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229cos(a)+14715sin(a)=3959.55

Solución

229 cos (a) + 14715 sin (a) = 3959.55

Solución

a = 0.25684 \dots + 2 πn, a = π - 0.28796 \dots + 2 πn

Grados

a = 14.71615 \dots^{\circ} + 36 0^{\circ} n, a = 163.50067 \dots^{\circ} + 36 0^{\circ} n

Pasos de solución

229 cos (a) + 14715 sin (a) = 3959.55

Restar

14715 sin (a)

de ambos lados

229 cos (a) = 3959.55 - 14715 sin (a)

Elevar al cuadrado ambos lados

(229 cos (a))^{2} = (3959.55 - 14715 sin (a))^{2}

Restar

(3959.55 - 14715 sin (a))^{2}

de ambos lados

52441 cos^{2} (a) - 15678036.2025 + 116529556.5 sin (a) - 216531225 sin^{2} (a) = 0

Re-escribir usando identidades trigonométricas

- 15678036.2025 + 116529556.5 sin (a) - 216531225 sin^{2} (a) + 52441 cos^{2} (a)

Utilizar la identidad pitagórica:

cos^{2} (x) + sin^{2} (x) = 1

cos^{2} (x) = 1 - sin^{2} (x)

= - 15678036.2025 + 116529556.5 sin (a) - 216531225 sin^{2} (a) + 52441 (1 - sin^{2} (a))

Simplificar

- 15678036.2025 + 116529556.5 sin (a) - 216531225 sin^{2} (a) + 52441 (1 - sin^{2} (a)) : 116529556.5 sin (a) - 216583666 sin^{2} (a) - 15625595.2025

- 15678036.2025 + 116529556.5 sin (a) - 216531225 sin^{2} (a) + 52441 (1 - sin^{2} (a))

Expandir

52441 (1 - sin^{2} (a)) : 52441 - 52441 sin^{2} (a)

52441 (1 - sin^{2} (a))

Poner los parentesis utilizando:

a (b - c) = ab - a c

a = 52441, b = 1, c = sin^{2} (a)

= 52441 \cdot 1 - 52441 sin^{2} (a)

Multiplicar los numeros:

52441 \cdot 1 = 52441

= 52441 - 52441 sin^{2} (a)

= - 15678036.2025 + 116529556.5 sin (a) - 216531225 sin^{2} (a) + 52441 - 52441 sin^{2} (a)

Simplificar

- 15678036.2025 + 116529556.5 sin (a) - 216531225 sin^{2} (a) + 52441 - 52441 sin^{2} (a) : 116529556.5 sin (a) - 216583666 sin^{2} (a) - 15625595.2025

- 15678036.2025 + 116529556.5 sin (a) - 216531225 sin^{2} (a) + 52441 - 52441 sin^{2} (a)

Agrupar términos semejantes

= 116529556.5 sin (a) - 216531225 sin^{2} (a) - 52441 sin^{2} (a) - 15678036.2025 + 52441

Sumar elementos similares:

- 216531225 sin^{2} (a) - 52441 sin^{2} (a) = - 216583666 sin^{2} (a)

= 116529556.5 sin (a) - 216583666 sin^{2} (a) - 15678036.2025 + 52441

Sumar/restar lo siguiente:

- 15678036.2025 + 52441 = - 15625595.2025

= 116529556.5 sin (a) - 216583666 sin^{2} (a) - 15625595.2025

= 116529556.5 sin (a) - 216583666 sin^{2} (a) - 15625595.2025

= 116529556.5 sin (a) - 216583666 sin^{2} (a) - 15625595.2025

- 15625595.2025 + 116529556.5 sin (a) - 216583666 sin^{2} (a) = 0

Usando el método de sustitución

- 15625595.2025 + 116529556.5 sin (a) - 216583666 sin^{2} (a) = 0

Sea:

sin (a) = u

- 15625595.2025 + 116529556.5 u - 216583666 u^{2} = 0

- 15625595.2025 + 116529556.5 u - 216583666 u^{2} = 0 : u = \frac{- 116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}, u = \frac{116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}

- 15625595.2025 + 116529556.5 u - 216583666 u^{2} = 0

Escribir en la forma binómica

a x^{2} + b x + c = 0

- 216583666 u^{2} + 116529556.5 u - 15625595.2025 = 0

Resolver con la fórmula general para ecuaciones de segundo grado:

