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8sin(x)=2+4/(csc(x))

Solución

8 sin (x) = 2 + \frac{4}{csc ( x )}

Solución

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

Grados

x = 3 0^{\circ} + 36 0^{\circ} n, x = 15 0^{\circ} + 36 0^{\circ} n

Pasos de solución

8 sin (x) = 2 + \frac{4}{csc ( x )}

Restar

2 + \frac{4}{csc ( x )}

de ambos lados

8 sin (x) - 2 - \frac{4}{csc ( x )} = 0

Simplificar

8 sin (x) - 2 - \frac{4}{csc ( x )} : \frac{8 sin ( x ) csc ( x ) - 2 csc ( x ) - 4}{csc ( x )}

8 sin (x) - 2 - \frac{4}{csc ( x )}

Convertir a fracción:

8 sin (x) = \frac{8 sin ( x ) csc ( x )}{csc ( x )}, 2 = \frac{2 csc ( x )}{csc ( x )}

= \frac{8 sin ( x ) csc ( x )}{csc ( x )} - \frac{2 csc ( x )}{csc ( x )} - \frac{4}{csc ( x )}

Ya que los denominadores son iguales, combinar las fracciones:

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

= \frac{8 sin ( x ) csc ( x ) - 2 csc ( x ) - 4}{csc ( x )}

\frac{8 sin ( x ) csc ( x ) - 2 csc ( x ) - 4}{csc ( x )} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

8 sin (x) csc (x) - 2 csc (x) - 4 = 0

Re-escribir usando identidades trigonométricas

8 sin (x) csc (x) - 2 csc (x) - 4

csc (x) sin (x) = 1

csc (x) sin (x)

Expresar con seno, coseno

csc (x) sin (x)

Utilizar la identidad trigonométrica básica:

csc (x) = \frac{1}{sin ( x )}

= \frac{1}{sin ( x )} sin (x)

\frac{1}{sin ( x )} sin (x) = 1

\frac{1}{sin ( x )} sin (x)

Multiplicar fracciones:

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

= \frac{1 sin ( x )}{sin ( x )}

Eliminar los terminos comunes:

sin (x)

= 1

= 1

= - 4 - 2 csc (x) + 8 \cdot 1

- 4 - 2 csc (x) + 8 \cdot 1 = 0

Multiplicar los numeros:

8 \cdot 1 = 8

- 4 - 2 csc (x) + 8 = 0

Agrupar términos semejantes

- 2 csc (x) - 4 + 8 = 0

Sumar/restar lo siguiente:

- 4 + 8 = 4

- 2 csc (x) + 4 = 0

Mover

4

al lado derecho

- 2 csc (x) + 4 = 0

Restar

4

de ambos lados

- 2 csc (x) + 4 - 4 = 0 - 4

Simplificar

- 2 csc (x) = - 4

- 2 csc (x) = - 4

Dividir ambos lados entre

- 2

- 2 csc (x) = - 4

Dividir ambos lados entre

- 2

\frac{- 2 csc ( x )}{- 2} = \frac{- 4}{- 2}

Simplificar

\frac{- 2 csc ( x )}{- 2} = \frac{- 4}{- 2}

Simplificar

\frac{- 2 csc ( x )}{- 2} : csc (x)

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

Aplicar las propiedades de las fracciones:

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

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

Dividir:

\frac{2}{2} = 1

= csc (x)

Simplificar

\frac{- 4}{- 2} : 2

\frac{- 4}{- 2}

Aplicar las propiedades de las fracciones:

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

= \frac{4}{2}

Dividir:

\frac{4}{2} = 2

= 2

csc (x) = 2

csc (x) = 2

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

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