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tan(x-45)-tan(x+45)=4

Solución

tan (x - 4 5^{\circ}) - tan (x + 4 5^{\circ}) = 4

Solución

x = 12 0^{\circ} + 18 0^{\circ} n, x = 6 0^{\circ} + 18 0^{\circ} n

radianes

x = \frac{2 π}{3} + πn, x = \frac{π}{3} + πn

Pasos de solución

tan (x - 4 5^{\circ}) - tan (x + 4 5^{\circ}) = 4

Re-escribir usando identidades trigonométricas

tan (x - 4 5^{\circ}) - tan (x + 4 5^{\circ}) = 4

Re-escribir usando identidades trigonométricas

tan (x - 4 5^{\circ})

Utilizar la identidad trigonométrica básica:

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

= \frac{sin ( x - 4 5 ^{\circ} )}{cos ( x - 4 5 ^{\circ} )}

Utilizar la identidad de diferencia de ángulos:

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

= \frac{sin ( x ) cos ( 4 5 ^{\circ} ) - cos ( x ) sin ( 4 5 ^{\circ} )}{cos ( x - 4 5 ^{\circ} )}

Utilizar la identidad de diferencia de ángulos:

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

= \frac{sin ( x ) cos ( 4 5 ^{\circ} ) - cos ( x ) sin ( 4 5 ^{\circ} )}{cos ( x ) cos ( 4 5 ^{\circ} ) + sin ( x ) sin ( 4 5 ^{\circ} )}

Simplificar

\frac{sin ( x ) cos ( 4 5 ^{\circ} ) - cos ( x ) sin ( 4 5 ^{\circ} )}{cos ( x ) cos ( 4 5 ^{\circ} ) + sin ( x ) sin ( 4 5 ^{\circ} )} : \frac{sin ( x ) - cos ( x )}{cos ( x ) + sin ( x )}

\frac{sin ( x ) cos ( 4 5 ^{\circ} ) - cos ( x ) sin ( 4 5 ^{\circ} )}{cos ( x ) cos ( 4 5 ^{\circ} ) + sin ( x ) sin ( 4 5 ^{\circ} )}

sin (x) cos (4 5^{\circ}) - cos (x) sin (4 5^{\circ}) = \frac{2}{2} sin (x) - \frac{2}{2} cos (x)

sin (x) cos (4 5^{\circ}) - cos (x) sin (4 5^{\circ})

Simplificar

cos (4 5^{\circ}) : \frac{2}{2}

cos (4 5^{\circ})

Utilizar la siguiente identidad trivial:

cos (4 5^{\circ}) = \frac{2}{2}

cos (x)

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

= \frac{2}{2}

= \frac{2}{2} sin (x) - sin (4 5^{\circ}) cos (x)

Simplificar

sin (4 5^{\circ}) : \frac{2}{2}

sin (4 5^{\circ})

Utilizar la siguiente identidad trivial:

sin (4 5^{\circ}) = \frac{2}{2}

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{2}{2}

= \frac{2}{2} sin (x) - \frac{2}{2} cos (x)

= \frac{\frac{2}{2} sin ( x ) - \frac{2}{2} cos ( x )}{cos ( 4 5 ^{\circ} ) cos ( x ) + sin ( 4 5 ^{\circ} ) sin ( x )}

cos (x) cos (4 5^{\circ}) + sin (x) sin (4 5^{\circ}) = \frac{2}{2} cos (x) + \frac{2}{2} sin (x)

cos (x) cos (4 5^{\circ}) + sin (x) sin (4 5^{\circ})

Simplificar

cos (4 5^{\circ}) : \frac{2}{2}

cos (4 5^{\circ})

Utilizar la siguiente identidad trivial:

cos (4 5^{\circ}) = \frac{2}{2}

cos (x)

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

= \frac{2}{2}

= \frac{2}{2} cos (x) + sin (4 5^{\circ}) sin (x)

Simplificar

sin (4 5^{\circ}) : \frac{2}{2}

sin (4 5^{\circ})

