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Popular >

3csc^2(x)+cot^2(x)-4=0

Solución

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

Solución

x = 1.10714 \dots + 2 πn, x = π - 1.10714 \dots + 2 πn, x = - 1.10714 \dots + 2 πn, x = π + 1.10714 \dots + 2 πn

Grados

x = 63.43494 \dots^{\circ} + 36 0^{\circ} n, x = 116.56505 \dots^{\circ} + 36 0^{\circ} n, x = - 63.43494 \dots^{\circ} + 36 0^{\circ} n, x = 243.43494 \dots^{\circ} + 36 0^{\circ} n

Pasos de solución

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

Re-escribir usando identidades trigonométricas

- 4 + cot^{2} (x) + 3 csc^{2} (x)

Utilizar la identidad pitagórica:

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

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

= - 4 + csc^{2} (x) - 1 + 3 csc^{2} (x)

Simplificar

- 4 + csc^{2} (x) - 1 + 3 csc^{2} (x) : 4 csc^{2} (x) - 5

- 4 + csc^{2} (x) - 1 + 3 csc^{2} (x)

Agrupar términos semejantes

= csc^{2} (x) + 3 csc^{2} (x) - 4 - 1

Sumar elementos similares:

csc^{2} (x) + 3 csc^{2} (x) = 4 csc^{2} (x)

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

Restar:

- 4 - 1 = - 5

= 4 csc^{2} (x) - 5

= 4 csc^{2} (x) - 5

- 5 + 4 csc^{2} (x) = 0

Usando el método de sustitución

- 5 + 4 csc^{2} (x) = 0

Sea:

csc (x) = u

- 5 + 4 u^{2} = 0

- 5 + 4 u^{2} = 0 : u = \frac{5}{2}, u = - \frac{5}{2}

- 5 + 4 u^{2} = 0

Mover

5

al lado derecho

- 5 + 4 u^{2} = 0

Sumar

5

a ambos lados

- 5 + 4 u^{2} + 5 = 0 + 5

Simplificar

4 u^{2} = 5

4 u^{2} = 5

Dividir ambos lados entre

4

4 u^{2} = 5

Dividir ambos lados entre

4

\frac{4 u ^{2}}{4} = \frac{5}{4}

Simplificar

u^{2} = \frac{5}{4}

u^{2} = \frac{5}{4}

Para

x^{2} = f (a)

las soluciones son

x = f (a), - f (a)

u = \frac{5}{4}, u = - \frac{5}{4}

\frac{5}{4} = \frac{5}{2}

\frac{5}{4}

Aplicar la siguiente propiedad de los radicales:

asumiendo que

a \geq 0, b \geq 0

= \frac{5}{4}

4 = 2

4

Descomponer el número en factores primos:

4 = 2^{2}

= 2^{2}

Aplicar las leyes de los exponentes:

2^{2} = 2

= 2

= \frac{5}{2}

- \frac{5}{4} = - \frac{5}{2}

- \frac{5}{4}

Simplificar

\frac{5}{4} : \frac{5}{2}

\frac{5}{4}

Aplicar la siguiente propiedad de los radicales:

asumiendo que

a \geq 0, b \geq 0

= \frac{5}{4}

4 = 2

4

Descomponer el número en factores primos:

4 = 2^{2}

= 2^{2}

Aplicar las leyes de los exponentes:

2^{2} = 2

= 2

= \frac{5}{2}

= - \frac{5}{2}

u = \frac{5}{2}, u = - \frac{5}{2}

Sustituir en la ecuación

u = csc (x)

csc (x) = \frac{5}{2}, csc (x) = - \frac{5}{2}

csc (x) = \frac{5}{2}, csc (x) = - \frac{5}{2}

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

csc (x) = \frac{5}{2}

Aplicar propiedades trigonométricas inversas

csc (x) = \frac{5}{2}

Soluciones generales para

csc (x) = \frac{5}{2}

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

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

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

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

csc (x) = - \frac{5}{2}

Aplicar propiedades trigonométricas inversas

csc (x) = - \frac{5}{2}

Soluciones generales para

csc (x) = - \frac{5}{2}

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

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

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

Combinar toda las soluciones

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

Mostrar soluciones en forma decimal

x = 1.10714 \dots + 2 πn, x = π - 1.10714 \dots + 2 πn, x = - 1.10714 \dots + 2 πn, x = π + 1.10714 \dots + 2 πn

Ejemplos populares

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sqrt(3)sec(3x)=-2

3 sec (3 x) = - 2

cos(x)-cos(3x)=-sin(x)

cos (x) - cos (3 x) = - sin (x)

tan(x)= 1/0

tan (x) = \frac{1}{0}

cos(x)=0.342

cos (x) = 0.342