((tan(x)+2sin(x)))/((tan(x)-2sin(x)))=3

Solución

\frac{( tan ( x ) + 2 sin ( x ) )}{( tan ( x ) - 2 sin ( x ) )} = 3

x = 1.31811 \dots + 2 πn, x = 2 π - 1.31811 \dots + 2 πn

Pasos de solución

\frac{( tan ( x ) + 2 sin ( x ) )}{( tan ( x ) - 2 sin ( x ) )} = 3

Restar

3

de ambos lados

\frac{tan ( x ) + 2 sin ( x )}{tan ( x ) - 2 sin ( x )} - 3 = 0

Simplificar

\frac{tan ( x ) + 2 sin ( x )}{tan ( x ) - 2 sin ( x )} - 3 : \frac{- 2 tan ( x ) + 8 sin ( x )}{tan ( x ) - 2 sin ( x )}

\frac{- 2 tan ( x ) + 8 sin ( x )}{tan ( x ) - 2 sin ( x )} = 0

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

- 2 tan (x) + 8 sin (x) = 0

Expresar con seno, coseno

\frac{- 2 sin ( x ) + 8 cos ( x ) sin ( x )}{cos ( x )} = 0

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

- 2 sin (x) + 8 cos (x) sin (x) = 0

Factorizar

- 2 sin (x) + 8 cos (x) sin (x) : 2 sin (x) (4 cos (x) - 1)

2 sin (x) (4 cos (x) - 1) = 0

Resolver cada parte por separado

sin (x) = 0 or 4 cos (x) - 1 = 0

sin (x) = 0 : x = 2 πn, x = π + 2 πn

4 cos (x) - 1 = 0 : x = arccos (\frac{1}{4}) + 2 πn, x = 2 π - arccos (\frac{1}{4}) + 2 πn

Combinar toda las soluciones

x = 2 πn, x = π + 2 πn, x = arccos (\frac{1}{4}) + 2 πn, x = 2 π - arccos (\frac{1}{4}) + 2 πn

Siendo que la ecuación esta indefinida para:

2 πn, π + 2 πn

x = arccos (\frac{1}{4}) + 2 πn, x = 2 π - arccos (\frac{1}{4}) + 2 πn

Mostrar soluciones en forma decimal

x = 1.31811 \dots + 2 πn, x = 2 π - 1.31811 \dots + 2 πn

sin^{(2)} (x) + cos (x) + 1 = 0

sin^{2} (θ) = 2 sin (θ)

1 - sin (2 x) = 0

cos (p) - sin (p) = 1

x, y = 2 sin (x)