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Popular Trigonometría >

400+290cos(x)+290sin(x)=0

Solución

400 + 290 cos (x) + 290 sin (x) = 0

Solución

x = - 2.13356 \dots + 2 πn, x = - 2.57882 \dots + 2 πn

Grados

x = - 122.24412 \dots^{\circ} + 36 0^{\circ} n, x = - 147.75587 \dots^{\circ} + 36 0^{\circ} n

Pasos de solución

400 + 290 cos (x) + 290 sin (x) = 0

Restar

290 sin (x)

de ambos lados

400 + 290 cos (x) = - 290 sin (x)

Elevar al cuadrado ambos lados

(400 + 290 cos (x))^{2} = (- 290 sin (x))^{2}

Restar

(- 290 sin (x))^{2}

de ambos lados

(400 + 290 cos (x))^{2} - 84100 sin^{2} (x) = 0

Re-escribir usando identidades trigonométricas

(400 + 290 cos (x))^{2} - 84100 sin^{2} (x)

Utilizar la identidad pitagórica:

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

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

= (400 + 290 cos (x))^{2} - 84100 (1 - cos^{2} (x))

Simplificar

(400 + 290 cos (x))^{2} - 84100 (1 - cos^{2} (x)) : 168200 cos^{2} (x) + 232000 cos (x) + 75900

(400 + 290 cos (x))^{2} - 84100 (1 - cos^{2} (x))

(400 + 290 cos (x))^{2} : 160000 + 232000 cos (x) + 84100 cos^{2} (x)

Aplicar la formula del binomio al cuadrado:

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

a = 400, b = 290 cos (x)

= 40 0^{2} + 2 \cdot 400 \cdot 290 cos (x) + (290 cos (x))^{2}

Simplificar

40 0^{2} + 2 \cdot 400 \cdot 290 cos (x) + (290 cos (x))^{2} : 160000 + 232000 cos (x) + 84100 cos^{2} (x)

40 0^{2} + 2 \cdot 400 \cdot 290 cos (x) + (290 cos (x))^{2}

40 0^{2} = 160000

40 0^{2}

40 0^{2} = 160000

= 160000

2 \cdot 400 \cdot 290 cos (x) = 232000 cos (x)

2 \cdot 400 \cdot 290 cos (x)

Multiplicar los numeros:

2 \cdot 400 \cdot 290 = 232000

= 232000 cos (x)

(290 cos (x))^{2} = 84100 cos^{2} (x)

(290 cos (x))^{2}

Aplicar las leyes de los exponentes:

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

= 29 0^{2} cos^{2} (x)

29 0^{2} = 84100

= 84100 cos^{2} (x)

= 160000 + 232000 cos (x) + 84100 cos^{2} (x)

= 160000 + 232000 cos (x) + 84100 cos^{2} (x)

= 160000 + 232000 cos (x) + 84100 cos^{2} (x) - 84100 (1 - cos^{2} (x))

Expandir

- 84100 (1 - cos^{2} (x)) : - 84100 + 84100 cos^{2} (x)

- 84100 (1 - cos^{2} (x))

Poner los parentesis utilizando:

a (b - c) = ab - a c

a = - 84100, b = 1, c = cos^{2} (x)

= - 84100 \cdot 1 - (- 84100) cos^{2} (x)

Aplicar las reglas de los signos

- (- a) = a

= - 84100 \cdot 1 + 84100 cos^{2} (x)

Multiplicar los numeros:

84100 \cdot 1 = 84100

= - 84100 + 84100 cos^{2} (x)

= 160000 + 232000 cos (x) + 84100 cos^{2} (x) - 84100 + 84100 cos^{2} (x)

Simplificar

160000 + 232000 cos (x) + 84100 cos^{2} (x) - 84100 + 84100 cos^{2} (x) : 168200 cos^{2} (x) + 232000 cos (x) + 75900

160000 + 232000 cos (x) + 84100 cos^{2} (x) - 84100 + 84100 cos^{2} (x)

