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Popular Trigonometría >

2cot^2(x)-csc^2(x)+csc(x)=4

Solución

2 cot^{2} (x) - csc^{2} (x) + csc (x) = 4

Solución

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn, x = - 0.33983 \dots + 2 πn, x = π + 0.33983 \dots + 2 πn

Grados

x = 3 0^{\circ} + 36 0^{\circ} n, x = 15 0^{\circ} + 36 0^{\circ} n, x = - 19.47122 \dots^{\circ} + 36 0^{\circ} n, x = 199.47122 \dots^{\circ} + 36 0^{\circ} n

Pasos de solución

2 cot^{2} (x) - csc^{2} (x) + csc (x) = 4

Restar

4

de ambos lados

2 cot^{2} (x) - csc^{2} (x) + csc (x) - 4 = 0

Re-escribir usando identidades trigonométricas

- 4 + csc (x) - csc^{2} (x) + 2 cot^{2} (x)

Utilizar la identidad pitagórica:

1 + cot^{2} (x) = csc^{2} (x)

cot^{2} (x) = csc^{2} (x) - 1

= - 4 + csc (x) - csc^{2} (x) + 2 (csc^{2} (x) - 1)

Simplificar

- 4 + csc (x) - csc^{2} (x) + 2 (csc^{2} (x) - 1) : csc^{2} (x) + csc (x) - 6

- 4 + csc (x) - csc^{2} (x) + 2 (csc^{2} (x) - 1)

Expandir

2 (csc^{2} (x) - 1) : 2 csc^{2} (x) - 2

2 (csc^{2} (x) - 1)

Poner los parentesis utilizando:

a (b - c) = ab - a c

a = 2, b = csc^{2} (x), c = 1

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

Multiplicar los numeros:

2 \cdot 1 = 2

= 2 csc^{2} (x) - 2

= - 4 + csc (x) - csc^{2} (x) + 2 csc^{2} (x) - 2

Simplificar

- 4 + csc (x) - csc^{2} (x) + 2 csc^{2} (x) - 2 : csc^{2} (x) + csc (x) - 6

- 4 + csc (x) - csc^{2} (x) + 2 csc^{2} (x) - 2

Sumar elementos similares:

- csc^{2} (x) + 2 csc^{2} (x) = csc^{2} (x)

= - 4 + csc (x) + csc^{2} (x) - 2

Agrupar términos semejantes

= csc (x) + csc^{2} (x) - 4 - 2

Restar:

- 4 - 2 = - 6

= csc^{2} (x) + csc (x) - 6

= csc^{2} (x) + csc (x) - 6

= csc^{2} (x) + csc (x) - 6

- 6 + csc (x) + csc^{2} (x) = 0

Usando el método de sustitución

- 6 + csc (x) + csc^{2} (x) = 0

Sea:

csc (x) = u

- 6 + u + u^{2} = 0

- 6 + u + u^{2} = 0 : u = 2, u = - 3

- 6 + u + u^{2} = 0

Escribir en la forma binómica

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

u^{2} + u - 6 = 0

Resolver con la fórmula general para ecuaciones de segundo grado:

u^{2} + u - 6 = 0

Formula general para ecuaciones de segundo grado:

Para

a = 1, b = 1, c = - 6

u_{1, 2} = \frac{- 1 \pm 1 ^{2} - 4 \cdot 1 \cdot ( - 6 )}{2 \cdot 1}

u_{1, 2} = \frac{- 1 \pm 1 ^{2} - 4 \cdot 1 \cdot ( - 6 )}{2 \cdot 1}

1^{2} - 4 \cdot 1 \cdot (- 6) = 5

1^{2} - 4 \cdot 1 \cdot (- 6)

Aplicar la regla

1^{a} = 1

1^{2} = 1

= 1 - 4 \cdot 1 \cdot (- 6)

Aplicar la regla

- (- a) = a

= 1 + 4 \cdot 1 \cdot 6

Multiplicar los numeros:

4 \cdot 1 \cdot 6 = 24

= 1 + 24

Sumar:

1 + 24 = 25

= 25

Descomponer el número en factores primos:

25 = 5^{2}

= 5^{2}

Aplicar las leyes de los exponentes:

5^{2} = 5

= 5

u_{1, 2} = \frac{- 1 \pm 5}{2 \cdot 1}

Separar las soluciones

u_{1} = \frac{- 1 + 5}{2 \cdot 1}, u_{2} = \frac{- 1 - 5}{2 \cdot 1}

u = \frac{- 1 + 5}{2 \cdot 1} : 2

\frac{- 1 + 5}{2 \cdot 1}

Sumar/restar lo siguiente:

- 1 + 5 = 4

= \frac{4}{2 \cdot 1}

Multiplicar los numeros:

2 \cdot 1 = 2

= \frac{4}{2}

Dividir:

\frac{4}{2} = 2

= 2

u = \frac{- 1 - 5}{2 \cdot 1} : - 3

\frac{- 1 - 5}{2 \cdot 1}

Restar:

- 1 - 5 = - 6

= \frac{- 6}{2 \cdot 1}

Multiplicar los numeros:

2 \cdot 1 = 2

= \frac{- 6}{2}

Aplicar las propiedades de las fracciones:

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

= - \frac{6}{2}

Dividir:

\frac{6}{2} = 3

= - 3

Las soluciones a la ecuación de segundo grado son:

u = 2, u = - 3

Sustituir en la ecuación

u = csc (x)

csc (x) = 2, csc (x) = - 3

csc (x) = 2, csc (x) = - 3

csc (x) = 2 : x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

csc (x) = 2

Soluciones generales para

csc (x) = 2

csc (x)

tabla de valores periódicos con

2 πn

intervalos:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} csc (x) Undefiend 22 \frac{2 3}{3} 1 \frac{2 3}{3} 22 x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} csc (x) Undefiend - 2 - 2 - \frac{2 3}{3} - 1 - \frac{2 3}{3} - 2 - 2

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

csc (x) = - 3 : x = arccsc (- 3) + 2 πn, x = π + arccsc (3) + 2 πn

csc (x) = - 3

Aplicar propiedades trigonométricas inversas

csc (x) = - 3

Soluciones generales para

csc (x) = - 3

csc (x) = - a \Rightarrow x = arccsc (- a) + 2 πn, x = π + arccsc (a) + 2 πn

x = arccsc (- 3) + 2 πn, x = π + arccsc (3) + 2 πn

x = arccsc (- 3) + 2 πn, x = π + arccsc (3) + 2 πn

Combinar toda las soluciones

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn, x = arccsc (- 3) + 2 πn, x = π + arccsc (3) + 2 πn

Mostrar soluciones en forma decimal

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn, x = - 0.33983 \dots + 2 πn, x = π + 0.33983 \dots + 2 πn

Gráfica

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