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sin(5x)-sin(3x)-sin(x)=0

Solución

sin (5 x) - sin (3 x) - sin (x) = 0

Solución

x = 2 πn, x = π + 2 πn, x = \frac{π}{12} + \frac{πn}{2}, x = \frac{5 π}{12} + \frac{πn}{2}

Grados

x = 0^{\circ} + 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n, x = 1 5^{\circ} + 9 0^{\circ} n, x = 7 5^{\circ} + 9 0^{\circ} n

Pasos de solución

sin (5 x) - sin (3 x) - sin (x) = 0

Re-escribir usando identidades trigonométricas

- sin (3 x) + sin (5 x) - sin (x)

Utilizar la identidad suma-producto:

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

= - sin (x) + 2 sin (\frac{5 x - 3 x}{2}) cos (\frac{5 x + 3 x}{2})

2 sin (\frac{5 x - 3 x}{2}) cos (\frac{5 x + 3 x}{2}) = 2 sin (x) cos (4 x)

2 sin (\frac{5 x - 3 x}{2}) cos (\frac{5 x + 3 x}{2})

\frac{5 x - 3 x}{2} = x

\frac{5 x - 3 x}{2}

Sumar elementos similares:

5 x - 3 x = 2 x

= \frac{2 x}{2}

Dividir:

\frac{2}{2} = 1

= x

= 2 sin (x) cos (\frac{5 x + 3 x}{2})

\frac{5 x + 3 x}{2} = 4 x

\frac{5 x + 3 x}{2}

Sumar elementos similares:

5 x + 3 x = 8 x

= \frac{8 x}{2}

Dividir:

\frac{8}{2} = 4

= 4 x

= 2 sin (x) cos (4 x)

= - sin (x) + 2 sin (x) cos (4 x)

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

Factorizar

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

- sin (x) + 2 cos (4 x) sin (x)

Factorizar el termino común

sin (x)

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

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

Resolver cada parte por separado

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

2 cos (4 x) - 1 = 0 : x = \frac{π}{12} + \frac{πn}{2}, x = \frac{5 π}{12} + \frac{πn}{2}

2 cos (4 x) - 1 = 0

Mover

1

al lado derecho

2 cos (4 x) - 1 = 0

Sumar

1

a ambos lados

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

Simplificar

2 cos (4 x) = 1

2 cos (4 x) = 1

Dividir ambos lados entre

2

2 cos (4 x) = 1

Dividir ambos lados entre

2

\frac{2 cos ( 4 x )}{2} = \frac{1}{2}

Simplificar

cos (4 x) = \frac{1}{2}

cos (4 x) = \frac{1}{2}

Soluciones generales para

cos (4 x) = \frac{1}{2}

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}

4 x = \frac{π}{3} + 2 πn, 4 x = \frac{5 π}{3} + 2 πn

4 x = \frac{π}{3} + 2 πn, 4 x = \frac{5 π}{3} + 2 πn

Resolver

4 x = \frac{π}{3} + 2 πn : x = \frac{π}{12} + \frac{πn}{2}

4 x = \frac{π}{3} + 2 πn

Dividir ambos lados entre

4

4 x = \frac{π}{3} + 2 πn

Dividir ambos lados entre

4

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

Simplificar

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

Simplificar

\frac{4 x}{4} : x

\frac{4 x}{4}

Dividir:

\frac{4}{4} = 1

= x

Simplificar

\frac{\frac{π}{3}}{4} + \frac{2 πn}{4} : \frac{π}{12} + \frac{πn}{2}

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

\frac{\frac{π}{3}}{4} = \frac{π}{12}

\frac{\frac{π}{3}}{4}

Aplicar las propiedades de las fracciones:

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

= \frac{π}{3 \cdot 4}

Multiplicar los numeros:

3 \cdot 4 = 12

= \frac{π}{12}

\frac{2 πn}{4} = \frac{πn}{2}

\frac{2 πn}{4}

Eliminar los terminos comunes:

2

= \frac{πn}{2}

= \frac{π}{12} + \frac{πn}{2}

x = \frac{π}{12} + \frac{πn}{2}

x = \frac{π}{12} + \frac{πn}{2}

x = \frac{π}{12} + \frac{πn}{2}

Resolver

4 x = \frac{5 π}{3} + 2 πn : x = \frac{5 π}{12} + \frac{πn}{2}

4 x = \frac{5 π}{3} + 2 πn

Dividir ambos lados entre

4

4 x = \frac{5 π}{3} + 2 πn

Dividir ambos lados entre

4

\frac{4 x}{4} = \frac{\frac{5 π}{3}}{4} + \frac{2 πn}{4}

Simplificar

\frac{4 x}{4} = \frac{\frac{5 π}{3}}{4} + \frac{2 πn}{4}

Simplificar

\frac{4 x}{4} : x

\frac{4 x}{4}

Dividir:

\frac{4}{4} = 1

= x

Simplificar

\frac{\frac{5 π}{3}}{4} + \frac{2 πn}{4} : \frac{5 π}{12} + \frac{πn}{2}

\frac{\frac{5 π}{3}}{4} + \frac{2 πn}{4}

\frac{\frac{5 π}{3}}{4} = \frac{5 π}{12}

\frac{\frac{5 π}{3}}{4}

Aplicar las propiedades de las fracciones:

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

= \frac{5 π}{3 \cdot 4}

Multiplicar los numeros:

3 \cdot 4 = 12

= \frac{5 π}{12}

\frac{2 πn}{4} = \frac{πn}{2}

\frac{2 πn}{4}

Eliminar los terminos comunes:

2

= \frac{πn}{2}

= \frac{5 π}{12} + \frac{πn}{2}

x = \frac{5 π}{12} + \frac{πn}{2}

x = \frac{5 π}{12} + \frac{πn}{2}

x = \frac{5 π}{12} + \frac{πn}{2}

x = \frac{π}{12} + \frac{πn}{2}, x = \frac{5 π}{12} + \frac{πn}{2}

Combinar toda las soluciones

x = 2 πn, x = π + 2 πn, x = \frac{π}{12} + \frac{πn}{2}, x = \frac{5 π}{12} + \frac{πn}{2}

Ejemplos populares

4arccos(2x)=pi

4 arccos (2 x) = π

tan(x)=sqrt(2)-1

tan (x) = 2 - 1

5+3cos(1/2 x)=7

5 + 3 cos (\frac{1}{2} x) = 7

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

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

sin(2x-0.35)=cos(3x)

sin (2 x - 0.35) = cos (3 x)