Agrupar términos semejantes

= 232000 cos (x) + 84100 cos^{2} (x) + 84100 cos^{2} (x) + 160000 - 84100

Sumar elementos similares:

84100 cos^{2} (x) + 84100 cos^{2} (x) = 168200 cos^{2} (x)

= 232000 cos (x) + 168200 cos^{2} (x) + 160000 - 84100

Sumar/restar lo siguiente:

160000 - 84100 = 75900

= 168200 cos^{2} (x) + 232000 cos (x) + 75900

= 168200 cos^{2} (x) + 232000 cos (x) + 75900

= 168200 cos^{2} (x) + 232000 cos (x) + 75900

75900 + 168200 cos^{2} (x) + 232000 cos (x) = 0

Usando el método de sustitución

75900 + 168200 cos^{2} (x) + 232000 cos (x) = 0

Sea:

cos (x) = u

75900 + 168200 u^{2} + 232000 u = 0

75900 + 168200 u^{2} + 232000 u = 0 : u = \frac{- 40 + 82}{58}, u = \frac{- 40 - 82}{58}

75900 + 168200 u^{2} + 232000 u = 0

Dividir ambos lados entre

168200

\frac{75900}{168200} + \frac{168200 u ^{2}}{168200} + \frac{232000 u}{168200} = \frac{0}{168200}

Escribir en la forma binómica

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

u^{2} + \frac{40 u}{29} + \frac{759}{1682} = 0

Resolver con la fórmula general para ecuaciones de segundo grado:

u^{2} + \frac{40 u}{29} + \frac{759}{1682} = 0

Formula general para ecuaciones de segundo grado:

Para

a = 1, b = \frac{40}{29}, c = \frac{759}{1682}

u_{1, 2} = \frac{- \frac{40}{29} \pm ( \frac{40}{29} ) ^{2} - 4 \cdot 1 \cdot \frac{759}{1682}}{2 \cdot 1}

u_{1, 2} = \frac{- \frac{40}{29} \pm ( \frac{40}{29} ) ^{2} - 4 \cdot 1 \cdot \frac{759}{1682}}{2 \cdot 1}

(\frac{40}{29})^{2} - 4 \cdot 1 \cdot \frac{759}{1682} = \frac{82}{29}

(\frac{40}{29})^{2} - 4 \cdot 1 \cdot \frac{759}{1682}

(\frac{40}{29})^{2} = \frac{4 0 ^{2}}{2 9 ^{2}}

(\frac{40}{29})^{2}

Aplicar las leyes de los exponentes:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

= \frac{4 0 ^{2}}{2 9 ^{2}}

4 \cdot 1 \cdot \frac{759}{1682} = \frac{1518}{841}

4 \cdot 1 \cdot \frac{759}{1682}

Multiplicar fracciones:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= 1 \cdot \frac{759 \cdot 4}{1682}

\frac{759 \cdot 4}{1682} = \frac{1518}{841}

\frac{759 \cdot 4}{1682}

Multiplicar los numeros:

759 \cdot 4 = 3036

= \frac{3036}{1682}

Eliminar los terminos comunes:

2

= \frac{1518}{841}

= 1 \cdot \frac{1518}{841}

Multiplicar:

1 \cdot \frac{1518}{841} = \frac{1518}{841}

= \frac{1518}{841}

= \frac{4 0 ^{2}}{2 9 ^{2}} - \frac{1518}{841}

\frac{4 0 ^{2}}{2 9 ^{2}} = \frac{1600}{841}

\frac{4 0 ^{2}}{2 9 ^{2}}

4 0^{2} = 1600

= \frac{1600}{2 9 ^{2}}

2 9^{2} = 841

= \frac{1600}{841}

= \frac{1600}{841} - \frac{1518}{841}

Combinar las fracciones usando el mínimo común denominador:

\frac{82}{841}

Aplicar la regla

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{1600 - 1518}{841}

Restar:

