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tan(pi/4-x)+tan(x)-1=0

Solución

tan (\frac{π}{4} - x) + tan (x) - 1 = 0

Solución

x = 2 πn, x = π + 2 πn, x = \frac{π}{4} + πn

Grados

x = 0^{\circ} + 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n, x = 4 5^{\circ} + 18 0^{\circ} n

Pasos de solución

tan (\frac{π}{4} - x) + tan (x) - 1 = 0

Re-escribir usando identidades trigonométricas

tan (\frac{π}{4} - x) + tan (x) - 1 = 0

Re-escribir usando identidades trigonométricas

tan (\frac{π}{4} - x)

Utilizar la identidad trigonométrica básica:

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

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

Utilizar la identidad de diferencia de ángulos:

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

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

Utilizar la identidad de diferencia de ángulos:

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

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

Simplificar

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

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

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

sin (\frac{π}{4}) cos (x) - cos (\frac{π}{4}) sin (x)

Simplificar

sin (\frac{π}{4}) : \frac{2}{2}

sin (\frac{π}{4})

Utilizar la siguiente identidad trivial:

sin (\frac{π}{4}) = \frac{2}{2}

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} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{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} 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) - cos (\frac{π}{4}) sin (x)

Simplificar

cos (\frac{π}{4}) : \frac{2}{2}

cos (\frac{π}{4})

Utilizar la siguiente identidad trivial:

cos (\frac{π}{4}) = \frac{2}{2}

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}

= \frac{2}{2}

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

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

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

cos (\frac{π}{4}) cos (x) + sin (\frac{π}{4}) sin (x)

Simplificar

cos (\frac{π}{4}) : \frac{2}{2}

cos (\frac{π}{4})

Utilizar la siguiente identidad trivial:

cos (\frac{π}{4}) = \frac{2}{2}

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}

= \frac{2}{2}

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

Simplificar

sin (\frac{π}{4}) : \frac{2}{2}

sin (\frac{π}{4})

Utilizar la siguiente identidad trivial:

sin (\frac{π}{4}) = \frac{2}{2}

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

Multiplicar

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

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

Multiplicar fracciones:

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

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

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

Multiplicar

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

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

Multiplicar fracciones:

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

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

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

Multiplicar

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

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

Multiplicar fracciones:

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

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

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

Multiplicar

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

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

Multiplicar fracciones:

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

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

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

Eliminar los terminos comunes:

2

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

Factorizar el termino común

2

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

Factorizar el termino común

2

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

Eliminar los terminos comunes:

2

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

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

\frac{cos ( x ) - sin ( x )}{cos ( x ) + sin ( x )} + tan (x) - 1 = 0

\frac{cos ( x ) - sin ( x )}{cos ( x ) + sin ( x )} + tan (x) - 1 = 0

Simplificar

\frac{cos ( x ) - sin ( x )}{cos ( x ) + sin ( x )} + tan (x) - 1 : \frac{- 2 sin ( x ) + tan ( x ) cos ( x ) + tan ( x ) sin ( x )}{cos ( x ) + sin ( x )}

\frac{cos ( x ) - sin ( x )}{cos ( x ) + sin ( x )} + tan (x) - 1

Convertir a fracción:

tan (x) = \frac{tan ( x ) ( cos ( x ) + sin ( x ) )}{cos ( x ) + sin ( x )}, 1 = \frac{1 ( cos ( x ) + sin ( x ) )}{cos ( x ) + sin ( x )}

= \frac{cos ( x ) - sin ( x )}{cos ( x ) + sin ( x )} + \frac{tan ( x ) ( cos ( x ) + sin ( x ) )}{cos ( x ) + sin ( x )} - \frac{1 \cdot ( 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{cos ( x ) - sin ( x ) + tan ( x ) ( cos ( x ) + sin ( x ) ) - 1 \cdot ( cos ( x ) + sin ( x ) )}{cos ( x ) + sin ( x )}

Multiplicar:

1 \cdot (cos (x) + sin (x)) = (cos (x) + sin (x))

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

Expandir

cos (x) - sin (x) + tan (x) (cos (x) + sin (x)) - (cos (x) + sin (x)) : - 2 sin (x) + tan (x) cos (x) + tan (x) sin (x)

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

Expandir

tan (x) (cos (x) + sin (x)) : tan (x) cos (x) + tan (x) sin (x)

tan (x) (cos (x) + sin (x))

Poner los parentesis utilizando:

a (b + c) = ab + a c

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

= tan (x) cos (x) + tan (x) sin (x)

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

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

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

Poner los parentesis

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

Aplicar las reglas de los signos

+ (- a) = - a

= - cos (x) - sin (x)

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

Simplificar

cos (x) - sin (x) + tan (x) cos (x) + tan (x) sin (x) - cos (x) - sin (x) : - 2 sin (x) + tan (x) cos (x) + tan (x) sin (x)

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

Sumar elementos similares:

cos (x) - cos (x) = 0

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

Sumar elementos similares:

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

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

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

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

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

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

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

Re-escribir usando identidades trigonométricas

- 2 sin (x) + cos (x) tan (x) + sin (x) tan (x)

cos (x) tan (x) = sin (x)

cos (x) tan (x)

Utilizar la identidad trigonométrica básica:

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

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

Simplificar

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

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

Multiplicar fracciones:

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

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

Eliminar los terminos comunes:

cos (x)

= sin (x)

= sin (x)

= - 2 sin (x) + sin (x) + sin (x) tan (x)

Simplificar

= - sin (x) + sin (x) tan (x)

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

Factorizar

- sin (x) + sin (x) tan (x) : sin (x) (tan (x) - 1)

- sin (x) + sin (x) tan (x)

Factorizar el termino común

sin (x)

= sin (x) (- 1 + tan (x))

sin (x) (tan (x) - 1) = 0

Resolver cada parte por separado

sin (x) = 0 or tan (x) - 1 = 0

sin (x) = 0 : x = 2 πn, x = π + 2 πn

sin (x) = 0

Soluciones generales para

sin (x) = 0

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

x = 0 + 2 πn, x = π + 2 πn

x = 0 + 2 πn, x = π + 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, x = π + 2 πn

tan (x) - 1 = 0 : x = \frac{π}{4} + πn

tan (x) - 1 = 0

Mover

1

al lado derecho

tan (x) - 1 = 0

Sumar

1

a ambos lados

tan (x) - 1 + 1 = 0 + 1

Simplificar

tan (x) = 1

tan (x) = 1

Soluciones generales para

tan (x) = 1

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

x = \frac{π}{4} + πn

x = \frac{π}{4} + πn

Combinar toda las soluciones

x = 2 πn, x = π + 2 πn, x = \frac{π}{4} + πn

Ejemplos populares

csc((2θ)/3)-1=0

csc (\frac{2 θ}{3}) - 1 = 0

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-0.5sin(t)=0,0<,=t<= 4pi

- 0.5 sin (t) = 0, 0 <, = t \leq 4 π

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

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

solvefor a,cos(ax)+(b/x)sin(ax)=0 resolver para

a, cos (a x) + (\frac{b}{x}) sin (a x) = 0