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sin(2x)=sin(0.5x)

Solución

sin (2 x) = sin (0.5 x)

Solución

x = \frac{2 π}{5} + \frac{8 πn}{5}, x = \frac{6 π}{5} + \frac{8 πn}{5}, x = \frac{8 πn}{3}, x = \frac{4 π}{3} + \frac{8 πn}{3}

Grados

x = 7 2^{\circ} + 28 8^{\circ} n, x = 21 6^{\circ} + 28 8^{\circ} n, x = 0^{\circ} + 48 0^{\circ} n, x = 24 0^{\circ} + 48 0^{\circ} n

Pasos de solución

sin (2 x) = sin (0.5 x)

Restar

sin (0.5 x)

de ambos lados

sin (2 x) - sin (0.5 x) = 0

Re-escribir usando identidades trigonométricas

- sin (0.5 x) + sin (2 x)

Utilizar la identidad suma-producto:

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

= 2 sin (\frac{2 x - 0.5 x}{2}) cos (\frac{2 x + 0.5 x}{2})

Simplificar

2 sin (\frac{2 x - 0.5 x}{2}) cos (\frac{2 x + 0.5 x}{2}) : 2 sin (0.75 x) cos (1.25 x)

2 sin (\frac{2 x - 0.5 x}{2}) cos (\frac{2 x + 0.5 x}{2})

Sumar elementos similares:

2 x - 0.5 x = 1.5 x

= 2 sin (\frac{1.5 x}{2}) cos (\frac{2 x + 0.5 x}{2})

Sumar elementos similares:

2 x + 0.5 x = 2.5 x

= 2 sin (\frac{1.5 x}{2}) cos (\frac{2.5 x}{2})

Dividir:

\frac{1.5}{2} = 0.75

= 2 sin (0.75 x) cos (\frac{2.5 x}{2})

Dividir:

\frac{2.5}{2} = 1.25

= 2 sin (0.75 x) cos (1.25 x)

= 2 sin (0.75 x) cos (1.25 x)

2 cos (1.25 x) sin (0.75 x) = 0

Resolver cada parte por separado

cos (1.25 x) = 0 or sin (0.75 x) = 0

cos (1.25 x) = 0 : x = \frac{2 π}{5} + \frac{8 πn}{5}, x = \frac{6 π}{5} + \frac{8 πn}{5}

cos (1.25 x) = 0

Soluciones generales para

cos (1.25 x) = 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}

1.25 x = \frac{π}{2} + 2 πn, 1.25 x = \frac{3 π}{2} + 2 πn

1.25 x = \frac{π}{2} + 2 πn, 1.25 x = \frac{3 π}{2} + 2 πn

Resolver

1.25 x = \frac{π}{2} + 2 πn : x = \frac{2 π}{5} + \frac{8 πn}{5}

1.25 x = \frac{π}{2} + 2 πn

Multiplicar ambos lados por

100

1.25 x = \frac{π}{2} + 2 πn

To eliminate decimal points, multiply by 10 for every digit after the decimal pointThere are

2

digits to the right of the decimal point, therefore multiply by

100

1.25 x \cdot 100 = \frac{π}{2} \cdot 100 + 2 πn \cdot 100

Simplificar

125 x = 50 π + 200 πn

125 x = 50 π + 200 πn

Dividir ambos lados entre

125

125 x = 50 π + 200 πn

Dividir ambos lados entre

125

\frac{125 x}{125} = \frac{50 π}{125} + \frac{200 πn}{125}

Simplificar

\frac{125 x}{125} = \frac{50 π}{125} + \frac{200 πn}{125}

Simplificar

\frac{125 x}{125} : x

\frac{125 x}{125}

Dividir:

\frac{125}{125} = 1

= x

Simplificar

\frac{50 π}{125} + \frac{200 πn}{125} : \frac{2 π}{5} + \frac{8 πn}{5}

\frac{50 π}{125} + \frac{200 πn}{125}

Cancelar

\frac{50 π}{125} : \frac{2 π}{5}

\frac{50 π}{125}

Eliminar los terminos comunes:

25

= \frac{2 π}{5}

= \frac{2 π}{5} + \frac{200 πn}{125}

Cancelar

\frac{200 πn}{125} : \frac{8 πn}{5}

\frac{200 πn}{125}

Eliminar los terminos comunes:

25

= \frac{8 πn}{5}

= \frac{2 π}{5} + \frac{8 πn}{5}

x = \frac{2 π}{5} + \frac{8 πn}{5}

x = \frac{2 π}{5} + \frac{8 πn}{5}

x = \frac{2 π}{5} + \frac{8 πn}{5}

Resolver

1.25 x = \frac{3 π}{2} + 2 πn : x = \frac{6 π}{5} + \frac{8 πn}{5}

1.25 x = \frac{3 π}{2} + 2 πn

Multiplicar ambos lados por

100

1.25 x = \frac{3 π}{2} + 2 πn

To eliminate decimal points, multiply by 10 for every digit after the decimal pointThere are