- 216583666 u^{2} + 116529556.5 u - 15625595.2025 = 0

Formula general para ecuaciones de segundo grado:

Para

a = - 216583666, b = 116529556.5, c = - 15625595.2025

u_{1, 2} = \frac{- 116529556.5 \pm 116529556. 5 ^{2} - 4 ( - 216583666 ) ( - 15625595.2025 )}{2 ( - 216583666 )}

u_{1, 2} = \frac{- 116529556.5 \pm 116529556. 5 ^{2} - 4 ( - 216583666 ) ( - 15625595.2025 )}{2 ( - 216583666 )}

116529556. 5^{2} - 4 (- 216583666) (- 15625595.2025) = 116529556. 5^{2} - 1.3537 E 16

116529556. 5^{2} - 4 (- 216583666) (- 15625595.2025)

Aplicar la regla

- (- a) = a

= 116529556. 5^{2} - 4 \cdot 216583666 \cdot 15625595.2025

Multiplicar los numeros:

4 \cdot 216583666 \cdot 15625595.2025 = 1.3537 E 16

= 116529556. 5^{2} - 1.3537 E 16

u_{1, 2} = \frac{- 116529556.5 \pm 116529556. 5 ^{2} - 1.3537 E 16}{2 ( - 216583666 )}

Separar las soluciones

u_{1} = \frac{- 116529556.5 + 116529556. 5 ^{2} - 1.3537 E 16}{2 ( - 216583666 )}, u_{2} = \frac{- 116529556.5 - 116529556. 5 ^{2} - 1.3537 E 16}{2 ( - 216583666 )}

u = \frac{- 116529556.5 + 116529556. 5 ^{2} - 1.3537 E 16}{2 ( - 216583666 )} : \frac{- 116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}

\frac{- 116529556.5 + 116529556. 5 ^{2} - 1.3537 E 16}{2 ( - 216583666 )}

Quitar los parentesis:

(- a) = - a

= \frac{- 116529556.5 + 116529556. 5 ^{2} - 1.3537 E 16}{- 2 \cdot 216583666}

Multiplicar los numeros:

2 \cdot 216583666 = 433167332

= \frac{- 116529556.5 + 116529556. 5 ^{2} - 1.3537 E 16}{- 433167332}

Aplicar las propiedades de las fracciones:

\frac{- a}{- b} = \frac{a}{b}

- 116529556.5 + 116529556. 5^{2} - 1.3537 E 16 = - (- 116529556. 5^{2} - 1.3537 E 16 + 116529556.5)

= \frac{- 116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}

u = \frac{- 116529556.5 - 116529556. 5 ^{2} - 1.3537 E 16}{2 ( - 216583666 )} : \frac{116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}

\frac{- 116529556.5 - 116529556. 5 ^{2} - 1.3537 E 16}{2 ( - 216583666 )}

Quitar los parentesis:

(- a) = - a

= \frac{- 116529556.5 - 116529556. 5 ^{2} - 1.3537 E 16}{- 2 \cdot 216583666}

Multiplicar los numeros:

2 \cdot 216583666 = 433167332

= \frac{- 116529556.5 - 116529556. 5 ^{2} - 1.3537 E 16}{- 433167332}

Aplicar las propiedades de las fracciones:

\frac{- a}{- b} = \frac{a}{b}

- 116529556.5 - 116529556. 5^{2} - 1.3537 E 16 = - (116529556. 5^{2} - 1.3537 E 16 + 116529556.5)

= \frac{116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}

Las soluciones a la ecuación de segundo grado son:

u = \frac{- 116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}, u = \frac{116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}

