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

Solución

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

Solución

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

Grados

x = 6 0^{\circ} + 24 0^{\circ} n, x = 18 0^{\circ} + 24 0^{\circ} n

Pasos de solución

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

Restar

- 2

de ambos lados

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

Simplificar

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

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

Convertir a fracción:

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

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

Ya que los denominadores son iguales, combinar las fracciones:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \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

Expresar con seno, coseno

tan (2 x) + 2 sin (x)

Utilizar la identidad trigonométrica básica:

tan (x) = \frac{sin ( x )}{cos ( x )}

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

Simplificar

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

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

Convertir a fracción:

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

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

Ya que los denominadores son iguales, combinar las fracciones:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

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

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

\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

Re-escribir usando identidades trigonométricas

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

Utilizar la identidad trigonométrica del ángulo doble:

sin (2 x) = 2 sin (x) cos (x)

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

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

Factorizar

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

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

Reescribir como

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

Factorizar el termino común

2 sin (x)

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

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

Resolver cada parte por separado

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

sin (x) = 0 : x = 2 πn, x = π + 2 πn

sin (x) = 0

Soluciones generales para

sin (x) = 0

tabla de valores periódicos con

2 πn

intervalos:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

x = 0 + 2 πn, x = π + 2 πn

x = 0 + 2 πn, x = π + 2 πn

Resolver

x = 0 + 2 πn : x = 2 πn

x = 0 + 2 πn

0 + 2 πn = 2 πn

x = 2 πn

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}

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

Re-escribir usando identidades trigonométricas

cos (2 x) + cos (x)

Utilizar la identidad suma-producto:

cos (s) + cos (t) = 2 cos (\frac{s + t}{2}) cos (\frac{s - t}{2})

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

Simplificar

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

2 cos (\frac{2 x + x}{2}) cos (\frac{2 x - x}{2})

Sumar elementos similares:

2 x + x = 3 x

= 2 cos (\frac{3 x}{2}) cos (\frac{2 x - x}{2})

Sumar elementos similares:

2 x - x = x

= 2 cos (\frac{3 x}{2}) cos (\frac{x}{2})

= 2 cos (\frac{3 x}{2}) cos (\frac{x}{2})

2 cos (\frac{3 x}{2}) cos (\frac{x}{2}) = 0

Resolver cada parte por separado

cos (\frac{3 x}{2}) = 0 or cos (\frac{x}{2}) = 0

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

cos (\frac{3 x}{2}) = 0

Soluciones generales para

cos (\frac{3 x}{2}) = 0

cos (x)

tabla de valores periódicos con

2 πn

intervalos:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

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

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

Resolver

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

\frac{3 x}{2} = \frac{π}{2} + 2 πn

Multiplicar ambos lados por

2

\frac{3 x}{2} = \frac{π}{2} + 2 πn

Multiplicar ambos lados por

2

\frac{2 \cdot 3 x}{2} = 2 \cdot \frac{π}{2} + 2 \cdot 2 πn

Simplificar

\frac{2 \cdot 3 x}{2} = 2 \cdot \frac{π}{2} + 2 \cdot 2 πn

Simplificar

\frac{2 \cdot 3 x}{2} : 3 x

\frac{2 \cdot 3 x}{2}

Multiplicar los numeros:

2 \cdot 3 = 6

= \frac{6 x}{2}

Dividir:

\frac{6}{2} = 3

= 3 x

Simplificar

2 \cdot \frac{π}{2} + 2 \cdot 2 πn : π + 4 πn

2 \cdot \frac{π}{2} + 2 \cdot 2 πn

2 \cdot \frac{π}{2} = π

2 \cdot \frac{π}{2}

Multiplicar fracciones:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{π 2}{2}

Eliminar los terminos comunes:

2

= π

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

Multiplicar los numeros:

2 \cdot 2 = 4

= 4 πn

= π + 4 πn

3 x = π + 4 πn

3 x = π + 4 πn

3 x = π + 4 πn

Dividir ambos lados entre

3

3 x = π + 4 πn

Dividir ambos lados entre

3

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

Simplificar

x = \frac{π}{3} + \frac{4 πn}{3}

x = \frac{π}{3} + \frac{4 πn}{3}

Resolver

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

\frac{3 x}{2} = \frac{3 π}{2} + 2 πn

Multiplicar ambos lados por

2

\frac{3 x}{2} = \frac{3 π}{2} + 2 πn

Multiplicar ambos lados por

2

\frac{2 \cdot 3 x}{2} = 2 \cdot \frac{3 π}{2} + 2 \cdot 2 πn

Simplificar

\frac{2 \cdot 3 x}{2} = 2 \cdot \frac{3 π}{2} + 2 \cdot 2 πn

Simplificar

\frac{2 \cdot 3 x}{2} : 3 x

\frac{2 \cdot 3 x}{2}

Multiplicar los numeros:

