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r^2=a^2(sin(x)+cos(x))

Solución

r^{2} = a^{2} (sin (x) + cos (x))

Solución

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

Pasos de solución

r^{2} = a^{2} (sin (x) + cos (x))

Intercambiar lados

a^{2} (sin (x) + cos (x)) = r^{2}

Re-escribir usando identidades trigonométricas

a^{2} (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})

= a^{2} 2 sin (x + \frac{π}{4})

a^{2} 2 sin (x + \frac{π}{4}) = r^{2}

Dividir ambos lados entre

a^{2} 2; a \neq = 0

a^{2} 2 sin (x + \frac{π}{4}) = r^{2}

Dividir ambos lados entre

a^{2} 2; a \neq = 0

\frac{a ^{2} 2 sin ( x + \frac{π}{4} )}{a ^{2} 2} = \frac{r ^{2}}{a ^{2} 2}; a \neq = 0

Simplificar

\frac{a ^{2} 2 sin ( x + \frac{π}{4} )}{a ^{2} 2} = \frac{r ^{2}}{a ^{2} 2}

Simplificar

\frac{a ^{2} 2 sin ( x + \frac{π}{4} )}{a ^{2} 2} : sin (x + \frac{π}{4})

\frac{a ^{2} 2 sin ( x + \frac{π}{4} )}{a ^{2} 2}

Eliminar los terminos comunes:

a^{2}

= \frac{2 sin ( x + \frac{π}{4} )}{2}

Eliminar los terminos comunes:

2

= sin (x + \frac{π}{4})

Simplificar

\frac{r ^{2}}{a ^{2} 2} : \frac{2 r ^{2}}{2 a ^{2}}

\frac{r ^{2}}{a ^{2} 2}

Multiplicar por el conjugado

\frac{2}{2}

= \frac{r ^{2} 2}{a ^{2} 2 2}

a^{2} 22 = 2 a^{2}

a^{2} 22

Aplicar las leyes de los exponentes:

a a = a

22 = 2

= 2 a^{2}

= \frac{2 r ^{2}}{2 a ^{2}}

sin (x + \frac{π}{4}) = \frac{2 r ^{2}}{2 a ^{2}}; a \neq = 0

sin (x + \frac{π}{4}) = \frac{2 r ^{2}}{2 a ^{2}}; a \neq = 0

sin (x + \frac{π}{4}) = \frac{2 r ^{2}}{2 a ^{2}}; a \neq = 0

Aplicar propiedades trigonométricas inversas

sin (x + \frac{π}{4}) = \frac{2 r ^{2}}{2 a ^{2}}

Soluciones generales para

sin (x + \frac{π}{4}) = \frac{2 r ^{2}}{2 a ^{2}}

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

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

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

Resolver

x + \frac{π}{4} = arcsin (\frac{2 r ^{2}}{2 a ^{2}}) + 2 πn : x = arcsin (\frac{r ^{2}}{2 a ^{2}}) + 2 πn - \frac{π}{4}

x + \frac{π}{4} = arcsin (\frac{2 r ^{2}}{2 a ^{2}}) + 2 πn

Simplificar

arcsin (\frac{2 r ^{2}}{2 a ^{2}}) + 2 πn : arcsin (\frac{r ^{2}}{2 a ^{2}}) + 2 πn

arcsin (\frac{2 r ^{2}}{2 a ^{2}}) + 2 πn

\frac{2 r ^{2}}{2 a ^{2}} = \frac{r ^{2}}{2 a ^{2}}

\frac{2 r ^{2}}{2 a ^{2}}

Aplicar las leyes de los exponentes:

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

= \frac{2 ^{\frac{1}{2}} r ^{2}}{2 a ^{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{r ^{2}}{2 ^{- \frac{1}{2} + 1} a ^{2}}

Restar:

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

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

Aplicar las leyes de los exponentes:

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

= \frac{r ^{2}}{2 a ^{2}}

= arcsin (\frac{r ^{2}}{2 a ^{2}}) + 2 πn

x + \frac{π}{4} = arcsin (\frac{r ^{2}}{2 a ^{2}}) + 2 πn

Mover

\frac{π}{4}

al lado derecho

x + \frac{π}{4} = arcsin (\frac{r ^{2}}{2 a ^{2}}) + 2 πn

Restar

\frac{π}{4}

de ambos lados

x + \frac{π}{4} - \frac{π}{4} = arcsin (\frac{r ^{2}}{2 a ^{2}}) + 2 πn - \frac{π}{4}

Simplificar

x = arcsin (\frac{r ^{2}}{2 a ^{2}}) + 2 πn - \frac{π}{4}

x = arcsin (\frac{r ^{2}}{2 a ^{2}}) + 2 πn - \frac{π}{4}

Resolver

x + \frac{π}{4} = π + arcsin (- \frac{2 r ^{2}}{2 a ^{2}}) + 2 πn : x = π + arcsin (- \frac{r ^{2}}{2 a ^{2}}) + 2 πn - \frac{π}{4}

x + \frac{π}{4} = π + arcsin (- \frac{2 r ^{2}}{2 a ^{2}}) + 2 πn

Simplificar

π + arcsin (- \frac{2 r ^{2}}{2 a ^{2}}) + 2 πn : π + arcsin (- \frac{r ^{2}}{2 a ^{2}}) + 2 πn

π + arcsin (- \frac{2 r ^{2}}{2 a ^{2}}) + 2 πn

\frac{2 r ^{2}}{2 a ^{2}} = \frac{r ^{2}}{2 a ^{2}}

\frac{2 r ^{2}}{2 a ^{2}}

Aplicar las leyes de los exponentes:

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

= \frac{2 ^{\frac{1}{2}} r ^{2}}{2 a ^{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{r ^{2}}{2 ^{- \frac{1}{2} + 1} a ^{2}}

Restar:

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

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

Aplicar las leyes de los exponentes:

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

= \frac{r ^{2}}{2 a ^{2}}

= π + arcsin (- \frac{r ^{2}}{2 a ^{2}}) + 2 πn

x + \frac{π}{4} = π + arcsin (- \frac{r ^{2}}{2 a ^{2}}) + 2 πn

Mover

\frac{π}{4}

al lado derecho

x + \frac{π}{4} = π + arcsin (- \frac{r ^{2}}{2 a ^{2}}) + 2 πn

Restar

\frac{π}{4}

de ambos lados

x + \frac{π}{4} - \frac{π}{4} = π + arcsin (- \frac{r ^{2}}{2 a ^{2}}) + 2 πn - \frac{π}{4}

Simplificar

x = π + arcsin (- \frac{r ^{2}}{2 a ^{2}}) + 2 πn - \frac{π}{4}

x = π + arcsin (- \frac{r ^{2}}{2 a ^{2}}) + 2 πn - \frac{π}{4}

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

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