1600 - 1518 = 82

= \frac{82}{841}

= \frac{82}{841}

Aplicar la siguiente propiedad de los radicales:

asumiendo que

a \geq 0, b \geq 0

= \frac{82}{841}

841 = 29

841

Descomponer el número en factores primos:

841 = 2 9^{2}

= 2 9^{2}

Aplicar las leyes de los exponentes:

2 9^{2} = 29

= 29

= \frac{82}{29}

u_{1, 2} = \frac{- \frac{40}{29} \pm \frac{82}{29}}{2 \cdot 1}

Separar las soluciones

u_{1} = \frac{- \frac{40}{29} + \frac{82}{29}}{2 \cdot 1}, u_{2} = \frac{- \frac{40}{29} - \frac{82}{29}}{2 \cdot 1}

u = \frac{- \frac{40}{29} + \frac{82}{29}}{2 \cdot 1} : \frac{- 40 + 82}{58}

\frac{- \frac{40}{29} + \frac{82}{29}}{2 \cdot 1}

Combinar las fracciones usando el mínimo común denominador:

\frac{- 40 + 82}{29}

Aplicar la regla

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{- 40 + 82}{29}

= \frac{\frac{- 40 + 82}{29}}{2 \cdot 1}

Multiplicar los numeros:

2 \cdot 1 = 2

= \frac{\frac{- 40 + 82}{29}}{2}

Aplicar las propiedades de las fracciones:

\frac{\frac{b}{c}}{a} = \frac{b}{c \cdot a}

= \frac{- 40 + 82}{29 \cdot 2}

Multiplicar los numeros:

29 \cdot 2 = 58

= \frac{- 40 + 82}{58}

u = \frac{- \frac{40}{29} - \frac{82}{29}}{2 \cdot 1} : \frac{- 40 - 82}{58}

\frac{- \frac{40}{29} - \frac{82}{29}}{2 \cdot 1}

Combinar las fracciones usando el mínimo común denominador:

\frac{- 40 - 82}{29}

Aplicar la regla

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{- 40 - 82}{29}

= \frac{\frac{- 40 - 82}{29}}{2 \cdot 1}

Multiplicar los numeros:

2 \cdot 1 = 2

= \frac{\frac{- 40 - 82}{29}}{2}

Aplicar las propiedades de las fracciones:

\frac{\frac{b}{c}}{a} = \frac{b}{c \cdot a}

= \frac{- 40 - 82}{29 \cdot 2}

Multiplicar los numeros:

29 \cdot 2 = 58

= \frac{- 40 - 82}{58}

Las soluciones a la ecuación de segundo grado son:

u = \frac{- 40 + 82}{58}, u = \frac{- 40 - 82}{58}

Sustituir en la ecuación

u = cos (x)

cos (x) = \frac{- 40 + 82}{58}, cos (x) = \frac{- 40 - 82}{58}

cos (x) = \frac{- 40 + 82}{58}, cos (x) = \frac{- 40 - 82}{58}

cos (x) = \frac{- 40 + 82}{58} : x = arccos (\frac{- 40 + 82}{58}) + 2 πn, x = - arccos (\frac{- 40 + 82}{58}) + 2 πn

cos (x) = \frac{- 40 + 82}{58}

Aplicar propiedades trigonométricas inversas

cos (x) = \frac{- 40 + 82}{58}

Soluciones generales para

cos (x) = \frac{- 40 + 82}{58}

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

x = arccos (\frac{- 40 + 82}{58}) + 2 πn, x = - arccos (\frac{- 40 + 82}{58}) + 2 πn

x = arccos (\frac{- 40 + 82}{58}) + 2 πn, x = - arccos (\frac{- 40 + 82}{58}) + 2 πn

cos (x) = \frac{- 40 - 82}{58} : x = arccos (\frac{- 40 - 82}{58}) + 2 πn, x = - arccos (\frac{- 40 - 82}{58}) + 2 πn

cos (x) = \frac{- 40 - 82}{58}

Aplicar propiedades trigonométricas inversas

cos (x) = \frac{- 40 - 82}{58}

Soluciones generales para

cos (x) = \frac{- 40 - 82}{58}

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

x = arccos (\frac{- 40 - 82}{58}) + 2 πn, x = - arccos (\frac{- 40 - 82}{58}) + 2 πn

x = arccos (\frac{- 40 - 82}{58}) + 2 πn, x = - arccos (\frac{- 40 - 82}{58}) + 2 πn

