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196sin(θ)-49cos(θ)=160

Solución

196 sin (θ) - 49 cos (θ) = 160

Solución

θ = 2.47257 \dots + 2 πn, θ = 1.15897 \dots + 2 πn

Grados

θ = 141.66783 \dots^{\circ} + 36 0^{\circ} n, θ = 66.40464 \dots^{\circ} + 36 0^{\circ} n

Pasos de solución

196 sin (θ) - 49 cos (θ) = 160

Sumar

49 cos (θ)

a ambos lados

196 sin (θ) = 160 + 49 cos (θ)

Elevar al cuadrado ambos lados

(196 sin (θ))^{2} = (160 + 49 cos (θ))^{2}

Restar

(160 + 49 cos (θ))^{2}

de ambos lados

38416 sin^{2} (θ) - 25600 - 15680 cos (θ) - 2401 cos^{2} (θ) = 0

Re-escribir usando identidades trigonométricas

- 25600 - 15680 cos (θ) - 2401 cos^{2} (θ) + 38416 sin^{2} (θ)

Utilizar la identidad pitagórica:

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

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

= - 25600 - 15680 cos (θ) - 2401 cos^{2} (θ) + 38416 (1 - cos^{2} (θ))

Simplificar

- 25600 - 15680 cos (θ) - 2401 cos^{2} (θ) + 38416 (1 - cos^{2} (θ)) : - 40817 cos^{2} (θ) - 15680 cos (θ) + 12816

- 25600 - 15680 cos (θ) - 2401 cos^{2} (θ) + 38416 (1 - cos^{2} (θ))

Expandir

38416 (1 - cos^{2} (θ)) : 38416 - 38416 cos^{2} (θ)

38416 (1 - cos^{2} (θ))

Poner los parentesis utilizando:

a (b - c) = ab - a c

a = 38416, b = 1, c = cos^{2} (θ)

= 38416 \cdot 1 - 38416 cos^{2} (θ)

Multiplicar los numeros:

38416 \cdot 1 = 38416

= 38416 - 38416 cos^{2} (θ)

= - 25600 - 15680 cos (θ) - 2401 cos^{2} (θ) + 38416 - 38416 cos^{2} (θ)

Simplificar

- 25600 - 15680 cos (θ) - 2401 cos^{2} (θ) + 38416 - 38416 cos^{2} (θ) : - 40817 cos^{2} (θ) - 15680 cos (θ) + 12816

- 25600 - 15680 cos (θ) - 2401 cos^{2} (θ) + 38416 - 38416 cos^{2} (θ)

Agrupar términos semejantes

= - 15680 cos (θ) - 2401 cos^{2} (θ) - 38416 cos^{2} (θ) - 25600 + 38416

Sumar elementos similares:

- 2401 cos^{2} (θ) - 38416 cos^{2} (θ) = - 40817 cos^{2} (θ)

= - 15680 cos (θ) - 40817 cos^{2} (θ) - 25600 + 38416

Sumar/restar lo siguiente:

- 25600 + 38416 = 12816

= - 40817 cos^{2} (θ) - 15680 cos (θ) + 12816

= - 40817 cos^{2} (θ) - 15680 cos (θ) + 12816

= - 40817 cos^{2} (θ) - 15680 cos (θ) + 12816

12816 - 15680 cos (θ) - 40817 cos^{2} (θ) = 0

Usando el método de sustitución

12816 - 15680 cos (θ) - 40817 cos^{2} (θ) = 0

Sea:

cos (θ) = u

12816 - 15680 u - 40817 u^{2} = 0

12816 - 15680 u - 40817 u^{2} = 0 : u = - \frac{15680 + 2338305088}{81634}, u = \frac{2338305088 - 15680}{81634}

12816 - 15680 u - 40817 u^{2} = 0

Escribir en la forma binómica

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

- 40817 u^{2} - 15680 u + 12816 = 0

Resolver con la fórmula general para ecuaciones de segundo grado:

- 40817 u^{2} - 15680 u + 12816 = 0

Formula general para ecuaciones de segundo grado:

Para

a = - 40817, b = - 15680, c = 12816

u_{1, 2} = \frac{- ( - 15680 ) \pm ( - 15680 ) ^{2} - 4 ( - 40817 ) \cdot 12816}{2 ( - 40817 )}

u_{1, 2} = \frac{- ( - 15680 ) \pm ( - 15680 ) ^{2} - 4 ( - 40817 ) \cdot 12816}{2 ( - 40817 )}

