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Popular >

sin(2x)+3sin(4x)=0

Solución

sin (2 x) + 3 sin (4 x) = 0

Solución

x = πn, x = \frac{π + 2 πn}{2}, x = \frac{1.73824 \dots + 2 πn}{2}, x = \frac{- 1.73824 \dots + 2 πn}{2}

Grados

x = 0^{\circ} + 18 0^{\circ} n, x = 9 0^{\circ} + 18 0^{\circ} n, x = 49.79703 \dots^{\circ} + 18 0^{\circ} n, x = - 49.79703 \dots^{\circ} + 18 0^{\circ} n

Pasos de solución

sin (2 x) + 3 sin (4 x) = 0

Sea:

u = 2 x

sin (u) + 3 sin (2 u) = 0

Re-escribir usando identidades trigonométricas

sin (u) + 3 sin (2 u)

Utilizar la identidad trigonométrica del ángulo doble:

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

= sin (u) + 3 \cdot 2 sin (u) cos (u)

Simplificar

= sin (u) + 6 sin (u) cos (u)

sin (u) + 6 cos (u) sin (u) = 0

Factorizar

sin (u) + 6 cos (u) sin (u) : sin (u) (6 cos (u) + 1)

sin (u) + 6 cos (u) sin (u)

Factorizar el termino común

sin (u)

= sin (u) (1 + 6 cos (u))

sin (u) (6 cos (u) + 1) = 0

Resolver cada parte por separado

sin (u) = 0 or 6 cos (u) + 1 = 0

sin (u) = 0 : u = 2 πn, u = π + 2 πn

sin (u) = 0

Soluciones generales para

sin (u) = 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}

u = 0 + 2 πn, u = π + 2 πn

u = 0 + 2 πn, u = π + 2 πn

Resolver

u = 0 + 2 πn : u = 2 πn

u = 0 + 2 πn

0 + 2 πn = 2 πn

u = 2 πn

u = 2 πn, u = π + 2 πn

6 cos (u) + 1 = 0 : u = arccos (- \frac{1}{6}) + 2 πn, u = - arccos (- \frac{1}{6}) + 2 πn

6 cos (u) + 1 = 0

Mover

1

al lado derecho

6 cos (u) + 1 = 0

Restar

1

de ambos lados

6 cos (u) + 1 - 1 = 0 - 1

Simplificar

6 cos (u) = - 1

6 cos (u) = - 1

Dividir ambos lados entre

6

6 cos (u) = - 1

Dividir ambos lados entre

6

\frac{6 cos ( u )}{6} = \frac{- 1}{6}

Simplificar

cos (u) = - \frac{1}{6}

cos (u) = - \frac{1}{6}

Aplicar propiedades trigonométricas inversas

cos (u) = - \frac{1}{6}

Soluciones generales para

cos (u) = - \frac{1}{6}

cos (x) = - a \Rightarrow x = arccos (- a) + 2 πn, x = - arccos (- a) + 2 πn

u = arccos (- \frac{1}{6}) + 2 πn, u = - arccos (- \frac{1}{6}) + 2 πn

u = arccos (- \frac{1}{6}) + 2 πn, u = - arccos (- \frac{1}{6}) + 2 πn

Combinar toda las soluciones

u = 2 πn, u = π + 2 πn, u = arccos (- \frac{1}{6}) + 2 πn, u = - arccos (- \frac{1}{6}) + 2 πn

Sustituir en la ecuación

u = 2 x

2 x = 2 πn : x = πn

2 x = 2 πn

Dividir ambos lados entre

2

2 x = 2 πn

Dividir ambos lados entre

2

\frac{2 x}{2} = \frac{2 πn}{2}

Simplificar

x = πn

x = πn

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

2 x = π + 2 πn

Dividir ambos lados entre

2

2 x = π + 2 πn

Dividir ambos lados entre

2

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

Simplificar

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

Simplificar

\frac{2 x}{2} : x

\frac{2 x}{2}

Dividir:

\frac{2}{2} = 1

= x

Simplificar

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

\frac{π}{2} + \frac{2 πn}{2}

Aplicar la regla

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

= \frac{π + 2 πn}{2}

x = \frac{π + 2 πn}{2}

x = \frac{π + 2 πn}{2}

x = \frac{π + 2 πn}{2}

2 x = arccos (- \frac{1}{6}) + 2 πn : x = \frac{arccos ( - \frac{1}{6} ) + 2 πn}{2}

2 x = arccos (- \frac{1}{6}) + 2 πn

Dividir ambos lados entre

2

2 x = arccos (- \frac{1}{6}) + 2 πn

Dividir ambos lados entre

2

\frac{2 x}{2} = \frac{arccos ( - \frac{1}{6} )}{2} + \frac{2 πn}{2}

Simplificar

\frac{2 x}{2} = \frac{arccos ( - \frac{1}{6} )}{2} + \frac{2 πn}{2}

Simplificar

\frac{2 x}{2} : x

\frac{2 x}{2}

Dividir:

\frac{2}{2} = 1

= x

Simplificar

\frac{arccos ( - \frac{1}{6} )}{2} + \frac{2 πn}{2} : \frac{arccos ( - \frac{1}{6} ) + 2 πn}{2}

\frac{arccos ( - \frac{1}{6} )}{2} + \frac{2 πn}{2}

Aplicar la regla

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

= \frac{arccos ( - \frac{1}{6} ) + 2 πn}{2}

x = \frac{arccos ( - \frac{1}{6} ) + 2 πn}{2}

x = \frac{arccos ( - \frac{1}{6} ) + 2 πn}{2}

x = \frac{arccos ( - \frac{1}{6} ) + 2 πn}{2}

2 x = - arccos (- \frac{1}{6}) + 2 πn : x = \frac{- arccos ( - \frac{1}{6} ) + 2 πn}{2}

2 x = - arccos (- \frac{1}{6}) + 2 πn

Dividir ambos lados entre

2

2 x = - arccos (- \frac{1}{6}) + 2 πn

Dividir ambos lados entre

2

\frac{2 x}{2} = - \frac{arccos ( - \frac{1}{6} )}{2} + \frac{2 πn}{2}

Simplificar

\frac{2 x}{2} = - \frac{arccos ( - \frac{1}{6} )}{2} + \frac{2 πn}{2}

Simplificar

\frac{2 x}{2} : x

\frac{2 x}{2}

Dividir:

\frac{2}{2} = 1

= x

Simplificar

- \frac{arccos ( - \frac{1}{6} )}{2} + \frac{2 πn}{2} : \frac{- arccos ( - \frac{1}{6} ) + 2 πn}{2}

- \frac{arccos ( - \frac{1}{6} )}{2} + \frac{2 πn}{2}

Aplicar la regla

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

= \frac{- arccos ( - \frac{1}{6} ) + 2 πn}{2}

x = \frac{- arccos ( - \frac{1}{6} ) + 2 πn}{2}

x = \frac{- arccos ( - \frac{1}{6} ) + 2 πn}{2}

x = \frac{- arccos ( - \frac{1}{6} ) + 2 πn}{2}

x = πn, x = \frac{π + 2 πn}{2}, x = \frac{arccos ( - \frac{1}{6} ) + 2 πn}{2}, x = \frac{- arccos ( - \frac{1}{6} ) + 2 πn}{2}

Mostrar soluciones en forma decimal

x = πn, x = \frac{π + 2 πn}{2}, x = \frac{1.73824 \dots + 2 πn}{2}, x = \frac{- 1.73824 \dots + 2 πn}{2}

Ejemplos populares

-2sin^2(3x)=cos(3x)-2

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

2sin^2(θ)=cos(θ)+1

2 sin^{2} (θ) = cos (θ) + 1

arccsc(2x)+arcsin(0)=arctan(1/(sqrt(3)))

arccsc (2 x) + arcsin (0) = arctan (\frac{1}{3})

sin(t)+cos(2t)=0.75

sin (t) + cos (2 t) = 0.75

2cos(x)=cot(x)

2 cos (x) = cot (x)