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solvefor t,x=5(sin(t)+cos(t))

Solución

resolver para

t, x = 5 (sin (t) + cos (t))

Solución

t = arcsin (\frac{x}{5 2}) + 2 πn - \frac{π}{4}, t = π + arcsin (- \frac{x}{5 2}) + 2 πn - \frac{π}{4}

Pasos de solución

x = 5 (sin (t) + cos (t))

Intercambiar lados

5 (sin (t) + cos (t)) = x

Re-escribir usando identidades trigonométricas

5 (sin (t) + cos (t))

sin (t) + cos (t) = 2 sin (t + \frac{π}{4})

sin (t) + cos (t)

Reescribir como

= 2 (\frac{1}{2} sin (t) + \frac{1}{2} cos (t))

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 (t) + sin (\frac{π}{4}) cos (t))

Utilizar la identidad de suma de ángulos:

sin (s + t) = sin (s) cos (t) + cos (s) sin (t)

= 2 sin (t + \frac{π}{4})

= 52 sin (t + \frac{π}{4})

52 sin (t + \frac{π}{4}) = x

Dividir ambos lados entre

52

52 sin (t + \frac{π}{4}) = x

Dividir ambos lados entre

52

\frac{5 2 sin ( t + \frac{π}{4} )}{5 2} = \frac{x}{5 2}

Simplificar

\frac{5 2 sin ( t + \frac{π}{4} )}{5 2} = \frac{x}{5 2}

Simplificar

\frac{5 2 sin ( t + \frac{π}{4} )}{5 2} : sin (t + \frac{π}{4})

\frac{5 2 sin ( t + \frac{π}{4} )}{5 2}

Dividir:

\frac{5}{5} = 1

= \frac{2 sin ( t + \frac{π}{4} )}{2}

Eliminar los terminos comunes:

2

= sin (t + \frac{π}{4})

Simplificar

\frac{x}{5 2} : \frac{2 x}{10}

\frac{x}{5 2}

Multiplicar por el conjugado

\frac{2}{2}

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

522 = 10

522

Aplicar las leyes de los exponentes:

a a = a

22 = 2

= 5 \cdot 2

Multiplicar los numeros:

5 \cdot 2 = 10

= 10

= \frac{2 x}{10}

sin (t + \frac{π}{4}) = \frac{2 x}{10}

sin (t + \frac{π}{4}) = \frac{2 x}{10}

sin (t + \frac{π}{4}) = \frac{2 x}{10}

Aplicar propiedades trigonométricas inversas

sin (t + \frac{π}{4}) = \frac{2 x}{10}

Soluciones generales para

sin (t + \frac{π}{4}) = \frac{2 x}{10}

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

t + \frac{π}{4} = arcsin (\frac{2 x}{10}) + 2 πn, t + \frac{π}{4} = π + arcsin (- \frac{2 x}{10}) + 2 πn

t + \frac{π}{4} = arcsin (\frac{2 x}{10}) + 2 πn, t + \frac{π}{4} = π + arcsin (- \frac{2 x}{10}) + 2 πn

Resolver

t + \frac{π}{4} = arcsin (\frac{2 x}{10}) + 2 πn : t = arcsin (\frac{x}{5 2}) + 2 πn - \frac{π}{4}

t + \frac{π}{4} = arcsin (\frac{2 x}{10}) + 2 πn

Simplificar

arcsin (\frac{2 x}{10}) + 2 πn : arcsin (\frac{x}{5 2}) + 2 πn

arcsin (\frac{2 x}{10}) + 2 πn

\frac{2 x}{10} = \frac{x}{5 2}

\frac{2 x}{10}

Factorizar

10 : 2 \cdot 5

Factorizar

10 = 2 \cdot 5

= \frac{2 x}{2 \cdot 5}

Cancelar

\frac{2 x}{2 \cdot 5} : \frac{x}{2 \cdot 5}

\frac{2 x}{2 \cdot 5}

Aplicar las leyes de los exponentes:

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

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

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

Restar:

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

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

Aplicar las leyes de los exponentes:

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

= \frac{x}{5 2}

= \frac{x}{2 \cdot 5}

= arcsin (\frac{x}{5 2}) + 2 πn

t + \frac{π}{4} = arcsin (\frac{x}{5 2}) + 2 πn

Mover

\frac{π}{4}

al lado derecho

t + \frac{π}{4} = arcsin (\frac{x}{5 2}) + 2 πn

Restar

\frac{π}{4}

de ambos lados

t + \frac{π}{4} - \frac{π}{4} = arcsin (\frac{x}{5 2}) + 2 πn - \frac{π}{4}

Simplificar

t = arcsin (\frac{x}{5 2}) + 2 πn - \frac{π}{4}

t = arcsin (\frac{x}{5 2}) + 2 πn - \frac{π}{4}

Resolver

t + \frac{π}{4} = π + arcsin (- \frac{2 x}{10}) + 2 πn : t = π + arcsin (- \frac{x}{5 2}) + 2 πn - \frac{π}{4}

t + \frac{π}{4} = π + arcsin (- \frac{2 x}{10}) + 2 πn

Simplificar

π + arcsin (- \frac{2 x}{10}) + 2 πn : π + arcsin (- \frac{x}{5 2}) + 2 πn

π + arcsin (- \frac{2 x}{10}) + 2 πn

\frac{2 x}{10} = \frac{x}{5 2}

\frac{2 x}{10}

Factorizar

10 : 2 \cdot 5

Factorizar

10 = 2 \cdot 5

= \frac{2 x}{2 \cdot 5}

Cancelar

\frac{2 x}{2 \cdot 5} : \frac{x}{2 \cdot 5}

\frac{2 x}{2 \cdot 5}

Aplicar las leyes de los exponentes:

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

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

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

Restar:

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

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

Aplicar las leyes de los exponentes:

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

= \frac{x}{5 2}

= \frac{x}{2 \cdot 5}

= π + arcsin (- \frac{x}{5 2}) + 2 πn

t + \frac{π}{4} = π + arcsin (- \frac{x}{5 2}) + 2 πn

Mover

\frac{π}{4}

al lado derecho

t + \frac{π}{4} = π + arcsin (- \frac{x}{5 2}) + 2 πn

Restar

\frac{π}{4}

de ambos lados

t + \frac{π}{4} - \frac{π}{4} = π + arcsin (- \frac{x}{5 2}) + 2 πn - \frac{π}{4}

Simplificar

t = π + arcsin (- \frac{x}{5 2}) + 2 πn - \frac{π}{4}

t = π + arcsin (- \frac{x}{5 2}) + 2 πn - \frac{π}{4}

t = arcsin (\frac{x}{5 2}) + 2 πn - \frac{π}{4}, t = π + arcsin (- \frac{x}{5 2}) + 2 πn - \frac{π}{4}

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