(- 15680)^{2} - 4 (- 40817) \cdot 12816 = 2338305088

(- 15680)^{2} - 4 (- 40817) \cdot 12816

Aplicar la regla

- (- a) = a

= (- 15680)^{2} + 4 \cdot 40817 \cdot 12816

Aplicar las leyes de los exponentes:

(- a)^{n} = a^{n},

n

es par

(- 15680)^{2} = 1568 0^{2}

= 1568 0^{2} + 4 \cdot 40817 \cdot 12816

Multiplicar los numeros:

4 \cdot 40817 \cdot 12816 = 2092442688

= 1568 0^{2} + 2092442688

1568 0^{2} = 245862400

= 245862400 + 2092442688

Sumar:

245862400 + 2092442688 = 2338305088

= 2338305088

u_{1, 2} = \frac{- ( - 15680 ) \pm 2338305088}{2 ( - 40817 )}

Separar las soluciones

u_{1} = \frac{- ( - 15680 ) + 2338305088}{2 ( - 40817 )}, u_{2} = \frac{- ( - 15680 ) - 2338305088}{2 ( - 40817 )}

u = \frac{- ( - 15680 ) + 2338305088}{2 ( - 40817 )} : - \frac{15680 + 2338305088}{81634}

\frac{- ( - 15680 ) + 2338305088}{2 ( - 40817 )}

Quitar los parentesis:

(- a) = - a, - (- a) = a

= \frac{15680 + 2338305088}{- 2 \cdot 40817}

Multiplicar los numeros:

2 \cdot 40817 = 81634

= \frac{15680 + 2338305088}{- 81634}

Aplicar las propiedades de las fracciones:

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

= - \frac{15680 + 2338305088}{81634}

u = \frac{- ( - 15680 ) - 2338305088}{2 ( - 40817 )} : \frac{2338305088 - 15680}{81634}

\frac{- ( - 15680 ) - 2338305088}{2 ( - 40817 )}

Quitar los parentesis:

(- a) = - a, - (- a) = a

= \frac{15680 - 2338305088}{- 2 \cdot 40817}

Multiplicar los numeros:

2 \cdot 40817 = 81634

= \frac{15680 - 2338305088}{- 81634}

Aplicar las propiedades de las fracciones:

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

15680 - 2338305088 = - (2338305088 - 15680)

= \frac{2338305088 - 15680}{81634}

Las soluciones a la ecuación de segundo grado son:

u = - \frac{15680 + 2338305088}{81634}, u = \frac{2338305088 - 15680}{81634}

Sustituir en la ecuación

u = cos (θ)

cos (θ) = - \frac{15680 + 2338305088}{81634}, cos (θ) = \frac{2338305088 - 15680}{81634}

cos (θ) = - \frac{15680 + 2338305088}{81634}, cos (θ) = \frac{2338305088 - 15680}{81634}

cos (θ) = - \frac{15680 + 2338305088}{81634} : θ = arccos (- \frac{15680 + 2338305088}{81634}) + 2 πn, θ = - arccos (- \frac{15680 + 2338305088}{81634}) + 2 πn

cos (θ) = - \frac{15680 + 2338305088}{81634}

Aplicar propiedades trigonométricas inversas

cos (θ) = - \frac{15680 + 2338305088}{81634}

Soluciones generales para

cos (θ) = - \frac{15680 + 2338305088}{81634}

cos (x) = - a \Rightarrow x = arccos (- a) + 2 πn, x = - arccos (- a) + 2 πn

θ = arccos (- \frac{15680 + 2338305088}{81634}) + 2 πn, θ = - arccos (- \frac{15680 + 2338305088}{81634}) + 2 πn

θ = arccos (- \frac{15680 + 2338305088}{81634}) + 2 πn, θ = - arccos (- \frac{15680 + 2338305088}{81634}) + 2 πn

cos (θ) = \frac{2338305088 - 15680}{81634} : θ = arccos (\frac{2338305088 - 15680}{81634}) + 2 πn, θ = 2 π - arccos (\frac{2338305088 - 15680}{81634}) + 2 πn

cos (θ) = \frac{2338305088 - 15680}{81634}

Aplicar propiedades trigonométricas inversas

cos (θ) = \frac{2338305088 - 15680}{81634}

Soluciones generales para

cos (θ) = \frac{2338305088 - 15680}{81634}

cos (x) = a \Rightarrow x = arccos (a) + 2 πn, x = 2 π - arccos (a) + 2 πn

θ = arccos (\frac{2338305088 - 15680}{81634}) + 2 πn, θ = 2 π - arccos (\frac{2338305088 - 15680}{81634}) + 2 πn

