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Popular Trigonometría >

117.72sin(θ)-35.316cos(θ)-12.5=0

Solución

117.72 sin (θ) - 35.316 cos (θ) - 12.5 = 0

Solución

θ = 0.39333 \dots + 2 πn, θ = π + 0.18957 \dots + 2 πn

Grados

θ = 22.53666 \dots^{\circ} + 36 0^{\circ} n, θ = 190.86182 \dots^{\circ} + 36 0^{\circ} n

Pasos de solución

117.72 sin (θ) - 35.316 cos (θ) - 12.5 = 0

Sumar

35.316 cos (θ)

a ambos lados

117.72 sin (θ) - 12.5 = 35.316 cos (θ)

Elevar al cuadrado ambos lados

(117.72 sin (θ) - 12.5)^{2} = (35.316 cos (θ))^{2}

Restar

(35.316 cos (θ))^{2}

de ambos lados

(117.72 sin (θ) - 12.5)^{2} - 1247.219856 cos^{2} (θ) = 0

Re-escribir usando identidades trigonométricas

(- 12.5 + 117.72 sin (θ))^{2} - 1247.219856 cos^{2} (θ)

Utilizar la identidad pitagórica:

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

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

= (- 12.5 + 117.72 sin (θ))^{2} - 1247.219856 (1 - sin^{2} (θ))

Simplificar

(- 12.5 + 117.72 sin (θ))^{2} - 1247.219856 (1 - sin^{2} (θ)) : 15105.218256 sin^{2} (θ) - 2943 sin (θ) - 1090.969856

(- 12.5 + 117.72 sin (θ))^{2} - 1247.219856 (1 - sin^{2} (θ))

(- 12.5 + 117.72 sin (θ))^{2} : 156.25 - 2943 sin (θ) + 13857.9984 sin^{2} (θ)

Aplicar la formula del binomio al cuadrado:

(a + b)^{2} = a^{2} + 2 ab + b^{2}

a = - 12.5, b = 117.72 sin (θ)

= (- 12.5)^{2} + 2 (- 12.5) \cdot 117.72 sin (θ) + (117.72 sin (θ))^{2}

Simplificar

(- 12.5)^{2} + 2 (- 12.5) \cdot 117.72 sin (θ) + (117.72 sin (θ))^{2} : 156.25 - 2943 sin (θ) + 13857.9984 sin^{2} (θ)

(- 12.5)^{2} + 2 (- 12.5) \cdot 117.72 sin (θ) + (117.72 sin (θ))^{2}

Quitar los parentesis:

(- a) = - a

= (- 12.5)^{2} - 2 \cdot 12.5 \cdot 117.72 sin (θ) + (117.72 sin (θ))^{2}

(- 12.5)^{2} = 156.25

(- 12.5)^{2}

Aplicar las leyes de los exponentes:

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

n

es par

(- 12.5)^{2} = 12. 5^{2}

= 12. 5^{2}

12. 5^{2} = 156.25

= 156.25

2 \cdot 12.5 \cdot 117.72 sin (θ) = 2943 sin (θ)

2 \cdot 12.5 \cdot 117.72 sin (θ)

Multiplicar los numeros:

2 \cdot 12.5 \cdot 117.72 = 2943

= 2943 sin (θ)

(117.72 sin (θ))^{2} = 13857.9984 sin^{2} (θ)

(117.72 sin (θ))^{2}

Aplicar las leyes de los exponentes:

(a \cdot b)^{n} = a^{n} b^{n}

= 117.7 2^{2} sin^{2} (θ)

117.7 2^{2} = 13857.9984

= 13857.9984 sin^{2} (θ)

= 156.25 - 2943 sin (θ) + 13857.9984 sin^{2} (θ)

= 156.25 - 2943 sin (θ) + 13857.9984 sin^{2} (θ)

= 156.25 - 2943 sin (θ) + 13857.9984 sin^{2} (θ) - 1247.219856 (1 - sin^{2} (θ))

Expandir

- 1247.219856 (1 - sin^{2} (θ)) : - 1247.219856 + 1247.219856 sin^{2} (θ)

- 1247.219856 (1 - sin^{2} (θ))

Poner los parentesis utilizando:

a (b - c) = ab - a c

a = - 1247.219856, b = 1, c = sin^{2} (θ)

= - 1247.219856 \cdot 1 - (- 1247.219856) sin^{2} (θ)

Aplicar las reglas de los signos

- (- a) = a

= - 1 \cdot 1247.219856 + 1247.219856 sin^{2} (θ)