2

digits to the right of the decimal point, therefore multiply by

100

1.25 x \cdot 100 = \frac{3 π}{2} \cdot 100 + 2 πn \cdot 100

Simplificar

125 x = 150 π + 200 πn

125 x = 150 π + 200 πn

Dividir ambos lados entre

125

125 x = 150 π + 200 πn

Dividir ambos lados entre

125

\frac{125 x}{125} = \frac{150 π}{125} + \frac{200 πn}{125}

Simplificar

\frac{125 x}{125} = \frac{150 π}{125} + \frac{200 πn}{125}

Simplificar

\frac{125 x}{125} : x

\frac{125 x}{125}

Dividir:

\frac{125}{125} = 1

= x

Simplificar

\frac{150 π}{125} + \frac{200 πn}{125} : \frac{6 π}{5} + \frac{8 πn}{5}

\frac{150 π}{125} + \frac{200 πn}{125}

Cancelar

\frac{150 π}{125} : \frac{6 π}{5}

\frac{150 π}{125}

Eliminar los terminos comunes:

25

= \frac{6 π}{5}

= \frac{6 π}{5} + \frac{200 πn}{125}

Cancelar

\frac{200 πn}{125} : \frac{8 πn}{5}

\frac{200 πn}{125}

Eliminar los terminos comunes:

25

= \frac{8 πn}{5}

= \frac{6 π}{5} + \frac{8 πn}{5}

x = \frac{6 π}{5} + \frac{8 πn}{5}

x = \frac{6 π}{5} + \frac{8 πn}{5}

x = \frac{6 π}{5} + \frac{8 πn}{5}

x = \frac{2 π}{5} + \frac{8 πn}{5}, x = \frac{6 π}{5} + \frac{8 πn}{5}

sin (0.75 x) = 0 : x = \frac{8 πn}{3}, x = \frac{4 π}{3} + \frac{8 πn}{3}

sin (0.75 x) = 0

Soluciones generales para

sin (0.75 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}

0.75 x = 0 + 2 πn, 0.75 x = π + 2 πn

0.75 x = 0 + 2 πn, 0.75 x = π + 2 πn

Resolver

0.75 x = 0 + 2 πn : x = \frac{8 πn}{3}

0.75 x = 0 + 2 πn

0 + 2 πn = 2 πn

0.75 x = 2 πn

Multiplicar ambos lados por

100

0.75 x = 2 πn

To eliminate decimal points, multiply by 10 for every digit after the decimal pointThere are

2

digits to the right of the decimal point, therefore multiply by

100

0.75 x \cdot 100 = 2 πn \cdot 100

Simplificar

75 x = 200 πn

75 x = 200 πn

Dividir ambos lados entre

75

75 x = 200 πn

Dividir ambos lados entre

75

\frac{75 x}{75} = \frac{200 πn}{75}

Simplificar

x = \frac{8 πn}{3}

x = \frac{8 πn}{3}

Resolver

0.75 x = π + 2 πn : x = \frac{4 π}{3} + \frac{8 πn}{3}

0.75 x = π + 2 πn

Multiplicar ambos lados por

100

0.75 x = π + 2 πn

To eliminate decimal points, multiply by 10 for every digit after the decimal pointThere are

2

digits to the right of the decimal point, therefore multiply by

100

0.75 x \cdot 100 = π 100 + 2 πn \cdot 100

Simplificar

75 x = 100 π + 200 πn

75 x = 100 π + 200 πn

Dividir ambos lados entre

75

75 x = 100 π + 200 πn

Dividir ambos lados entre

75

\frac{75 x}{75} = \frac{100 π}{75} + \frac{200 πn}{75}

Simplificar

\frac{75 x}{75} = \frac{100 π}{75} + \frac{200 πn}{75}

Simplificar

\frac{75 x}{75} : x

\frac{75 x}{75}

Dividir:

\frac{75}{75} = 1

= x

Simplificar

\frac{100 π}{75} + \frac{200 πn}{75} : \frac{4 π}{3} + \frac{8 πn}{3}

\frac{100 π}{75} + \frac{200 πn}{75}

Cancelar

\frac{100 π}{75} : \frac{4 π}{3}

\frac{100 π}{75}

Eliminar los terminos comunes:

25

= \frac{4 π}{3}

= \frac{4 π}{3} + \frac{200 πn}{75}

Cancelar

\frac{200 πn}{75} : \frac{8 πn}{3}

\frac{200 πn}{75}

Eliminar los terminos comunes:

25

= \frac{8 πn}{3}

= \frac{4 π}{3} + \frac{8 πn}{3}

x = \frac{4 π}{3} + \frac{8 πn}{3}

x = \frac{4 π}{3} + \frac{8 πn}{3}

x = \frac{4 π}{3} + \frac{8 πn}{3}

x = \frac{8 πn}{3}, x = \frac{4 π}{3} + \frac{8 πn}{3}

Combinar toda las soluciones

x = \frac{2 π}{5} + \frac{8 πn}{5}, x = \frac{6 π}{5} + \frac{8 πn}{5}, x = \frac{8 πn}{3}, x = \frac{4 π}{3} + \frac{8 πn}{3}

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