Utilizar la siguiente identidad trivial:

sin (4 5^{\circ}) = \frac{2}{2}

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{2}{2}

= \frac{2}{2} cos (x) + \frac{2}{2} sin (x)

= \frac{\frac{2}{2} sin ( x ) - \frac{2}{2} cos ( x )}{\frac{2}{2} cos ( x ) + \frac{2}{2} sin ( x )}

Multiplicar

cos (x) \frac{2}{2} : \frac{2 cos ( x )}{2}

cos (x) \frac{2}{2}

Multiplicar fracciones:

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

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

= \frac{\frac{2}{2} sin ( x ) - \frac{2}{2} cos ( x )}{\frac{2 c o s ( x )}{2} + \frac{2}{2} sin ( x )}

Multiplicar

sin (x) \frac{2}{2} : \frac{2 sin ( x )}{2}

sin (x) \frac{2}{2}

Multiplicar fracciones:

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

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

= \frac{\frac{2}{2} sin ( x ) - \frac{2}{2} cos ( x )}{\frac{2 c o s ( x )}{2} + \frac{2 s i n ( x )}{2}}

Multiplicar

sin (x) \frac{2}{2} : \frac{2 sin ( x )}{2}

sin (x) \frac{2}{2}

Multiplicar fracciones:

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

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

= \frac{\frac{2 s i n ( x )}{2} - \frac{2}{2} cos ( x )}{\frac{2 c o s ( x )}{2} + \frac{2 s i n ( x )}{2}}

Multiplicar

cos (x) \frac{2}{2} : \frac{2 cos ( x )}{2}

cos (x) \frac{2}{2}

Multiplicar fracciones:

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

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

= \frac{\frac{2 s i n ( x )}{2} - \frac{2 c o s ( x )}{2}}{\frac{2 c o s ( x )}{2} + \frac{2 s i n ( x )}{2}}

Combinar las fracciones usando el mínimo común denominador:

\frac{2 cos ( x ) + 2 sin ( x )}{2}

Aplicar la regla

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

= \frac{2 cos ( x ) + 2 sin ( x )}{2}

= \frac{\frac{2 s i n ( x )}{2} - \frac{2 c o s ( x )}{2}}{\frac{2 c o s ( x ) + 2 s i n ( x )}{2}}

Combinar las fracciones usando el mínimo común denominador:

\frac{2 sin ( x ) - 2 cos ( x )}{2}

Aplicar la regla

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

= \frac{2 sin ( x ) - 2 cos ( x )}{2}

= \frac{\frac{2 s i n ( x ) - 2 c o s ( x )}{2}}{\frac{2 c o s ( x ) + 2 s i n ( x )}{2}}

Dividir fracciones:

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

= \frac{( 2 sin ( x ) - 2 cos ( x ) ) \cdot 2}{2 ( 2 cos ( x ) + 2 sin ( x ) )}

Eliminar los terminos comunes:

2

= \frac{2 sin ( x ) - 2 cos ( x )}{2 cos ( x ) + 2 sin ( x )}

Factorizar el termino común

2

= \frac{2 ( sin ( x ) - cos ( x ) )}{2 cos ( x ) + 2 sin ( x )}

Factorizar el termino común

2

= \frac{2 ( sin ( x ) - cos ( x ) )}{2 ( cos ( x ) + sin ( x ) )}

Eliminar los terminos comunes:

2

= \frac{sin ( x ) - cos ( x )}{cos ( x ) + sin ( x )}

= \frac{sin ( x ) - cos ( x )}{cos ( x ) + sin ( x )}

Utilizar la identidad trigonométrica básica:

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

= \frac{sin ( x + 4 5 ^{\circ} )}{cos ( x + 4 5 ^{\circ} )}

Utilizar la identidad de suma de ángulos:

sin (s + t) = sin (s) cos (t) + cos (s) sin (t)

= \frac{sin ( x ) cos ( 4 5 ^{\circ} ) + cos ( x ) sin ( 4 5 ^{\circ} )}{cos ( x + 4 5 ^{\circ} )}

