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

The general solution for (tan(2x))/(sin(x))=-2 is x= pi/3+(4pin)/3 ,x=pi+(4pin)/3

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(tan(2x))/(sin(x))=-2

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

x = \frac{π}{3} + \frac{4 πn}{3}, x = π + \frac{4 πn}{3}

Solution steps

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

Subtract

- 2

from both sides

\frac{tan ( 2 x )}{sin ( x )} + 2 = 0

Simplify

\frac{tan ( 2 x )}{sin ( x )} + 2 : \frac{tan ( 2 x ) + 2 sin ( x )}{sin ( x )}

\frac{tan ( 2 x ) + 2 sin ( x )}{sin ( x )} = 0

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

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

Express with sin, cos

\frac{sin ( 2 x ) + 2 cos ( 2 x ) sin ( x )}{cos ( 2 x )} = 0

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

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

Rewrite using trig identities

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

Factor

2 cos (2 x) sin (x) + 2 cos (x) sin (x) : 2 sin (x) (cos (2 x) + cos (x))

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

Solving each part separately

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

sin (x) = 0 : x = 2 πn, x = π + 2 πn

cos (2 x) + cos (x) = 0 : x = \frac{π}{3} + \frac{4 πn}{3}, x = π + \frac{4 πn}{3}

Combine all the solutions

x = 2 πn, x = π + 2 πn, x = \frac{π}{3} + \frac{4 πn}{3}, x = π + \frac{4 πn}{3}

Since the equation is undefined for:

2 πn, π + 2 πn

x = \frac{π}{3} + \frac{4 πn}{3}, x = π + \frac{4 πn}{3}

sin (A) = 0.5

cos (θ) = tan (θ)

tan (\frac{π}{4} + x) = \frac{1 - tan ( x )}{1 + tan ( x )}

cos (x) = \frac{45}{52}

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

What is the general solution for (tan(2x))/(sin(x))=-2 ?
The general solution for (tan(2x))/(sin(x))=-2 is x= pi/3+(4pin)/3 ,x=pi+(4pin)/3