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Popular Trigonometría >

3/(cot(x))=8-5tan(x)

Solución

\frac{3}{cot ( x )} = 8 - 5 tan (x)

Solución

x = \frac{π}{4} + πn

Grados

x = 4 5^{\circ} + 18 0^{\circ} n

Pasos de solución

\frac{3}{cot ( x )} = 8 - 5 tan (x)

Restar

8 - 5 tan (x)

de ambos lados

\frac{3}{cot ( x )} - 8 + 5 tan (x) = 0

Simplificar

\frac{3}{cot ( x )} - 8 + 5 tan (x) : \frac{3 - 8 cot ( x ) + 5 tan ( x ) cot ( x )}{cot ( x )}

\frac{3}{cot ( x )} - 8 + 5 tan (x)

Convertir a fracción:

8 = \frac{8 cot ( x )}{cot ( x )}, 5 tan (x) = \frac{5 tan ( x ) cot ( x )}{cot ( x )}

= \frac{3}{cot ( x )} - \frac{8 cot ( x )}{cot ( x )} + \frac{5 tan ( x ) cot ( x )}{cot ( x )}

Ya que los denominadores son iguales, combinar las fracciones:

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

= \frac{3 - 8 cot ( x ) + 5 tan ( x ) cot ( x )}{cot ( x )}

\frac{3 - 8 cot ( x ) + 5 tan ( x ) cot ( x )}{cot ( x )} = 0

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

3 - 8 cot (x) + 5 tan (x) cot (x) = 0

Re-escribir usando identidades trigonométricas

3 - 8 cot (x) + 5 cot (x) tan (x)

Utilizar la identidad trigonométrica básica:

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

= 3 - 8 cot (x) + 5 cot (x) \frac{1}{cot ( x )}

Simplificar

3 - 8 cot (x) + 5 cot (x) \frac{1}{cot ( x )} : - 8 cot (x) + 8

3 - 8 cot (x) + 5 cot (x) \frac{1}{cot ( x )}

5 cot (x) \frac{1}{cot ( x )} = 5

5 cot (x) \frac{1}{cot ( x )}

Multiplicar fracciones:

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

= \frac{1 \cdot 5 cot ( x )}{cot ( x )}

Eliminar los terminos comunes:

cot (x)

= 1 \cdot 5

Multiplicar los numeros:

1 \cdot 5 = 5

= 5

= 3 - 8 cot (x) + 5

Agrupar términos semejantes

= - 8 cot (x) + 3 + 5

Sumar:

3 + 5 = 8

= - 8 cot (x) + 8

= - 8 cot (x) + 8

8 - 8 cot (x) = 0

Desplace

8

a la derecha

8 - 8 cot (x) = 0

Restar

8

de ambos lados

8 - 8 cot (x) - 8 = 0 - 8

Simplificar

- 8 cot (x) = - 8

- 8 cot (x) = - 8

Dividir ambos lados entre

- 8

- 8 cot (x) = - 8

Dividir ambos lados entre

- 8

\frac{- 8 cot ( x )}{- 8} = \frac{- 8}{- 8}

Simplificar

cot (x) = 1

cot (x) = 1

Soluciones generales para

cot (x) = 1

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{π}{4} + πn

x = \frac{π}{4} + πn

Gráfica

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