Utilizar la identidad de suma de ángulos:

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

= \frac{sin ( x ) cos ( 4 5 ^{\circ} ) + cos ( x ) sin ( 4 5 ^{\circ} )}{cos ( x ) cos ( 4 5 ^{\circ} ) - sin ( x ) sin ( 4 5 ^{\circ} )}

Simplificar

\frac{sin ( x ) cos ( 4 5 ^{\circ} ) + cos ( x ) sin ( 4 5 ^{\circ} )}{cos ( x ) cos ( 4 5 ^{\circ} ) - sin ( x ) sin ( 4 5 ^{\circ} )} : \frac{sin ( x ) + cos ( x )}{cos ( x ) - sin ( x )}

\frac{sin ( x ) cos ( 4 5 ^{\circ} ) + cos ( x ) sin ( 4 5 ^{\circ} )}{cos ( x ) cos ( 4 5 ^{\circ} ) - sin ( x ) sin ( 4 5 ^{\circ} )}

sin (x) cos (4 5^{\circ}) + cos (x) sin (4 5^{\circ}) = \frac{2}{2} sin (x) + \frac{2}{2} cos (x)

sin (x) cos (4 5^{\circ}) + cos (x) sin (4 5^{\circ})

Simplificar

cos (4 5^{\circ}) : \frac{2}{2}

cos (4 5^{\circ})

Utilizar la siguiente identidad trivial:

cos (4 5^{\circ}) = \frac{2}{2}

cos (x)

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

= \frac{2}{2}

= \frac{2}{2} sin (x) + sin (4 5^{\circ}) cos (x)

Simplificar

sin (4 5^{\circ}) : \frac{2}{2}

sin (4 5^{\circ})

Utilizar la siguiente identidad trivial:

sin (4 5^{\circ}) = \frac{2}{2}

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{2}{2}

= \frac{2}{2} sin (x) + \frac{2}{2} cos (x)

= \frac{\frac{2}{2} sin ( x ) + \frac{2}{2} cos ( x )}{cos ( 4 5 ^{\circ} ) cos ( x ) - sin ( 4 5 ^{\circ} ) sin ( x )}

cos (x) cos (4 5^{\circ}) - sin (x) sin (4 5^{\circ}) = \frac{2}{2} cos (x) - \frac{2}{2} sin (x)

cos (x) cos (4 5^{\circ}) - sin (x) sin (4 5^{\circ})

Simplificar

cos (4 5^{\circ}) : \frac{2}{2}

cos (4 5^{\circ})

Utilizar la siguiente identidad trivial:

cos (4 5^{\circ}) = \frac{2}{2}

cos (x)

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

= \frac{2}{2}

= \frac{2}{2} cos (x) - sin (4 5^{\circ}) sin (x)

Simplificar

sin (4 5^{\circ}) : \frac{2}{2}

sin (4 5^{\circ})

Utilizar la siguiente identidad trivial:

sin (4 5^{\circ}) = \frac{2}{2}

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{2}{2}

= \frac{2}{2} cos (x) - \frac{2}{2} sin (x)

= \frac{\frac{2}{2} sin ( x ) + \frac{2}{2} cos ( x )}{\frac{2}{2} cos ( x ) - \frac{2}{2} sin ( x )}

Multiplicar

cos (x) \frac{2}{2} : \frac{2 cos ( x )}{2}

cos (x) \frac{2}{2}

Multiplicar fracciones:

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

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

= \frac{\frac{2}{2} sin ( x ) + \frac{2}{2} cos ( x )}{\frac{2 c o s ( x )}{2} - \frac{2}{2} sin ( x )}

Multiplicar

sin (x) \frac{2}{2} : \frac{2 sin ( x )}{2}

sin (x) \frac{2}{2}

Multiplicar fracciones:

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

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

= \frac{\frac{2}{2} sin ( x ) + \frac{2}{2} cos ( x )}{\frac{2 c o s ( x )}{2} - \frac{2 s i n ( x )}{2}}