2 \cdot 3 = 6

= \frac{6 x}{2}

Dividir:

\frac{6}{2} = 3

= 3 x

Simplificar

2 \cdot \frac{3 π}{2} + 2 \cdot 2 πn : 3 π + 4 πn

2 \cdot \frac{3 π}{2} + 2 \cdot 2 πn

2 \cdot \frac{3 π}{2} = 3 π

2 \cdot \frac{3 π}{2}

Multiplicar fracciones:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{3 π 2}{2}

Eliminar los terminos comunes:

2

= 3 π

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

Multiplicar los numeros:

2 \cdot 2 = 4

= 4 πn

= 3 π + 4 πn

3 x = 3 π + 4 πn

3 x = 3 π + 4 πn

3 x = 3 π + 4 πn

Dividir ambos lados entre

3

3 x = 3 π + 4 πn

Dividir ambos lados entre

3

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

Simplificar

x = π + \frac{4 πn}{3}

x = π + \frac{4 πn}{3}

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

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

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

Soluciones generales para

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

cos (x)

tabla de valores periódicos con

2 πn

intervalos:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

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

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

Resolver

\frac{x}{2} = \frac{π}{2} + 2 πn : x = π + 4 πn

\frac{x}{2} = \frac{π}{2} + 2 πn

Multiplicar ambos lados por

2

\frac{x}{2} = \frac{π}{2} + 2 πn

Multiplicar ambos lados por

2

\frac{2 x}{2} = 2 \cdot \frac{π}{2} + 2 \cdot 2 πn

Simplificar

\frac{2 x}{2} = 2 \cdot \frac{π}{2} + 2 \cdot 2 πn

Simplificar

\frac{2 x}{2} : x

\frac{2 x}{2}

Dividir:

\frac{2}{2} = 1

= x

Simplificar

2 \cdot \frac{π}{2} + 2 \cdot 2 πn : π + 4 πn

2 \cdot \frac{π}{2} + 2 \cdot 2 πn

2 \cdot \frac{π}{2} = π

2 \cdot \frac{π}{2}

Multiplicar fracciones:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{π 2}{2}

Eliminar los terminos comunes:

2

= π

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

Multiplicar los numeros:

2 \cdot 2 = 4

= 4 πn

= π + 4 πn

x = π + 4 πn

x = π + 4 πn

x = π + 4 πn

Resolver

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

\frac{x}{2} = \frac{3 π}{2} + 2 πn

Multiplicar ambos lados por

2

\frac{x}{2} = \frac{3 π}{2} + 2 πn

Multiplicar ambos lados por

2

\frac{2 x}{2} = 2 \cdot \frac{3 π}{2} + 2 \cdot 2 πn

Simplificar

\frac{2 x}{2} = 2 \cdot \frac{3 π}{2} + 2 \cdot 2 πn

Simplificar

\frac{2 x}{2} : x

\frac{2 x}{2}

Dividir:

\frac{2}{2} = 1

= x

Simplificar

2 \cdot \frac{3 π}{2} + 2 \cdot 2 πn : 3 π + 4 πn

2 \cdot \frac{3 π}{2} + 2 \cdot 2 πn

2 \cdot \frac{3 π}{2} = 3 π

2 \cdot \frac{3 π}{2}

Multiplicar fracciones:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{3 π 2}{2}

Eliminar los terminos comunes:

2

= 3 π

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

Multiplicar los numeros:

2 \cdot 2 = 4

= 4 πn

= 3 π + 4 πn

x = 3 π + 4 πn

x = 3 π + 4 πn

x = 3 π + 4 πn

x = π + 4 πn, x = 3 π + 4 πn

Combinar toda las soluciones

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

Mezclar intervalos sobrepuestos

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

Combinar toda las soluciones

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

Siendo que la ecuación esta indefinida para:

2 πn, π + 2 πn

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

Ejemplos populares

sin(A)=0.5

sin (A) = 0.5

cos(θ)=tan(θ)

cos (θ) = tan (θ)

tan(pi/4+x)=(1-tan(x))/(1+tan(x))

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

cos(x)= 45/52

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

solvefor x,cos(2x)=(m-1)/(m+1)resolver para

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