the general solution for (sin(x))/(sin(2x))= 1/2 ?

The general solution for (sin(x))/(sin(2x))= 1/2 is No Solution for x\in\mathbb{R}

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(sin(x))/(sin(2x))= 1/2

\frac{sin ( x )}{sin ( 2 x )} = \frac{1}{2}

No Solution for x \in R

Solution steps

\frac{sin ( x )}{sin ( 2 x )} = \frac{1}{2}

Subtract

\frac{1}{2}

from both sides

\frac{sin ( x )}{sin ( 2 x )} - \frac{1}{2} = 0

Simplify

\frac{sin ( x )}{sin ( 2 x )} - \frac{1}{2} : \frac{2 sin ( x ) - sin ( 2 x )}{2 sin ( 2 x )}

\frac{2 sin ( x ) - sin ( 2 x )}{2 sin ( 2 x )} = 0

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

2 sin (x) - sin (2 x) = 0

Rewrite using trig identities

2 sin (x) - 2 cos (x) sin (x) = 0

Factor

2 sin (x) - 2 cos (x) sin (x) : - 2 sin (x) (cos (x) - 1)

- 2 sin (x) (cos (x) - 1) = 0

Solving each part separately

sin (x) = 0 or cos (x) - 1 = 0

sin (x) = 0 : x = 2 πn, x = π + 2 πn

cos (x) - 1 = 0 : x = 2 πn

Combine all the solutions

x = 2 πn, x = π + 2 πn

Since the equation is undefined for:

2 πn, π + 2 πn

No Solution for x \in R

- 4 cos (2 θ) - 19 = - 26 cos (θ)

x, z = e^{y} - 3 sin (x y)

tan (4 x + 6) = - 0.272

7 sin (θ) + 22 = 0

\frac{1}{2} sinh (2 x) - \frac{4}{5} cosh (2 x) + 1 = 0

What is the general solution for (sin(x))/(sin(2x))= 1/2 ?
The general solution for (sin(x))/(sin(2x))= 1/2 is No Solution for x\in\mathbb{R}