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cot(x)=cot(3x-50),0<x<180

Solución

cot (x) = cot (3 x - 5 0^{\circ}), 0 < x < 18 0^{\circ}

Solución

x = 2 5^{\circ}, x = 11 5^{\circ}

radianes

x = \frac{5 π}{36}, x = \frac{23 π}{36}

Pasos de solución

cot (x) = cot (3 x - 5 0^{\circ}), 0 < x < 18 0^{\circ}

Restar

cot (3 x - 5 0^{\circ})

de ambos lados

cot (x) - cot (3 x - 5 0^{\circ}) = 0

Expresar con seno, coseno

- cot (- 5 0^{\circ} + 3 x) + cot (x)

Utilizar la identidad trigonométrica básica:

cot (x) = \frac{cos ( x )}{sin ( x )}

= - \frac{cos ( - 5 0 ^{\circ} + 3 x )}{sin ( - 5 0 ^{\circ} + 3 x )} + cot (x)

Utilizar la identidad trigonométrica básica:

cot (x) = \frac{cos ( x )}{sin ( x )}

= - \frac{cos ( - 5 0 ^{\circ} + 3 x )}{sin ( - 5 0 ^{\circ} + 3 x )} + \frac{cos ( x )}{sin ( x )}

Simplificar

- \frac{cos ( - 5 0 ^{\circ} + 3 x )}{sin ( - 5 0 ^{\circ} + 3 x )} + \frac{cos ( x )}{sin ( x )} : \frac{- cos ( \frac{- 90 0 ^{\circ} + 54 x}{18} ) sin ( x ) + cos ( x ) sin ( \frac{54 x - 90 0 ^{\circ}}{18} )}{sin ( \frac{54 x - 90 0 ^{\circ}}{18} ) sin ( x )}

- \frac{cos ( - 5 0 ^{\circ} + 3 x )}{sin ( - 5 0 ^{\circ} + 3 x )} + \frac{cos ( x )}{sin ( x )}

\frac{cos ( - 5 0 ^{\circ} + 3 x )}{sin ( - 5 0 ^{\circ} + 3 x )} = \frac{cos ( \frac{- 90 0 ^{\circ} + 54 x}{18} )}{sin ( \frac{- 90 0 ^{\circ} + 54 x}{18} )}

\frac{cos ( - 5 0 ^{\circ} + 3 x )}{sin ( - 5 0 ^{\circ} + 3 x )}

Simplificar

- 5 0^{\circ} + 3 x

en una fracción:

\frac{- 90 0 ^{\circ} + 54 x}{18}

- 5 0^{\circ} + 3 x

Convertir a fracción:

3 x = \frac{3 x 18}{18}

= - 5 0^{\circ} + \frac{3 x \cdot 18}{18}

Ya que los denominadores son iguales, combinar las fracciones:

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

= \frac{- 90 0 ^{\circ} + 3 x \cdot 18}{18}

Multiplicar los numeros:

3 \cdot 18 = 54

= \frac{- 90 0 ^{\circ} + 54 x}{18}

= \frac{cos ( - 5 0 ^{\circ} + 3 x )}{sin ( \frac{- 90 0 ^{\circ} + 54 x}{18} )}

Simplificar

- 5 0^{\circ} + 3 x

en una fracción:

\frac{- 90 0 ^{\circ} + 54 x}{18}

- 5 0^{\circ} + 3 x

Convertir a fracción:

3 x = \frac{3 x 18}{18}

= - 5 0^{\circ} + \frac{3 x \cdot 18}{18}

Ya que los denominadores son iguales, combinar las fracciones:

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

= \frac{- 90 0 ^{\circ} + 3 x \cdot 18}{18}

Multiplicar los numeros:

3 \cdot 18 = 54

= \frac{- 90 0 ^{\circ} + 54 x}{18}

= \frac{cos ( \frac{- 90 0 ^{\circ} + 54 x}{18} )}{sin ( \frac{- 90 0 ^{\circ} + 54 x}{18} )}

= - \frac{cos ( \frac{54 x - 90 0 ^{\circ}}{18} )}{sin ( \frac{54 x - 90 0 ^{\circ}}{18} )} + \frac{cos ( x )}{sin ( x )}

Mínimo común múltiplo de

sin (\frac{- 90 0 ^{\circ} + 54 x}{18}), sin (x) : sin (\frac{54 x - 90 0 ^{\circ}}{18}) sin (x)

sin (\frac{- 90 0 ^{\circ} + 54 x}{18}), sin (x)

Mínimo común múltiplo (MCM)

Calcular una expresión que este compuesta de factores que aparezcan tanto en

sin (\frac{- 90 0 ^{\circ} + 54 x}{18})

sin (x)

= sin (\frac{54 x - 90 0 ^{\circ}}{18}) sin (x)

Reescribir las fracciones basandose en el mínimo común denominador

Multiplicar cada numerador por la misma cantidad necesaria para multiplicar el denominador correspondiente y convertirlo en el mínimo común denominador