Sustituir en la ecuación

u = sin (a)

sin (a) = \frac{- 116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}, sin (a) = \frac{116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}

sin (a) = \frac{- 116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}, sin (a) = \frac{116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}

sin (a) = \frac{- 116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332} : a = arcsin (\frac{- 116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}) + 2 πn, a = π - arcsin (\frac{- 116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}) + 2 πn

sin (a) = \frac{- 116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}

Aplicar propiedades trigonométricas inversas

sin (a) = \frac{- 116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}

Soluciones generales para

sin (a) = \frac{- 116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}

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

a = arcsin (\frac{- 116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}) + 2 πn, a = π - arcsin (\frac{- 116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}) + 2 πn

a = arcsin (\frac{- 116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}) + 2 πn, a = π - arcsin (\frac{- 116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}) + 2 πn

sin (a) = \frac{116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332} : a = arcsin (\frac{116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}) + 2 πn, a = π - arcsin (\frac{116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}) + 2 πn

sin (a) = \frac{116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}

Aplicar propiedades trigonométricas inversas

sin (a) = \frac{116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}

Soluciones generales para

sin (a) = \frac{116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}

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

a = arcsin (\frac{116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}) + 2 πn, a = π - arcsin (\frac{116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}) + 2 πn

a = arcsin (\frac{116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}) + 2 πn, a = π - arcsin (\frac{116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}) + 2 πn

Combinar toda las soluciones

a = arcsin (\frac{- 116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}) + 2 πn, a = π - arcsin (\frac{- 116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}) + 2 πn, a = arcsin (\frac{116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}) + 2 πn, a = π - arcsin (\frac{116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}) + 2 πn

Verificar las soluciones sustituyendo en la ecuación original

Verificar las soluciones sustituyéndolas en

229 cos (a) + 14715 sin (a) = 3959.55

Quitar las que no concuerden con la ecuación.

Verificar la solución

arcsin (\frac{- 116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}) + 2 πn :

Verdadero

arcsin (\frac{- 116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}) + 2 πn

Sustituir

n = 1

arcsin (\frac{- 116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}) + 2 π 1

Multiplicar

229 cos (a) + 14715 sin (a) = 3959.55

por

a = arcsin (\frac{- 116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}) + 2 π 1

229 cos (arcsin (\frac{- 116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}) + 2 π 1) + 14715 sin (arcsin (\frac{- 116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}) + 2 π 1) = 3959.55

Simplificar

3959.54999 \dots = 3959.55

\Rightarrow Verdadero

Verificar la solución

π - arcsin (\frac{- 116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}) + 2 πn :

Falso

π - arcsin (\frac{- 116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}) + 2 πn

Sustituir

n = 1

π - arcsin (\frac{- 116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}) + 2 π 1

Multiplicar

229 cos (a) + 14715 sin (a) = 3959.55

por

a = π - arcsin (\frac{- 116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}) + 2 π 1

229 cos (π - arcsin (\frac{- 116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}) + 2 π 1) + 14715 sin (π - arcsin (\frac{- 116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}) + 2 π 1) = 3959.55

Simplificar

3516.57416 \dots = 3959.55

\Rightarrow Falso

Verificar la solución

arcsin (\frac{116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}) + 2 πn :

Falso

arcsin (\frac{116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}) + 2 πn

Sustituir

n = 1

arcsin (\frac{116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}) + 2 π 1

Multiplicar

229 cos (a) + 14715 sin (a) = 3959.55

por

a = arcsin (\frac{116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}) + 2 π 1

229 cos (arcsin (\frac{116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}) + 2 π 1) + 14715 sin (arcsin (\frac{116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}) + 2 π 1) = 3959.55

Simplificar

4398.69096 \dots = 3959.55

\Rightarrow Falso

Verificar la solución

π - arcsin (\frac{116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}) + 2 πn :

Verdadero

π - arcsin (\frac{116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}) + 2 πn

Sustituir

n = 1

π - arcsin (\frac{116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}) + 2 π 1

Multiplicar

229 cos (a) + 14715 sin (a) = 3959.55

por

a = π - arcsin (\frac{116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}) + 2 π 1

229 cos (π - arcsin (\frac{116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}) + 2 π 1) + 14715 sin (π - arcsin (\frac{116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}) + 2 π 1) = 3959.55

Simplificar

3959.54999 \dots = 3959.55

\Rightarrow Verdadero

a = arcsin (\frac{- 116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}) + 2 πn, a = π - arcsin (\frac{116529556. 5 ^{2} - 1.3537 E 16 + 116529556.5}{433167332}) + 2 πn

Mostrar soluciones en forma decimal

a = 0.25684 \dots + 2 πn, a = π - 0.28796 \dots + 2 πn

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