Multiplicar los numeros:

1 \cdot 1247.219856 = 1247.219856

= - 1247.219856 + 1247.219856 sin^{2} (θ)

= 156.25 - 2943 sin (θ) + 13857.9984 sin^{2} (θ) - 1247.219856 + 1247.219856 sin^{2} (θ)

Simplificar

156.25 - 2943 sin (θ) + 13857.9984 sin^{2} (θ) - 1247.219856 + 1247.219856 sin^{2} (θ) : 15105.218256 sin^{2} (θ) - 2943 sin (θ) - 1090.969856

156.25 - 2943 sin (θ) + 13857.9984 sin^{2} (θ) - 1247.219856 + 1247.219856 sin^{2} (θ)

Agrupar términos semejantes

= - 2943 sin (θ) + 13857.9984 sin^{2} (θ) + 1247.219856 sin^{2} (θ) + 156.25 - 1247.219856

Sumar elementos similares:

13857.9984 sin^{2} (θ) + 1247.219856 sin^{2} (θ) = 15105.218256 sin^{2} (θ)

= - 2943 sin (θ) + 15105.218256 sin^{2} (θ) + 156.25 - 1247.219856

Sumar/restar lo siguiente:

156.25 - 1247.219856 = - 1090.969856

= 15105.218256 sin^{2} (θ) - 2943 sin (θ) - 1090.969856

= 15105.218256 sin^{2} (θ) - 2943 sin (θ) - 1090.969856

= 15105.218256 sin^{2} (θ) - 2943 sin (θ) - 1090.969856

- 1090.969856 + 15105.218256 sin^{2} (θ) - 2943 sin (θ) = 0

Usando el método de sustitución

- 1090.969856 + 15105.218256 sin^{2} (θ) - 2943 sin (θ) = 0

Sea:

sin (θ) = u

- 1090.969856 + 15105.218256 u^{2} - 2943 u = 0

- 1090.969856 + 15105.218256 u^{2} - 2943 u = 0 : u = \frac{0.19483 \dots + 0.32685 \dots}{2}, u = \frac{0.19483 \dots - 0.32685 \dots}{2}

- 1090.969856 + 15105.218256 u^{2} - 2943 u = 0

Dividir ambos lados entre

15105.218256

- \frac{1090.969856}{15105.218256} + \frac{15105.218256 u ^{2}}{15105.218256} - \frac{2943 u}{15105.218256} = \frac{0}{15105.218256}

Escribir en la forma binómica

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

u^{2} - 0.19483 \dots u - 0.07222 \dots = 0

Resolver con la fórmula general para ecuaciones de segundo grado:

u^{2} - 0.19483 \dots u - 0.07222 \dots = 0

Formula general para ecuaciones de segundo grado:

Para

a = 1, b = - 0.19483 \dots, c = - 0.07222 \dots

u_{1, 2} = \frac{- ( - 0.19483 \dots ) \pm ( - 0.19483 \dots ) ^{2} - 4 \cdot 1 \cdot ( - 0.07222 \dots )}{2 \cdot 1}

u_{1, 2} = \frac{- ( - 0.19483 \dots ) \pm ( - 0.19483 \dots ) ^{2} - 4 \cdot 1 \cdot ( - 0.07222 \dots )}{2 \cdot 1}

(- 0.19483 \dots)^{2} - 4 \cdot 1 \cdot (- 0.07222 \dots) = 0.32685 \dots

(- 0.19483 \dots)^{2} - 4 \cdot 1 \cdot (- 0.07222 \dots)

Aplicar la regla

- (- a) = a

= (- 0.19483 \dots)^{2} + 4 \cdot 1 \cdot 0.07222 \dots

Aplicar las leyes de los exponentes:

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

n

es par

(- 0.19483 \dots)^{2} = 0.19483 \dots^{2}

= 0.19483 \dots^{2} + 4 \cdot 1 \cdot 0.07222 \dots

Multiplicar los numeros:

4 \cdot 1 \cdot 0.07222 \dots = 0.28889 \dots

= 0.19483 \dots^{2} + 0.28889 \dots

0.19483 \dots^{2} = 0.03796 \dots

= 0.03796 \dots + 0.28889 \dots

Sumar:

0.03796 \dots + 0.28889 \dots = 0.32685 \dots

= 0.32685 \dots

u_{1, 2} = \frac{- ( - 0.19483 \dots ) \pm 0.32685 \dots}{2 \cdot 1}

Separar las soluciones

u_{1} = \frac{- ( - 0.19483 \dots ) + 0.32685 \dots}{2 \cdot 1}, u_{2} = \frac{- ( - 0.19483 \dots ) - 0.32685 \dots}{2 \cdot 1}

u = \frac{- ( - 0.19483 \dots ) + 0.32685 \dots}{2 \cdot 1} : \frac{0.19483 \dots + 0.32685 \dots}{2}

\frac{- ( - 0.19483 \dots ) + 0.32685 \dots}{2 \cdot 1}

Aplicar la regla

- (- a) = a

= \frac{0.19483 \dots + 0.32685 \dots}{2 \cdot 1}

Multiplicar los numeros:

2 \cdot 1 = 2

= \frac{0.19483 \dots + 0.32685 \dots}{2}

u = \frac{- ( - 0.19483 \dots ) - 0.32685 \dots}{2 \cdot 1} : \frac{0.19483 \dots - 0.32685 \dots}{2}

\frac{- ( - 0.19483 \dots ) - 0.32685 \dots}{2 \cdot 1}

Aplicar la regla

- (- a) = a

= \frac{0.19483 \dots - 0.32685 \dots}{2 \cdot 1}

Multiplicar los numeros:

2 \cdot 1 = 2

= \frac{0.19483 \dots - 0.32685 \dots}{2}

Las soluciones a la ecuación de segundo grado son:

u = \frac{0.19483 \dots + 0.32685 \dots}{2}, u = \frac{0.19483 \dots - 0.32685 \dots}{2}

Sustituir en la ecuación

u = sin (θ)

sin (θ) = \frac{0.19483 \dots + 0.32685 \dots}{2}, sin (θ) = \frac{0.19483 \dots - 0.32685 \dots}{2}

sin (θ) = \frac{0.19483 \dots + 0.32685 \dots}{2}, sin (θ) = \frac{0.19483 \dots - 0.32685 \dots}{2}

sin (θ) = \frac{0.19483 \dots + 0.32685 \dots}{2} : θ = arcsin (\frac{0.19483 \dots + 0.32685 \dots}{2}) + 2 πn, θ = π - arcsin (\frac{0.19483 \dots + 0.32685 \dots}{2}) + 2 πn

sin (θ) = \frac{0.19483 \dots + 0.32685 \dots}{2}

Aplicar propiedades trigonométricas inversas

sin (θ) = \frac{0.19483 \dots + 0.32685 \dots}{2}

Soluciones generales para

sin (θ) = \frac{0.19483 \dots + 0.32685 \dots}{2}

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

θ = arcsin (\frac{0.19483 \dots + 0.32685 \dots}{2}) + 2 πn, θ = π - arcsin (\frac{0.19483 \dots + 0.32685 \dots}{2}) + 2 πn

θ = arcsin (\frac{0.19483 \dots + 0.32685 \dots}{2}) + 2 πn, θ = π - arcsin (\frac{0.19483 \dots + 0.32685 \dots}{2}) + 2 πn

sin (θ) = \frac{0.19483 \dots - 0.32685 \dots}{2} : θ = arcsin (\frac{0.19483 \dots - 0.32685 \dots}{2}) + 2 πn, θ = π + arcsin (- \frac{0.19483 \dots - 0.32685 \dots}{2}) + 2 πn

sin (θ) = \frac{0.19483 \dots - 0.32685 \dots}{2}

Aplicar propiedades trigonométricas inversas

sin (θ) = \frac{0.19483 \dots - 0.32685 \dots}{2}

Soluciones generales para

sin (θ) = \frac{0.19483 \dots - 0.32685 \dots}{2}

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

θ = arcsin (\frac{0.19483 \dots - 0.32685 \dots}{2}) + 2 πn, θ = π + arcsin (- \frac{0.19483 \dots - 0.32685 \dots}{2}) + 2 πn

θ = arcsin (\frac{0.19483 \dots - 0.32685 \dots}{2}) + 2 πn, θ = π + arcsin (- \frac{0.19483 \dots - 0.32685 \dots}{2}) + 2 πn

Combinar toda las soluciones

θ = arcsin (\frac{0.19483 \dots + 0.32685 \dots}{2}) + 2 πn, θ = π - arcsin (\frac{0.19483 \dots + 0.32685 \dots}{2}) + 2 πn, θ = arcsin (\frac{0.19483 \dots - 0.32685 \dots}{2}) + 2 πn, θ = π + arcsin (- \frac{0.19483 \dots - 0.32685 \dots}{2}) + 2 πn

Verificar las soluciones sustituyendo en la ecuación original

Verificar las soluciones sustituyéndolas en

117.72 sin (θ) - 35.316 cos (θ) - 12.5 = 0

Quitar las que no concuerden con la ecuación.