Multiplicar

sin (x) \frac{2}{2} : \frac{2 sin ( x )}{2}

sin (x) \frac{2}{2}

Multiplicar fracciones:

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

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

= \frac{\frac{2 s i n ( x )}{2} + \frac{2}{2} cos ( x )}{\frac{2 c o s ( x )}{2} - \frac{2 s i n ( x )}{2}}

Multiplicar

cos (x) \frac{2}{2} : \frac{2 cos ( x )}{2}

cos (x) \frac{2}{2}

Multiplicar fracciones:

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

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

= \frac{\frac{2 s i n ( x )}{2} + \frac{2 c o s ( x )}{2}}{\frac{2 c o s ( x )}{2} - \frac{2 s i n ( x )}{2}}

Combinar las fracciones usando el mínimo común denominador:

\frac{2 cos ( x ) - 2 sin ( x )}{2}

Aplicar la regla

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

= \frac{2 cos ( x ) - 2 sin ( x )}{2}

= \frac{\frac{2 s i n ( x )}{2} + \frac{2 c o s ( x )}{2}}{\frac{2 c o s ( x ) - 2 s i n ( x )}{2}}

Combinar las fracciones usando el mínimo común denominador:

\frac{2 sin ( x ) + 2 cos ( x )}{2}

Aplicar la regla

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

= \frac{2 sin ( x ) + 2 cos ( x )}{2}

= \frac{\frac{2 s i n ( x ) + 2 c o s ( x )}{2}}{\frac{2 c o s ( x ) - 2 s i n ( x )}{2}}

Dividir fracciones:

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

= \frac{( 2 sin ( x ) + 2 cos ( x ) ) \cdot 2}{2 ( 2 cos ( x ) - 2 sin ( x ) )}

Eliminar los terminos comunes:

2

= \frac{2 sin ( x ) + 2 cos ( x )}{2 cos ( x ) - 2 sin ( x )}

Factorizar el termino común

2

= \frac{2 ( sin ( x ) + cos ( x ) )}{2 cos ( x ) - 2 sin ( x )}

Factorizar el termino común

2

= \frac{2 ( sin ( x ) + cos ( x ) )}{2 ( cos ( x ) - sin ( x ) )}

Eliminar los terminos comunes:

2

= \frac{sin ( x ) + cos ( x )}{cos ( x ) - sin ( x )}

= \frac{sin ( x ) + cos ( x )}{cos ( x ) - sin ( x )}

\frac{sin ( x ) - cos ( x )}{cos ( x ) + sin ( x )} - \frac{sin ( x ) + cos ( x )}{cos ( x ) - sin ( x )} = 4

Simplificar

\frac{sin ( x ) - cos ( x )}{cos ( x ) + sin ( x )} - \frac{sin ( x ) + cos ( x )}{cos ( x ) - sin ( x )} : \frac{- 2 sin ^{2} ( x ) - 2 cos ^{2} ( x )}{( cos ( x ) + sin ( x ) ) ( cos ( x ) - sin ( x ) )}

\frac{sin ( x ) - cos ( x )}{cos ( x ) + sin ( x )} - \frac{sin ( x ) + cos ( x )}{cos ( x ) - sin ( x )}

Mínimo común múltiplo de

cos (x) + sin (x), cos (x) - sin (x) : (cos (x) + sin (x)) (cos (x) - sin (x))

cos (x) + sin (x), cos (x) - sin (x)

Mínimo común múltiplo (MCM)

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

cos (x) + sin (x)

cos (x) - sin (x)

= (cos (x) + sin (x)) (cos (x) - 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{sin ( x ) - cos ( x )}{cos ( x ) + sin ( x )} :

multiplicar el denominador y el numerador por

cos (x) - sin (x)

\frac{sin ( x ) - cos ( x )}{cos ( x ) + sin ( x )} = \frac{( sin ( x ) - cos ( x ) ) ( cos ( x ) - sin ( x ) )}{( cos ( x ) + sin ( x ) ) ( cos ( x ) - sin ( x ) )}

Para

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

multiplicar el denominador y el numerador por

cos (x) + sin (x)