Combinar toda las soluciones

x = arccos (\frac{- 40 + 82}{58}) + 2 πn, x = - arccos (\frac{- 40 + 82}{58}) + 2 πn, x = arccos (\frac{- 40 - 82}{58}) + 2 πn, x = - arccos (\frac{- 40 - 82}{58}) + 2 πn

Verificar las soluciones sustituyendo en la ecuación original

Verificar las soluciones sustituyéndolas en

400 + 290 cos (x) + 290 sin (x) = 0

Quitar las que no concuerden con la ecuación.

Verificar la solución

arccos (\frac{- 40 + 82}{58}) + 2 πn :

Falso

arccos (\frac{- 40 + 82}{58}) + 2 πn

Sustituir

n = 1

arccos (\frac{- 40 + 82}{58}) + 2 π 1

Multiplicar

400 + 290 cos (x) + 290 sin (x) = 0

por

x = arccos (\frac{- 40 + 82}{58}) + 2 π 1

400 + 290 cos (arccos (\frac{- 40 + 82}{58}) + 2 π 1) + 290 sin (arccos (\frac{- 40 + 82}{58}) + 2 π 1) = 0

Simplificar

490.55385 \dots = 0

\Rightarrow Falso

Verificar la solución

- arccos (\frac{- 40 + 82}{58}) + 2 πn :

Verdadero

- arccos (\frac{- 40 + 82}{58}) + 2 πn

Sustituir

n = 1

- arccos (\frac{- 40 + 82}{58}) + 2 π 1

Multiplicar

400 + 290 cos (x) + 290 sin (x) = 0

por

x = - arccos (\frac{- 40 + 82}{58}) + 2 π 1

400 + 290 cos (- arccos (\frac{- 40 + 82}{58}) + 2 π 1) + 290 sin (- arccos (\frac{- 40 + 82}{58}) + 2 π 1) = 0

Simplificar

0 = 0

\Rightarrow Verdadero

Verificar la solución

arccos (\frac{- 40 - 82}{58}) + 2 πn :

Falso

arccos (\frac{- 40 - 82}{58}) + 2 πn

Sustituir

n = 1

arccos (\frac{- 40 - 82}{58}) + 2 π 1

Multiplicar

400 + 290 cos (x) + 290 sin (x) = 0

por

x = arccos (\frac{- 40 - 82}{58}) + 2 π 1

400 + 290 cos (arccos (\frac{- 40 - 82}{58}) + 2 π 1) + 290 sin (arccos (\frac{- 40 - 82}{58}) + 2 π 1) = 0

Simplificar

309.44614 \dots = 0

\Rightarrow Falso

Verificar la solución

- arccos (\frac{- 40 - 82}{58}) + 2 πn :

Verdadero

- arccos (\frac{- 40 - 82}{58}) + 2 πn

Sustituir

n = 1

- arccos (\frac{- 40 - 82}{58}) + 2 π 1

Multiplicar

400 + 290 cos (x) + 290 sin (x) = 0

por

x = - arccos (\frac{- 40 - 82}{58}) + 2 π 1

400 + 290 cos (- arccos (\frac{- 40 - 82}{58}) + 2 π 1) + 290 sin (- arccos (\frac{- 40 - 82}{58}) + 2 π 1) = 0

Simplificar

0 = 0

\Rightarrow Verdadero

x = - arccos (\frac{- 40 + 82}{58}) + 2 πn, x = - arccos (\frac{- 40 - 82}{58}) + 2 πn

Mostrar soluciones en forma decimal

x = - 2.13356 \dots + 2 πn, x = - 2.57882 \dots + 2 πn

Gráfica

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