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sin(x)-cos(x)= 1/3

Solución

sin (x) - cos (x) = \frac{1}{3}

Solución

x = 0.23794 \dots + 2 πn + \frac{π}{4}, x = π - 0.23794 \dots + 2 πn + \frac{π}{4}

Grados

x = 58.63302 \dots^{\circ} + 36 0^{\circ} n, x = 211.36697 \dots^{\circ} + 36 0^{\circ} n

Pasos de solución

sin (x) - cos (x) = \frac{1}{3}

Re-escribir usando identidades trigonométricas

sin (x) - cos (x)

sin (x) - cos (x) = 2 sin (x - \frac{π}{4})

sin (x) - cos (x)

Reescribir como

= 2 (\frac{1}{2} sin (x) - \frac{1}{2} cos (x))

Utilizar la siguiente identidad trivial:

cos (\frac{π}{4}) = \frac{1}{2}

Utilizar la siguiente identidad trivial:

sin (\frac{π}{4}) = \frac{1}{2}

= 2 (cos (\frac{π}{4}) sin (x) - sin (\frac{π}{4}) cos (x))

Utilizar la identidad de suma de ángulos:

sin (s - t) = sin (s) cos (t) - cos (s) sin (t)

= 2 sin (x - \frac{π}{4})

= 2 sin (x - \frac{π}{4})

2 sin (x - \frac{π}{4}) = \frac{1}{3}

Dividir ambos lados entre

2

2 sin (x - \frac{π}{4}) = \frac{1}{3}

Dividir ambos lados entre

2

\frac{2 sin ( x - \frac{π}{4} )}{2} = \frac{\frac{1}{3}}{2}

Simplificar

\frac{2 sin ( x - \frac{π}{4} )}{2} = \frac{\frac{1}{3}}{2}

Simplificar

\frac{2 sin ( x - \frac{π}{4} )}{2} : sin (x - \frac{π}{4})

\frac{2 sin ( x - \frac{π}{4} )}{2}

Eliminar los terminos comunes:

2

= sin (x - \frac{π}{4})

Simplificar

\frac{\frac{1}{3}}{2} : \frac{2}{6}

\frac{\frac{1}{3}}{2}

Aplicar las propiedades de las fracciones:

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

= \frac{1}{3 2}

Racionalizar

\frac{1}{3 2} : \frac{2}{6}

\frac{1}{3 2}

Multiplicar por el conjugado

\frac{2}{2}

= \frac{1 \cdot 2}{3 2 2}

1 \cdot 2 = 2

322 = 6

322

Aplicar las leyes de los exponentes:

a a = a

22 = 2

= 3 \cdot 2

Multiplicar los numeros:

3 \cdot 2 = 6

= 6

= \frac{2}{6}

= \frac{2}{6}

sin (x - \frac{π}{4}) = \frac{2}{6}

sin (x - \frac{π}{4}) = \frac{2}{6}

sin (x - \frac{π}{4}) = \frac{2}{6}

Aplicar propiedades trigonométricas inversas

sin (x - \frac{π}{4}) = \frac{2}{6}

Soluciones generales para

sin (x - \frac{π}{4}) = \frac{2}{6}

sin (x) = a \Rightarrow x = arcsin (a) + 2 πn, x = π - arcsin (a) + 2 πn

x - \frac{π}{4} = arcsin (\frac{2}{6}) + 2 πn, x - \frac{π}{4} = π - arcsin (\frac{2}{6}) + 2 πn

x - \frac{π}{4} = arcsin (\frac{2}{6}) + 2 πn, x - \frac{π}{4} = π - arcsin (\frac{2}{6}) + 2 πn

Resolver

x - \frac{π}{4} = arcsin (\frac{2}{6}) + 2 πn : x = arcsin (\frac{1}{3 2}) + 2 πn + \frac{π}{4}

x - \frac{π}{4} = arcsin (\frac{2}{6}) + 2 πn

Simplificar

arcsin (\frac{2}{6}) + 2 πn : arcsin (\frac{1}{3 2}) + 2 πn

arcsin (\frac{2}{6}) + 2 πn

\frac{2}{6} = \frac{1}{3 2}

\frac{2}{6}

Factorizar

6 : 2 \cdot 3

Factorizar

6 = 2 \cdot 3

= \frac{2}{2 \cdot 3}

Cancelar

\frac{2}{2 \cdot 3} : \frac{1}{2 \cdot 3}

\frac{2}{2 \cdot 3}

Aplicar las leyes de los exponentes:

2 = 2^{\frac{1}{2}}

= \frac{2 ^{\frac{1}{2}}}{2 \cdot 3}

Aplicar las leyes de los exponentes:

\frac{x ^{a}}{x ^{b}} = \frac{1}{x ^{b - a}}

\frac{2 ^{\frac{1}{2}}}{2 ^{1}} = \frac{1}{2 ^{1 - \frac{1}{2}}}

= \frac{1}{3 \cdot 2 ^{- \frac{1}{2} + 1}}

Restar:

1 - \frac{1}{2} = \frac{1}{2}

= \frac{1}{3 \cdot 2 ^{\frac{1}{2}}}

Aplicar las leyes de los exponentes:

2^{\frac{1}{2}} = 2

= \frac{1}{3 2}

= \frac{1}{2 \cdot 3}

= arcsin (\frac{1}{3 2}) + 2 πn

x - \frac{π}{4} = arcsin (\frac{1}{3 2}) + 2 πn

Mover

\frac{π}{4}

al lado derecho

x - \frac{π}{4} = arcsin (\frac{1}{3 2}) + 2 πn

Sumar

\frac{π}{4}

a ambos lados

x - \frac{π}{4} + \frac{π}{4} = arcsin (\frac{1}{3 2}) + 2 πn + \frac{π}{4}

Simplificar

x - \frac{π}{4} + \frac{π}{4} = arcsin (\frac{1}{3 2}) + 2 πn + \frac{π}{4}

Simplificar

x - \frac{π}{4} + \frac{π}{4} : x

x - \frac{π}{4} + \frac{π}{4}

Sumar elementos similares:

- \frac{π}{4} + \frac{π}{4} = 0

= x

Simplificar

arcsin (\frac{1}{3 2}) + 2 πn + \frac{π}{4} : arcsin (\frac{1}{3 2}) + 2 πn + \frac{π}{4}

arcsin (\frac{1}{3 2}) + 2 πn + \frac{π}{4}

= arcsin (\frac{2}{6}) + 2 πn + \frac{π}{4}

\frac{2}{6} = \frac{1}{3 2}

\frac{2}{6}

Factorizar

6 : 2 \cdot 3

Factorizar

6 = 2 \cdot 3

= \frac{2}{2 \cdot 3}

Cancelar

\frac{2}{2 \cdot 3} : \frac{1}{2 \cdot 3}

\frac{2}{2 \cdot 3}

Aplicar las leyes de los exponentes:

2 = 2^{\frac{1}{2}}

= \frac{2 ^{\frac{1}{2}}}{2 \cdot 3}

Aplicar las leyes de los exponentes:

\frac{x ^{a}}{x ^{b}} = \frac{1}{x ^{b - a}}

\frac{2 ^{\frac{1}{2}}}{2 ^{1}} = \frac{1}{2 ^{1 - \frac{1}{2}}}

= \frac{1}{3 \cdot 2 ^{- \frac{1}{2} + 1}}

Restar:

1 - \frac{1}{2} = \frac{1}{2}

= \frac{1}{3 \cdot 2 ^{\frac{1}{2}}}

Aplicar las leyes de los exponentes:

2^{\frac{1}{2}} = 2

= \frac{1}{3 2}

= \frac{1}{2 \cdot 3}

= arcsin (\frac{1}{3 2}) + 2 πn + \frac{π}{4}

No se puede simplificar mas

= arcsin (\frac{1}{3 2}) + 2 πn + \frac{π}{4}

x = arcsin (\frac{1}{3 2}) + 2 πn + \frac{π}{4}

x = arcsin (\frac{1}{3 2}) + 2 πn + \frac{π}{4}

x = arcsin (\frac{1}{3 2}) + 2 πn + \frac{π}{4}

Resolver

x - \frac{π}{4} = π - arcsin (\frac{2}{6}) + 2 πn : x = π - arcsin (\frac{1}{3 2}) + 2 πn + \frac{π}{4}

x - \frac{π}{4} = π - arcsin (\frac{2}{6}) + 2 πn

Simplificar

π - arcsin (\frac{2}{6}) + 2 πn : π - arcsin (\frac{1}{3 2}) + 2 πn

π - arcsin (\frac{2}{6}) + 2 πn

\frac{2}{6} = \frac{1}{3 2}

\frac{2}{6}

Factorizar

6 : 2 \cdot 3

Factorizar

6 = 2 \cdot 3

= \frac{2}{2 \cdot 3}

Cancelar

\frac{2}{2 \cdot 3} : \frac{1}{2 \cdot 3}

\frac{2}{2 \cdot 3}

Aplicar las leyes de los exponentes:

2 = 2^{\frac{1}{2}}

= \frac{2 ^{\frac{1}{2}}}{2 \cdot 3}

Aplicar las leyes de los exponentes:

\frac{x ^{a}}{x ^{b}} = \frac{1}{x ^{b - a}}

\frac{2 ^{\frac{1}{2}}}{2 ^{1}} = \frac{1}{2 ^{1 - \frac{1}{2}}}

= \frac{1}{3 \cdot 2 ^{- \frac{1}{2} + 1}}

Restar:

1 - \frac{1}{2} = \frac{1}{2}

= \frac{1}{3 \cdot 2 ^{\frac{1}{2}}}

Aplicar las leyes de los exponentes:

2^{\frac{1}{2}} = 2

= \frac{1}{3 2}

= \frac{1}{2 \cdot 3}

= π - arcsin (\frac{1}{3 2}) + 2 πn

x - \frac{π}{4} = π - arcsin (\frac{1}{3 2}) + 2 πn

Mover

\frac{π}{4}

al lado derecho

x - \frac{π}{4} = π - arcsin (\frac{1}{3 2}) + 2 πn

Sumar

\frac{π}{4}

a ambos lados

x - \frac{π}{4} + \frac{π}{4} = π - arcsin (\frac{1}{3 2}) + 2 πn + \frac{π}{4}

Simplificar

x - \frac{π}{4} + \frac{π}{4} = π - arcsin (\frac{1}{3 2}) + 2 πn + \frac{π}{4}

Simplificar

x - \frac{π}{4} + \frac{π}{4} : x

x - \frac{π}{4} + \frac{π}{4}

Sumar elementos similares:

- \frac{π}{4} + \frac{π}{4} = 0

= x

Simplificar

π - arcsin (\frac{1}{3 2}) + 2 πn + \frac{π}{4} : π - arcsin (\frac{1}{3 2}) + 2 πn + \frac{π}{4}

π - arcsin (\frac{1}{3 2}) + 2 πn + \frac{π}{4}

= π - arcsin (\frac{2}{6}) + 2 πn + \frac{π}{4}

\frac{2}{6} = \frac{1}{3 2}

\frac{2}{6}

Factorizar

6 : 2 \cdot 3

Factorizar

6 = 2 \cdot 3

= \frac{2}{2 \cdot 3}

Cancelar

\frac{2}{2 \cdot 3} : \frac{1}{2 \cdot 3}

\frac{2}{2 \cdot 3}

Aplicar las leyes de los exponentes:

2 = 2^{\frac{1}{2}}

= \frac{2 ^{\frac{1}{2}}}{2 \cdot 3}

Aplicar las leyes de los exponentes:

\frac{x ^{a}}{x ^{b}} = \frac{1}{x ^{b - a}}

\frac{2 ^{\frac{1}{2}}}{2 ^{1}} = \frac{1}{2 ^{1 - \frac{1}{2}}}

= \frac{1}{3 \cdot 2 ^{- \frac{1}{2} + 1}}

Restar:

1 - \frac{1}{2} = \frac{1}{2}

= \frac{1}{3 \cdot 2 ^{\frac{1}{2}}}

Aplicar las leyes de los exponentes:

2^{\frac{1}{2}} = 2

= \frac{1}{3 2}

= \frac{1}{2 \cdot 3}

= π - arcsin (\frac{1}{3 2}) + 2 πn + \frac{π}{4}

No se puede simplificar mas

= π - arcsin (\frac{1}{3 2}) + 2 πn + \frac{π}{4}

x = π - arcsin (\frac{1}{3 2}) + 2 πn + \frac{π}{4}

x = π - arcsin (\frac{1}{3 2}) + 2 πn + \frac{π}{4}

x = π - arcsin (\frac{1}{3 2}) + 2 πn + \frac{π}{4}

x = arcsin (\frac{1}{3 2}) + 2 πn + \frac{π}{4}, x = π - arcsin (\frac{1}{3 2}) + 2 πn + \frac{π}{4}

Mostrar soluciones en forma decimal

x = 0.23794 \dots + 2 πn + \frac{π}{4}, x = π - 0.23794 \dots + 2 πn + \frac{π}{4}

Ejemplos populares

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