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Popular >

1/(cot^2(x))+sqrt(3)tan(x)=0

Solución

\frac{1}{cot ^{2} ( x )} + 3 tan (x) = 0

Solución

x = \frac{2 π}{3} + πn

Grados

x = 12 0^{\circ} + 18 0^{\circ} n

Pasos de solución

\frac{1}{cot ^{2} ( x )} + 3 tan (x) = 0

Simplificar

\frac{1}{cot ^{2} ( x )} + 3 tan (x) : \frac{1 + 3 cot ^{2} ( x ) tan ( x )}{cot ^{2} ( x )}

\frac{1}{cot ^{2} ( x )} + 3 tan (x)

Convertir a fracción:

3 tan (x) = \frac{3 tan ( x ) cot ^{2} ( x )}{cot ^{2} ( x )}

= \frac{1}{cot ^{2} ( x )} + \frac{3 tan ( x ) cot ^{2} ( x )}{cot ^{2} ( x )}

Ya que los denominadores son iguales, combinar las fracciones:

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

= \frac{1 + 3 tan ( x ) cot ^{2} ( x )}{cot ^{2} ( x )}

\frac{1 + 3 cot ^{2} ( x ) tan ( x )}{cot ^{2} ( x )} = 0

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

1 + 3 cot^{2} (x) tan (x) = 0

Re-escribir usando identidades trigonométricas

1 + cot^{2} (x) 3 tan (x)

Utilizar la identidad trigonométrica básica:

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

= 1 + cot^{2} (x) 3 \frac{1}{cot ( x )}

cot^{2} (x) 3 \frac{1}{cot ( x )} = 3 cot (x)

cot^{2} (x) 3 \frac{1}{cot ( x )}

Multiplicar fracciones:

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

= \frac{1 \cdot cot ^{2} ( x ) 3}{cot ( x )}

Multiplicar:

1 \cdot cot^{2} (x) = cot^{2} (x)

= \frac{3 cot ^{2} ( x )}{cot ( x )}

Eliminar los terminos comunes:

cot (x)

= 3 cot (x)

= 1 + 3 cot (x)

1 + cot (x) 3 = 0

Mover

1

al lado derecho

1 + cot (x) 3 = 0

Restar

1

de ambos lados

1 + cot (x) 3 - 1 = 0 - 1

Simplificar

cot (x) 3 = - 1

cot (x) 3 = - 1

Dividir ambos lados entre

3

cot (x) 3 = - 1

Dividir ambos lados entre

3

\frac{cot ( x ) 3}{3} = \frac{- 1}{3}

Simplificar

\frac{cot ( x ) 3}{3} = \frac{- 1}{3}

Simplificar

\frac{cot ( x ) 3}{3} : cot (x)

\frac{cot ( x ) 3}{3}

Eliminar los terminos comunes:

3

= cot (x)

Simplificar

\frac{- 1}{3} : - \frac{3}{3}

\frac{- 1}{3}

Aplicar las propiedades de las fracciones:

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

= - \frac{1}{3}

Racionalizar

- \frac{1}{3} : - \frac{3}{3}

- \frac{1}{3}

Multiplicar por el conjugado

\frac{3}{3}

= - \frac{1 \cdot 3}{3 3}

1 \cdot 3 = 3

33 = 3

33

Aplicar las leyes de los exponentes:

a a = a

33 = 3

= 3

= - \frac{3}{3}

= - \frac{3}{3}

cot (x) = - \frac{3}{3}

cot (x) = - \frac{3}{3}

cot (x) = - \frac{3}{3}

Soluciones generales para

cot (x) = - \frac{3}{3}

cot (x)

tabla de valores periódicos con

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} cot (x) \mp \infty 31 \frac{3}{3} 0 - \frac{3}{3} - 1 - 3

x = \frac{2 π}{3} + πn

x = \frac{2 π}{3} + πn

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