\frac{sin ( x ) + cos ( x )}{cos ( x ) - sin ( x )} = \frac{( sin ( x ) + cos ( x ) ) ( cos ( x ) + sin ( x ) )}{( cos ( x ) - sin ( x ) ) ( cos ( x ) + sin ( x ) )} = \frac{( sin ( x ) + cos ( x ) ) ^{2}}{( cos ( x ) + sin ( x ) ) ( cos ( x ) - sin ( x ) )}

= \frac{( sin ( x ) - cos ( x ) ) ( cos ( x ) - sin ( x ) )}{( cos ( x ) + sin ( x ) ) ( cos ( x ) - sin ( x ) )} - \frac{( sin ( x ) + cos ( x ) ) ^{2}}{( cos ( x ) + sin ( x ) ) ( cos ( x ) - sin ( x ) )}

Ya que los denominadores son iguales, combinar las fracciones:

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

= \frac{( sin ( x ) - cos ( x ) ) ( cos ( x ) - sin ( x ) ) - ( sin ( x ) + cos ( x ) ) ^{2}}{( cos ( x ) + sin ( x ) ) ( cos ( x ) - sin ( x ) )}

Expandir

(sin (x) - cos (x)) (cos (x) - sin (x)) - (sin (x) + cos (x))^{2} : - 2 sin^{2} (x) - 2 cos^{2} (x)

(sin (x) - cos (x)) (cos (x) - sin (x)) - (sin (x) + cos (x))^{2}

(sin (x) + cos (x))^{2} : sin^{2} (x) + 2 sin (x) cos (x) + cos^{2} (x)

Aplicar la formula del binomio al cuadrado:

(a + b)^{2} = a^{2} + 2 ab + b^{2}

a = sin (x), b = cos (x)

= sin^{2} (x) + 2 sin (x) cos (x) + cos^{2} (x)

= (sin (x) - cos (x)) (cos (x) - sin (x)) - (sin^{2} (x) + 2 sin (x) cos (x) + cos^{2} (x))

Expandir

(sin (x) - cos (x)) (cos (x) - sin (x)) : 2 cos (x) sin (x) - sin^{2} (x) - cos^{2} (x)

(sin (x) - cos (x)) (cos (x) - sin (x))

Aplicar la propiedad distributiva:

(a + b) (c + d) = a c + a d + b c + b d

a = sin (x), b = - cos (x), c = cos (x), d = - sin (x)

= sin (x) cos (x) + sin (x) (- sin (x)) + (- cos (x)) cos (x) + (- cos (x)) (- sin (x))

Aplicar las reglas de los signos

+ (- a) = - a, (- a) (- b) = ab

= sin (x) cos (x) - sin (x) sin (x) - cos (x) cos (x) + cos (x) sin (x)

Simplificar

sin (x) cos (x) - sin (x) sin (x) - cos (x) cos (x) + cos (x) sin (x) : 2 cos (x) sin (x) - sin^{2} (x) - cos^{2} (x)

sin (x) cos (x) - sin (x) sin (x) - cos (x) cos (x) + cos (x) sin (x)

Sumar elementos similares:

sin (x) cos (x) + cos (x) sin (x) = 2 cos (x) sin (x)

= 2 cos (x) sin (x) - sin (x) sin (x) - cos (x) cos (x)

sin (x) sin (x) = sin^{2} (x)

sin (x) sin (x)

Aplicar las leyes de los exponentes:

a^{b} \cdot a^{c} = a^{b + c}

sin (x) sin (x) = sin^{1 + 1} (x)

= sin^{1 + 1} (x)

Sumar:

1 + 1 = 2

= sin^{2} (x)

cos (x) cos (x) = cos^{2} (x)

cos (x) cos (x)

Aplicar las leyes de los exponentes:

a^{b} \cdot a^{c} = a^{b + c}

cos (x) cos (x) = cos^{1 + 1} (x)

= cos^{1 + 1} (x)

Sumar:

1 + 1 = 2

= cos^{2} (x)

= 2 cos (x) sin (x) - sin^{2} (x) - cos^{2} (x)