Verificar la solución

arcsin (\frac{0.19483 \dots + 0.32685 \dots}{2}) + 2 πn :

Verdadero

arcsin (\frac{0.19483 \dots + 0.32685 \dots}{2}) + 2 πn

Sustituir

n = 1

arcsin (\frac{0.19483 \dots + 0.32685 \dots}{2}) + 2 π 1

Multiplicar

117.72 sin (θ) - 35.316 cos (θ) - 12.5 = 0

por

θ = arcsin (\frac{0.19483 \dots + 0.32685 \dots}{2}) + 2 π 1

117.72 sin (arcsin (\frac{0.19483 \dots + 0.32685 \dots}{2}) + 2 π 1) - 35.316 cos (arcsin (\frac{0.19483 \dots + 0.32685 \dots}{2}) + 2 π 1) - 12.5 = 0

Simplificar

0 = 0

\Rightarrow Verdadero

Verificar la solución

π - arcsin (\frac{0.19483 \dots + 0.32685 \dots}{2}) + 2 πn :

Falso

π - arcsin (\frac{0.19483 \dots + 0.32685 \dots}{2}) + 2 πn

Sustituir

n = 1

π - arcsin (\frac{0.19483 \dots + 0.32685 \dots}{2}) + 2 π 1

Multiplicar

117.72 sin (θ) - 35.316 cos (θ) - 12.5 = 0

por

θ = π - arcsin (\frac{0.19483 \dots + 0.32685 \dots}{2}) + 2 π 1

117.72 sin (π - arcsin (\frac{0.19483 \dots + 0.32685 \dots}{2}) + 2 π 1) - 35.316 cos (π - arcsin (\frac{0.19483 \dots + 0.32685 \dots}{2}) + 2 π 1) - 12.5 = 0

Simplificar

65.23815 \dots = 0

\Rightarrow Falso

Verificar la solución

arcsin (\frac{0.19483 \dots - 0.32685 \dots}{2}) + 2 πn :

Falso

arcsin (\frac{0.19483 \dots - 0.32685 \dots}{2}) + 2 πn

Sustituir

n = 1

arcsin (\frac{0.19483 \dots - 0.32685 \dots}{2}) + 2 π 1

Multiplicar

117.72 sin (θ) - 35.316 cos (θ) - 12.5 = 0

por

θ = arcsin (\frac{0.19483 \dots - 0.32685 \dots}{2}) + 2 π 1

117.72 sin (arcsin (\frac{0.19483 \dots - 0.32685 \dots}{2}) + 2 π 1) - 35.316 cos (arcsin (\frac{0.19483 \dots - 0.32685 \dots}{2}) + 2 π 1) - 12.5 = 0

Simplificar

- 69.36659 \dots = 0

\Rightarrow Falso

Verificar la solución

π + arcsin (- \frac{0.19483 \dots - 0.32685 \dots}{2}) + 2 πn :

Verdadero

π + arcsin (- \frac{0.19483 \dots - 0.32685 \dots}{2}) + 2 πn

Sustituir

n = 1

π + arcsin (- \frac{0.19483 \dots - 0.32685 \dots}{2}) + 2 π 1

Multiplicar

117.72 sin (θ) - 35.316 cos (θ) - 12.5 = 0

por

θ = π + arcsin (- \frac{0.19483 \dots - 0.32685 \dots}{2}) + 2 π 1

117.72 sin (π + arcsin (- \frac{0.19483 \dots - 0.32685 \dots}{2}) + 2 π 1) - 35.316 cos (π + arcsin (- \frac{0.19483 \dots - 0.32685 \dots}{2}) + 2 π 1) - 12.5 = 0

Simplificar

0 = 0

\Rightarrow Verdadero

θ = arcsin (\frac{0.19483 \dots + 0.32685 \dots}{2}) + 2 πn, θ = π + arcsin (- \frac{0.19483 \dots - 0.32685 \dots}{2}) + 2 πn

Mostrar soluciones en forma decimal

θ = 0.39333 \dots + 2 πn, θ = π + 0.18957 \dots + 2 πn

Gráfica

Ejemplos populares

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