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Popular >

((1-cos(x))(csc^2(x)-4))/(1+cos(x))=0

Solución

\frac{( 1 - cos ( x ) ) ( csc ^{2} ( x ) - 4 )}{1 + cos ( x )} = 0

Solución

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn, x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

Grados

x = 3 0^{\circ} + 36 0^{\circ} n, x = 15 0^{\circ} + 36 0^{\circ} n, x = 21 0^{\circ} + 36 0^{\circ} n, x = 33 0^{\circ} + 36 0^{\circ} n

Pasos de solución

\frac{( 1 - cos ( x ) ) ( csc ^{2} ( x ) - 4 )}{1 + cos ( x )} = 0

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

(1 - cos (x)) (csc^{2} (x) - 4) = 0

Resolver cada parte por separado

1 - cos (x) = 0 or csc^{2} (x) - 4 = 0

1 - cos (x) = 0 : x = 2 πn

1 - cos (x) = 0

Mover

1

al lado derecho

1 - cos (x) = 0

Restar

1

de ambos lados

1 - cos (x) - 1 = 0 - 1

Simplificar

- cos (x) = - 1

- cos (x) = - 1

Dividir ambos lados entre

- 1

- cos (x) = - 1

Dividir ambos lados entre

- 1

\frac{- cos ( x )}{- 1} = \frac{- 1}{- 1}

Simplificar

cos (x) = 1

cos (x) = 1

Soluciones generales para

cos (x) = 1

cos (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} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} 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} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

x = 0 + 2 πn

x = 0 + 2 πn

Resolver

x = 0 + 2 πn : x = 2 πn

x = 0 + 2 πn

0 + 2 πn = 2 πn

x = 2 πn

x = 2 πn

csc^{2} (x) - 4 = 0 : x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn, x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

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

Usando el método de sustitución

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

Sea:

csc (x) = u

u^{2} - 4 = 0

u^{2} - 4 = 0 : u = 2, u = - 2

u^{2} - 4 = 0

Mover

4

al lado derecho

u^{2} - 4 = 0

Sumar

4

a ambos lados

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

Simplificar

u^{2} = 4

u^{2} = 4

Para

x^{2} = f (a)

las soluciones son

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

u = 4, u = - 4

4 = 2

4

Descomponer el número en factores primos:

4 = 2^{2}

= 2^{2}

Aplicar las leyes de los exponentes:

n a^{n} = a

2^{2} = 2

= 2

- 4 = - 2

- 4

4 = 2

4

Descomponer el número en factores primos:

4 = 2^{2}

= 2^{2}

Aplicar las leyes de los exponentes:

n a^{n} = a

2^{2} = 2

= 2

= - 2

u = 2, u = - 2

Sustituir en la ecuación

u = csc (x)

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

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

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) = - 2 : x = \frac{7 π}{6} + 2 πn, x = \frac{11 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{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

Combinar toda las soluciones

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn, x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

Combinar toda las soluciones

x = 2 πn, x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn, x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

Siendo que la ecuación esta indefinida para:

2 πn

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn, x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

Gráfica

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tan (x) = \frac{2}{6}

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