= 2 cos (x) sin (x) - sin^{2} (x) - cos^{2} (x)

= 2 cos (x) sin (x) - sin^{2} (x) - cos^{2} (x) - (sin^{2} (x) + 2 sin (x) cos (x) + cos^{2} (x))

- (sin^{2} (x) + 2 sin (x) cos (x) + cos^{2} (x)) : - sin^{2} (x) - 2 sin (x) cos (x) - cos^{2} (x)

- (sin^{2} (x) + 2 sin (x) cos (x) + cos^{2} (x))

Poner los parentesis

= - (sin^{2} (x)) - (2 sin (x) cos (x)) - (cos^{2} (x))

Aplicar las reglas de los signos

+ (- a) = - a

= - sin^{2} (x) - 2 sin (x) cos (x) - cos^{2} (x)

= 2 cos (x) sin (x) - sin^{2} (x) - cos^{2} (x) - sin^{2} (x) - 2 sin (x) cos (x) - cos^{2} (x)

Simplificar

2 cos (x) sin (x) - sin^{2} (x) - cos^{2} (x) - sin^{2} (x) - 2 sin (x) cos (x) - cos^{2} (x) : - 2 sin^{2} (x) - 2 cos^{2} (x)

2 cos (x) sin (x) - sin^{2} (x) - cos^{2} (x) - sin^{2} (x) - 2 sin (x) cos (x) - cos^{2} (x)

Sumar elementos similares:

2 cos (x) sin (x) - 2 sin (x) cos (x) = 0

= - sin^{2} (x) - cos^{2} (x) - sin^{2} (x) - cos^{2} (x)

Sumar elementos similares:

- cos^{2} (x) - cos^{2} (x) = - 2 cos^{2} (x)

= - sin^{2} (x) - 2 cos^{2} (x) - sin^{2} (x)

Sumar elementos similares:

- sin^{2} (x) - sin^{2} (x) = - 2 sin^{2} (x)

= - 2 sin^{2} (x) - 2 cos^{2} (x)

= - 2 sin^{2} (x) - 2 cos^{2} (x)

= \frac{- 2 sin ^{2} ( x ) - 2 cos ^{2} ( x )}{( cos ( x ) + sin ( x ) ) ( cos ( x ) - sin ( x ) )}

\frac{- 2 sin ^{2} ( x ) - 2 cos ^{2} ( x )}{( cos ( x ) + sin ( x ) ) ( cos ( x ) - sin ( x ) )} = 4

\frac{- 2 sin ^{2} ( x ) - 2 cos ^{2} ( x )}{( cos ( x ) + sin ( x ) ) ( cos ( x ) - sin ( x ) )} = 4

Restar

4

de ambos lados

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

Simplificar

\frac{- 2 sin ^{2} ( x ) - 2 cos ^{2} ( x )}{( cos ( x ) + sin ( x ) ) ( cos ( x ) - sin ( x ) )} - 4 : \frac{2 sin ^{2} ( x ) - 6 cos ^{2} ( x )}{( cos ( x ) + sin ( x ) ) ( cos ( x ) - sin ( x ) )}

\frac{- 2 sin ^{2} ( x ) - 2 cos ^{2} ( x )}{( cos ( x ) + sin ( x ) ) ( cos ( x ) - sin ( x ) )} - 4

Convertir a fracción:

4 = \frac{4 ( cos ( x ) + sin ( x ) ) ( cos ( x ) - sin ( x ) )}{( cos ( x ) + sin ( x ) ) ( cos ( x ) - sin ( x ) )}

= \frac{- 2 sin ^{2} ( x ) - 2 cos ^{2} ( x )}{( cos ( x ) + sin ( x ) ) ( cos ( x ) - sin ( x ) )} - \frac{4 ( cos ( x ) + sin ( x ) ) ( cos ( x ) - sin ( x ) )}{( cos ( x ) + sin ( x ) ) ( cos ( x ) - sin ( x ) )}

Ya que los denominadores son iguales, combinar las fracciones:

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

= \frac{- 2 sin ^{2} ( x ) - 2 cos ^{2} ( x ) - 4 ( cos ( x ) + sin ( x ) ) ( cos ( x ) - sin ( x ) )}{( cos ( x ) + sin ( x ) ) ( cos ( x ) - sin ( x ) )}

Expandir

- 2 sin^{2} (x) - 2 cos^{2} (x) - 4 (cos (x) + sin (x)) (cos (x) - sin (x)) : 2 sin^{2} (x) - 6 cos^{2} (x)

- 2 sin^{2} (x) - 2 cos^{2} (x) - 4 (cos (x) + sin (x)) (cos (x) - sin (x))

Expandir

- 4 (cos (x) + sin (x)) (cos (x) - sin (x)) : - 4 cos^{2} (x) + 4 sin^{2} (x)

Expandir

(cos (x) + sin (x)) (cos (x) - sin (x)) : cos^{2} (x) - sin^{2} (x)

(cos (x) + sin (x)) (cos (x) - sin (x))

Aplicar la siguiente regla para binomios al cuadrado:

(a + b) (a - b) = a^{2} - b^{2}

a = cos (x), b = sin (x)

= cos^{2} (x) - sin^{2} (x)

= - 4 (cos^{2} (x) - sin^{2} (x))

Expandir

- 4 (cos^{2} (x) - sin^{2} (x)) : - 4 cos^{2} (x) + 4 sin^{2} (x)

- 4 (cos^{2} (x) - sin^{2} (x))

Poner los parentesis utilizando:

a (b - c) = ab - a c

a = - 4, b = cos^{2} (x), c = sin^{2} (x)

= - 4 cos^{2} (x) - (- 4) sin^{2} (x)

Aplicar las reglas de los signos

- (- a) = a

= - 4 cos^{2} (x) + 4 sin^{2} (x)

= - 4 cos^{2} (x) + 4 sin^{2} (x)

= - 2 sin^{2} (x) - 2 cos^{2} (x) - 4 cos^{2} (x) + 4 sin^{2} (x)

Simplificar

- 2 sin^{2} (x) - 2 cos^{2} (x) - 4 cos^{2} (x) + 4 sin^{2} (x) : 2 sin^{2} (x) - 6 cos^{2} (x)

- 2 sin^{2} (x) - 2 cos^{2} (x) - 4 cos^{2} (x) + 4 sin^{2} (x)

Sumar elementos similares:

- 2 cos^{2} (x) - 4 cos^{2} (x) = - 6 cos^{2} (x)

= - 2 sin^{2} (x) - 6 cos^{2} (x) + 4 sin^{2} (x)

Sumar elementos similares:

- 2 sin^{2} (x) + 4 sin^{2} (x) = 2 sin^{2} (x)

= 2 sin^{2} (x) - 6 cos^{2} (x)

= 2 sin^{2} (x) - 6 cos^{2} (x)

= \frac{2 sin ^{2} ( x ) - 6 cos ^{2} ( x )}{( cos ( x ) + sin ( x ) ) ( cos ( x ) - sin ( x ) )}

\frac{2 sin ^{2} ( x ) - 6 cos ^{2} ( x )}{( cos ( x ) + sin ( x ) ) ( cos ( x ) - sin ( x ) )} = 0

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

2 sin^{2} (x) - 6 cos^{2} (x) = 0

Factorizar

2 sin^{2} (x) - 6 cos^{2} (x) : 2 (sin (x) + 3 cos (x)) (sin (x) - 3 cos (x))

2 sin^{2} (x) - 6 cos^{2} (x)

Reescribir

- 6

como

3 \cdot 2

= 2 sin^{2} (x) + 3 \cdot 2 cos^{2} (x)

Factorizar el termino común

2

= 2 (sin^{2} (x) - 3 cos^{2} (x))

Factorizar

sin^{2} (x) - 3 cos^{2} (x) : (sin (x) + 3 cos (x)) (sin (x) - 3 cos (x))

sin^{2} (x) - 3 cos^{2} (x)