Para

\frac{cos ( \frac{- 90 0 ^{\circ} + 54 x}{18} )}{sin ( \frac{- 90 0 ^{\circ} + 54 x}{18} )} :

multiplicar el denominador y el numerador por

sin (x)

\frac{cos ( \frac{- 90 0 ^{\circ} + 54 x}{18} )}{sin ( \frac{- 90 0 ^{\circ} + 54 x}{18} )} = \frac{cos ( \frac{- 90 0 ^{\circ} + 54 x}{18} ) sin ( x )}{sin ( \frac{- 90 0 ^{\circ} + 54 x}{18} ) sin ( x )}

Para

\frac{cos ( x )}{sin ( x )} :

multiplicar el denominador y el numerador por

sin (\frac{54 x - 90 0 ^{\circ}}{18})

\frac{cos ( x )}{sin ( x )} = \frac{cos ( x ) sin ( \frac{54 x - 90 0 ^{\circ}}{18} )}{sin ( x ) sin ( \frac{54 x - 90 0 ^{\circ}}{18} )}

= - \frac{cos ( \frac{- 90 0 ^{\circ} + 54 x}{18} ) sin ( x )}{sin ( \frac{- 90 0 ^{\circ} + 54 x}{18} ) sin ( x )} + \frac{cos ( x ) sin ( \frac{54 x - 90 0 ^{\circ}}{18} )}{sin ( x ) sin ( \frac{54 x - 90 0 ^{\circ}}{18} )}

Ya que los denominadores son iguales, combinar las fracciones:

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

= \frac{- cos ( \frac{- 90 0 ^{\circ} + 54 x}{18} ) sin ( x ) + cos ( x ) sin ( \frac{54 x - 90 0 ^{\circ}}{18} )}{sin ( \frac{54 x - 90 0 ^{\circ}}{18} ) sin ( x )}

= \frac{- cos ( \frac{- 90 0 ^{\circ} + 54 x}{18} ) sin ( x ) + cos ( x ) sin ( \frac{54 x - 90 0 ^{\circ}}{18} )}{sin ( \frac{54 x - 90 0 ^{\circ}}{18} ) sin ( x )}

\frac{- cos ( \frac{54 x - 90 0 ^{\circ}}{18} ) sin ( x ) + cos ( x ) sin ( \frac{54 x - 90 0 ^{\circ}}{18} )}{sin ( \frac{54 x - 90 0 ^{\circ}}{18} ) sin ( x )} = 0

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

- cos (\frac{54 x - 90 0 ^{\circ}}{18}) sin (x) + cos (x) sin (\frac{54 x - 90 0 ^{\circ}}{18}) = 0

Re-escribir usando identidades trigonométricas

- cos (\frac{54 x - 90 0 ^{\circ}}{18}) sin (x) + cos (x) sin (\frac{54 x - 90 0 ^{\circ}}{18})

Utilizar la identidad de diferencia de ángulos:

sin (s) cos (t) - cos (s) sin (t) = sin (s - t)

= sin (\frac{54 x - 90 0 ^{\circ}}{18} - x)

sin (\frac{54 x - 90 0 ^{\circ}}{18} - x) = 0

Soluciones generales para

sin (\frac{54 x - 90 0 ^{\circ}}{18} - x) = 0

tabla de valores periódicos con

36 0^{\circ} n

intervalos:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

\frac{54 x - 90 0 ^{\circ}}{18} - x = 0 + 36 0^{\circ} n, \frac{54 x - 90 0 ^{\circ}}{18} - x = 18 0^{\circ} + 36 0^{\circ} n

\frac{54 x - 90 0 ^{\circ}}{18} - x = 0 + 36 0^{\circ} n, \frac{54 x - 90 0 ^{\circ}}{18} - x = 18 0^{\circ} + 36 0^{\circ} n

Resolver

\frac{54 x - 90 0 ^{\circ}}{18} - x = 0 + 36 0^{\circ} n : x = 18 0^{\circ} n + 2 5^{\circ}

\frac{54 x - 90 0 ^{\circ}}{18} - x = 0 + 36 0^{\circ} n

0 + 36 0^{\circ} n = 36 0^{\circ} n

\frac{54 x - 90 0 ^{\circ}}{18} - x = 36 0^{\circ} n

Multiplicar ambos lados por

18

\frac{54 x - 90 0 ^{\circ}}{18} - x = 36 0^{\circ} n

Multiplicar ambos lados por

18

\frac{54 x - 90 0 ^{\circ}}{18} \cdot 18 - x \cdot 18 = 36 0^{\circ} n \cdot 18

Simplificar

\frac{54 x - 90 0 ^{\circ}}{18} \cdot 18 - x \cdot 18 = 36 0^{\circ} n \cdot 18

Simplificar

\frac{54 x - 90 0 ^{\circ}}{18} \cdot 18 : 54 x - 90 0^{\circ}

\frac{54 x - 90 0 ^{\circ}}{18} \cdot 18

Multiplicar fracciones:

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

= \frac{( 54 x - 90 0 ^{\circ} ) \cdot 18}{18}

Eliminar los terminos comunes:

18

= 54 x - 90 0^{\circ}

Simplificar

x \cdot 18 : 18 x

x \cdot 18

Aplicar regla conmutativa:

x \cdot 18 = 18 x

18 x

Simplificar

36 0^{\circ} n \cdot 18 : 648 0^{\circ} n

36 0^{\circ} n \cdot 18

Multiplicar los numeros:

2 \cdot 18 = 36

= 648 0^{\circ} n

54 x - 90 0^{\circ} - 18 x = 648 0^{\circ} n

36 x - 90 0^{\circ} = 648 0^{\circ} n

36 x - 90 0^{\circ} = 648 0^{\circ} n

36 x - 90 0^{\circ} = 648 0^{\circ} n

Mover

90 0^{\circ}

al lado derecho

36 x - 90 0^{\circ} = 648 0^{\circ} n

Sumar

90 0^{\circ}

a ambos lados

36 x - 90 0^{\circ} + 90 0^{\circ} = 648 0^{\circ} n + 90 0^{\circ}

Simplificar

36 x = 648 0^{\circ} n + 90 0^{\circ}

36 x = 648 0^{\circ} n + 90 0^{\circ}

Dividir ambos lados entre

36

36 x = 648 0^{\circ} n + 90 0^{\circ}

Dividir ambos lados entre

36

\frac{36 x}{36} = \frac{648 0 ^{\circ} n}{36} + 2 5^{\circ}

Simplificar

x = 18 0^{\circ} n + 2 5^{\circ}

x = 18 0^{\circ} n + 2 5^{\circ}

Resolver

\frac{54 x - 90 0 ^{\circ}}{18} - x = 18 0^{\circ} + 36 0^{\circ} n : x = 11 5^{\circ} + 18 0^{\circ} n

\frac{54 x - 90 0 ^{\circ}}{18} - x = 18 0^{\circ} + 36 0^{\circ} n

Multiplicar ambos lados por

18

\frac{54 x - 90 0 ^{\circ}}{18} - x = 18 0^{\circ} + 36 0^{\circ} n

Multiplicar ambos lados por

18

\frac{54 x - 90 0 ^{\circ}}{18} \cdot 18 - x \cdot 18 = 18 0^{\circ} 18 + 36 0^{\circ} n \cdot 18

Simplificar

\frac{54 x - 90 0 ^{\circ}}{18} \cdot 18 - x \cdot 18 = 18 0^{\circ} 18 + 36 0^{\circ} n \cdot 18

Simplificar

\frac{54 x - 90 0 ^{\circ}}{18} \cdot 18 : 54 x - 90 0^{\circ}

\frac{54 x - 90 0 ^{\circ}}{18} \cdot 18

Multiplicar fracciones:

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

= \frac{( 54 x - 90 0 ^{\circ} ) \cdot 18}{18}

Eliminar los terminos comunes:

18

= 54 x - 90 0^{\circ}

Simplificar

x \cdot 18 : 18 x

x \cdot 18

Aplicar regla conmutativa:

x \cdot 18 = 18 x

18 x

Simplificar

18 0^{\circ} 18 : 324 0^{\circ}

18 0^{\circ} 18

Aplicar regla conmutativa:

18 0^{\circ} 18 = 324 0^{\circ}

324 0^{\circ}

Simplificar

36 0^{\circ} n \cdot 18 : 648 0^{\circ} n

36 0^{\circ} n \cdot 18

Multiplicar los numeros:

2 \cdot 18 = 36

= 648 0^{\circ} n

54 x - 90 0^{\circ} - 18 x = 324 0^{\circ} + 648 0^{\circ} n

36 x - 90 0^{\circ} = 324 0^{\circ} + 648 0^{\circ} n

36 x - 90 0^{\circ} = 324 0^{\circ} + 648 0^{\circ} n

36 x - 90 0^{\circ} = 324 0^{\circ} + 648 0^{\circ} n

Mover

90 0^{\circ}

al lado derecho

36 x - 90 0^{\circ} = 324 0^{\circ} + 648 0^{\circ} n

Sumar

90 0^{\circ}

a ambos lados

36 x - 90 0^{\circ} + 90 0^{\circ} = 324 0^{\circ} + 648 0^{\circ} n + 90 0^{\circ}

Simplificar

36 x = 414 0^{\circ} + 648 0^{\circ} n

36 x = 414 0^{\circ} + 648 0^{\circ} n

Dividir ambos lados entre

36

36 x = 414 0^{\circ} + 648 0^{\circ} n

Dividir ambos lados entre

36

\frac{36 x}{36} = 11 5^{\circ} + \frac{648 0 ^{\circ} n}{36}

Simplificar

x = 11 5^{\circ} + 18 0^{\circ} n

x = 11 5^{\circ} + 18 0^{\circ} n

x = 18 0^{\circ} n + 2 5^{\circ}, x = 11 5^{\circ} + 18 0^{\circ} n

Soluciones para el rango

0 < x < 18 0^{\circ}

x = 2 5^{\circ}, x = 11 5^{\circ}

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