θ = arccos (\frac{2338305088 - 15680}{81634}) + 2 πn, θ = 2 π - arccos (\frac{2338305088 - 15680}{81634}) + 2 πn

Combinar toda las soluciones

θ = arccos (- \frac{15680 + 2338305088}{81634}) + 2 πn, θ = - arccos (- \frac{15680 + 2338305088}{81634}) + 2 πn, θ = arccos (\frac{2338305088 - 15680}{81634}) + 2 πn, θ = 2 π - arccos (\frac{2338305088 - 15680}{81634}) + 2 πn

Verificar las soluciones sustituyendo en la ecuación original

Verificar las soluciones sustituyéndolas en

196 sin (θ) - 49 cos (θ) = 160

Quitar las que no concuerden con la ecuación.

Verificar la solución

arccos (- \frac{15680 + 2338305088}{81634}) + 2 πn :

Verdadero

arccos (- \frac{15680 + 2338305088}{81634}) + 2 πn

Sustituir

n = 1

arccos (- \frac{15680 + 2338305088}{81634}) + 2 π 1

Multiplicar

196 sin (θ) - 49 cos (θ) = 160

por

θ = arccos (- \frac{15680 + 2338305088}{81634}) + 2 π 1

196 sin (arccos (- \frac{15680 + 2338305088}{81634}) + 2 π 1) - 49 cos (arccos (- \frac{15680 + 2338305088}{81634}) + 2 π 1) = 160

Simplificar

160 = 160

\Rightarrow Verdadero

Verificar la solución

- arccos (- \frac{15680 + 2338305088}{81634}) + 2 πn :

Falso

- arccos (- \frac{15680 + 2338305088}{81634}) + 2 πn

Sustituir

n = 1

- arccos (- \frac{15680 + 2338305088}{81634}) + 2 π 1

Multiplicar

196 sin (θ) - 49 cos (θ) = 160

por

θ = - arccos (- \frac{15680 + 2338305088}{81634}) + 2 π 1

196 sin (- arccos (- \frac{15680 + 2338305088}{81634}) + 2 π 1) - 49 cos (- arccos (- \frac{15680 + 2338305088}{81634}) + 2 π 1) = 160

Simplificar

- 83.12602 \dots = 160

\Rightarrow Falso

Verificar la solución

arccos (\frac{2338305088 - 15680}{81634}) + 2 πn :

Verdadero

arccos (\frac{2338305088 - 15680}{81634}) + 2 πn

Sustituir

n = 1

arccos (\frac{2338305088 - 15680}{81634}) + 2 π 1

Multiplicar

196 sin (θ) - 49 cos (θ) = 160

por

θ = arccos (\frac{2338305088 - 15680}{81634}) + 2 π 1

196 sin (arccos (\frac{2338305088 - 15680}{81634}) + 2 π 1) - 49 cos (arccos (\frac{2338305088 - 15680}{81634}) + 2 π 1) = 160

Simplificar

160 = 160

\Rightarrow Verdadero

Verificar la solución

2 π - arccos (\frac{2338305088 - 15680}{81634}) + 2 πn :

Falso

2 π - arccos (\frac{2338305088 - 15680}{81634}) + 2 πn

Sustituir

n = 1

2 π - arccos (\frac{2338305088 - 15680}{81634}) + 2 π 1

Multiplicar

196 sin (θ) - 49 cos (θ) = 160

por

θ = 2 π - arccos (\frac{2338305088 - 15680}{81634}) + 2 π 1

196 sin (2 π - arccos (\frac{2338305088 - 15680}{81634}) + 2 π 1) - 49 cos (2 π - arccos (\frac{2338305088 - 15680}{81634}) + 2 π 1) = 160

Simplificar

- 199.22691 \dots = 160

\Rightarrow Falso

θ = arccos (- \frac{15680 + 2338305088}{81634}) + 2 πn, θ = arccos (\frac{2338305088 - 15680}{81634}) + 2 πn

Mostrar soluciones en forma decimal

θ = 2.47257 \dots + 2 πn, θ = 1.15897 \dots + 2 πn

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