Reescribir

sin^{2} (x) - 3 cos^{2} (x)

como

sin^{2} (x) - (3 cos (x))^{2}

sin^{2} (x) - 3 cos^{2} (x)

Aplicar las leyes de los exponentes:

a = (a)^{2}

3 = (3)^{2}

= sin^{2} (x) - (3)^{2} cos^{2} (x)

Aplicar las leyes de los exponentes:

a^{m} b^{m} = (ab)^{m}

(3)^{2} cos^{2} (x) = (3 cos (x))^{2}

= sin^{2} (x) - (3 cos (x))^{2}

= sin^{2} (x) - (3 cos (x))^{2}

Aplicar la siguiente regla para binomios al cuadrado:

x^{2} - y^{2} = (x + y) (x - y)

sin^{2} (x) - (3 cos (x))^{2} = (sin (x) + 3 cos (x)) (sin (x) - 3 cos (x))

= (sin (x) + 3 cos (x)) (sin (x) - 3 cos (x))

= 2 (sin (x) + 3 cos (x)) (sin (x) - 3 cos (x))

2 (sin (x) + 3 cos (x)) (sin (x) - 3 cos (x)) = 0

Resolver cada parte por separado

sin (x) + 3 cos (x) = 0 or sin (x) - 3 cos (x) = 0

sin (x) + 3 cos (x) = 0 : x = 12 0^{\circ} + 18 0^{\circ} n

sin (x) + 3 cos (x) = 0

Re-escribir usando identidades trigonométricas

sin (x) + 3 cos (x) = 0

Dividir ambos lados entre

cos (x), cos (x) \neq = 0

\frac{sin ( x ) + 3 cos ( x )}{cos ( x )} = \frac{0}{cos ( x )}

Simplificar

\frac{sin ( x )}{cos ( x )} + 3 = 0

Utilizar la identidad trigonométrica básica:

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

tan (x) + 3 = 0

tan (x) + 3 = 0

Mover

3

al lado derecho

tan (x) + 3 = 0

Restar

3

de ambos lados

tan (x) + 3 - 3 = 0 - 3

Simplificar

tan (x) = - 3

tan (x) = - 3

Soluciones generales para

tan (x) = - 3

tan (x)

tabla de valores periódicos con

18 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} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

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

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

sin (x) - 3 cos (x) = 0 : x = 6 0^{\circ} + 18 0^{\circ} n

sin (x) - 3 cos (x) = 0

Re-escribir usando identidades trigonométricas

sin (x) - 3 cos (x) = 0

Dividir ambos lados entre

cos (x), cos (x) \neq = 0

\frac{sin ( x ) - 3 cos ( x )}{cos ( x )} = \frac{0}{cos ( x )}

Simplificar

\frac{sin ( x )}{cos ( x )} - 3 = 0

Utilizar la identidad trigonométrica básica:

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

tan (x) - 3 = 0

tan (x) - 3 = 0

Mover

3

al lado derecho

tan (x) - 3 = 0

Sumar

3

a ambos lados

tan (x) - 3 + 3 = 0 + 3

Simplificar

tan (x) = 3

tan (x) = 3

Soluciones generales para

tan (x) = 3

tan (x)

tabla de valores periódicos con

18 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} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

x = 6 0^{\circ} + 18 0^{\circ} n

x = 6 0^{\circ} + 18 0^{\circ} n

Combinar toda las soluciones

x = 12 0^{\circ} + 18 0^{\circ} n, x = 6 0^{\circ} + 18 0^{\circ} n

Ejemplos populares

sin(x)(2sin^2(x)-1)=0

sin (x) (2 sin^{2} (x) - 1) = 0

sin(2x)=10cos(x)

sin (2 x) = 10 cos (x)

-2cos(3θ)=1,0<= θ<= 2pi

- 2 cos (3 θ) = 1, 0 \leq θ \leq 2 π

cos(a)=0.998482

cos (a) = 0.998482

3-4sin(θ)=4-6sin(θ)

3 - 4 sin (θ) = 4 - 6 sin (θ)