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solvefor a,E=25(sin(a)-cos(a))

Solución

resolver para

a, E = 25 (sin (a) - cos (a))

Solución

a = arcsin (\frac{E}{25 2}) + 2 πn + \frac{π}{4}, a = π + arcsin (- \frac{E}{25 2}) + 2 πn + \frac{π}{4}

Pasos de solución

E = 25 (sin (a) - cos (a))

Intercambiar lados

25 (sin (a) - cos (a)) = E

Re-escribir usando identidades trigonométricas

25 (sin (a) - cos (a))

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

sin (a) - cos (a)

Reescribir como

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

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 (a) - sin (\frac{π}{4}) cos (a))

Utilizar la identidad de suma de ángulos:

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

= 2 sin (a - \frac{π}{4})

= 252 sin (a - \frac{π}{4})

252 sin (a - \frac{π}{4}) = E

Dividir ambos lados entre

252

252 sin (a - \frac{π}{4}) = E

Dividir ambos lados entre

252

\frac{25 2 sin ( a - \frac{π}{4} )}{25 2} = \frac{E}{25 2}

Simplificar

\frac{25 2 sin ( a - \frac{π}{4} )}{25 2} = \frac{E}{25 2}

Simplificar

\frac{25 2 sin ( a - \frac{π}{4} )}{25 2} : sin (a - \frac{π}{4})

\frac{25 2 sin ( a - \frac{π}{4} )}{25 2}

Dividir:

\frac{25}{25} = 1

= \frac{2 sin ( a - \frac{π}{4} )}{2}

Eliminar los terminos comunes:

2

= sin (a - \frac{π}{4})

Simplificar

\frac{E}{25 2} : \frac{2 E}{50}

\frac{E}{25 2}

Multiplicar por el conjugado

\frac{2}{2}

= \frac{E 2}{25 2 2}

2522 = 50

2522

Aplicar las leyes de los exponentes:

a a = a

22 = 2

= 25 \cdot 2

Multiplicar los numeros:

25 \cdot 2 = 50

= 50

= \frac{2 E}{50}

sin (a - \frac{π}{4}) = \frac{2 E}{50}

sin (a - \frac{π}{4}) = \frac{2 E}{50}

sin (a - \frac{π}{4}) = \frac{2 E}{50}

Aplicar propiedades trigonométricas inversas

sin (a - \frac{π}{4}) = \frac{2 E}{50}

Soluciones generales para

sin (a - \frac{π}{4}) = \frac{2 E}{50}

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

a - \frac{π}{4} = arcsin (\frac{2 E}{50}) + 2 πn, a - \frac{π}{4} = π + arcsin (- \frac{2 E}{50}) + 2 πn

a - \frac{π}{4} = arcsin (\frac{2 E}{50}) + 2 πn, a - \frac{π}{4} = π + arcsin (- \frac{2 E}{50}) + 2 πn

Resolver

a - \frac{π}{4} = arcsin (\frac{2 E}{50}) + 2 πn : a = arcsin (\frac{E}{25 2}) + 2 πn + \frac{π}{4}

a - \frac{π}{4} = arcsin (\frac{2 E}{50}) + 2 πn

Simplificar

arcsin (\frac{2 E}{50}) + 2 πn : arcsin (\frac{E}{25 2}) + 2 πn

arcsin (\frac{2 E}{50}) + 2 πn

\frac{2 E}{50} = \frac{E}{5 ^{2} 2}

\frac{2 E}{50}

Factorizar

50 : 5^{2} \cdot 2

Factorizar

50 = 5^{2} \cdot 2

= \frac{2 E}{5 ^{2} \cdot 2}

Cancelar

\frac{2 E}{2 \cdot 5 ^{2}} : \frac{E}{2 \cdot 5 ^{2}}

\frac{2 E}{2 \cdot 5 ^{2}}

Aplicar las leyes de los exponentes:

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

= \frac{2 ^{\frac{1}{2}} E}{5 ^{2} \cdot 2}

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{E}{5 ^{2} \cdot 2 ^{- \frac{1}{2} + 1}}

Restar:

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

= \frac{E}{5 ^{2} \cdot 2 ^{\frac{1}{2}}}

Aplicar las leyes de los exponentes:

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

= \frac{E}{5 ^{2} 2}

= \frac{E}{2 \cdot 5 ^{2}}

= arcsin (\frac{E}{5 ^{2} 2}) + 2 πn

5^{2} = 25

= arcsin (\frac{E}{25 2}) + 2 πn

a - \frac{π}{4} = arcsin (\frac{E}{25 2}) + 2 πn

Mover

\frac{π}{4}

al lado derecho

a - \frac{π}{4} = arcsin (\frac{E}{25 2}) + 2 πn

Sumar

\frac{π}{4}

a ambos lados

a - \frac{π}{4} + \frac{π}{4} = arcsin (\frac{E}{25 2}) + 2 πn + \frac{π}{4}

Simplificar

a = arcsin (\frac{E}{25 2}) + 2 πn + \frac{π}{4}

a = arcsin (\frac{E}{25 2}) + 2 πn + \frac{π}{4}

Resolver

a - \frac{π}{4} = π + arcsin (- \frac{2 E}{50}) + 2 πn : a = π + arcsin (- \frac{E}{25 2}) + 2 πn + \frac{π}{4}

a - \frac{π}{4} = π + arcsin (- \frac{2 E}{50}) + 2 πn

Simplificar

π + arcsin (- \frac{2 E}{50}) + 2 πn : π + arcsin (- \frac{E}{25 2}) + 2 πn

π + arcsin (- \frac{2 E}{50}) + 2 πn

\frac{2 E}{50} = \frac{E}{5 ^{2} 2}

\frac{2 E}{50}

Factorizar

50 : 5^{2} \cdot 2

Factorizar

50 = 5^{2} \cdot 2

= \frac{2 E}{5 ^{2} \cdot 2}

Cancelar

\frac{2 E}{2 \cdot 5 ^{2}} : \frac{E}{2 \cdot 5 ^{2}}

\frac{2 E}{2 \cdot 5 ^{2}}

Aplicar las leyes de los exponentes:

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

= \frac{2 ^{\frac{1}{2}} E}{5 ^{2} \cdot 2}

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{E}{5 ^{2} \cdot 2 ^{- \frac{1}{2} + 1}}

Restar:

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

= \frac{E}{5 ^{2} \cdot 2 ^{\frac{1}{2}}}

Aplicar las leyes de los exponentes:

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

= \frac{E}{5 ^{2} 2}

= \frac{E}{2 \cdot 5 ^{2}}

= π + arcsin (- \frac{E}{5 ^{2} 2}) + 2 πn

5^{2} = 25

= π + arcsin (- \frac{E}{25 2}) + 2 πn

a - \frac{π}{4} = π + arcsin (- \frac{E}{25 2}) + 2 πn

Mover

\frac{π}{4}

al lado derecho

a - \frac{π}{4} = π + arcsin (- \frac{E}{25 2}) + 2 πn

Sumar

\frac{π}{4}

a ambos lados

a - \frac{π}{4} + \frac{π}{4} = π + arcsin (- \frac{E}{25 2}) + 2 πn + \frac{π}{4}

Simplificar

a = π + arcsin (- \frac{E}{25 2}) + 2 πn + \frac{π}{4}

a = π + arcsin (- \frac{E}{25 2}) + 2 πn + \frac{π}{4}

a = arcsin (\frac{E}{25 2}) + 2 πn + \frac{π}{4}, a = π + arcsin (- \frac{E}{25 2}) + 2 πn + \frac{π}{4}

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