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Popular Trigonometria >

tan(45-x)+tan(x)=1

Solução

tan (4 5^{\circ} - x) + tan (x) = 1

Solução

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

Radianos

x = 0 + 2 πn, x = π + 2 πn, x = \frac{π}{4} + πn

Passos da solução

tan (4 5^{\circ} - x) + tan (x) = 1

Reeecreva usando identidades trigonométricas

tan (4 5^{\circ} - x) + tan (x) = 1

Reeecreva usando identidades trigonométricas

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

Utilizar a Identidade Básica da Trigonometria:

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

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

Use a identidade de diferença de ângulos:

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

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

Use a identidade de diferença de ângulos:

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

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

Simplificar

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

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

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

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

Simplificar

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

sin (4 5^{\circ})

Utilizar a seguinte identidade trivial:

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

sin (x)

tabela de periodicidade com ciclo de

36 0^{\circ} n

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

Simplificar

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

cos (4 5^{\circ})

Utilizar a seguinte identidade trivial:

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

cos (x)

tabela de periodicidade com ciclo de

36 0^{\circ} n

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

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

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

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

Simplificar

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

cos (4 5^{\circ})

Utilizar a seguinte identidade trivial:

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

cos (x)

tabela de periodicidade com ciclo de

36 0^{\circ} n

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 a seguinte identidade trivial:

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

sin (x)

tabela de periodicidade com ciclo de

36 0^{\circ} n

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} 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 frações:

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 frações:

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 frações:

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 frações:

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 as frações usando o mínimo múltiplo comum:

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

Aplicar a regra

\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 as frações usando o mínimo múltiplo comum:

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

Aplicar a regra

\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 frações:

\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 o fator comum:

2

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

Fatorar o termo comum

2

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

Fatorar o termo comum

2

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

Eliminar o fator comum:

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

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

Subtrair

1

de ambos os lados

\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

Converter para fração:

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 )}

Já que os denominadores são iguais, combinar as frações:

\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))

Colocar os parênteses 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))

Colocar os parênteses

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

Aplicar as regras dos sinais

+ (- 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)

Somar elementos similares:

cos (x) - cos (x) = 0

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

Somar 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

Reeecreva 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 a Identidade Básica da Trigonometria:

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 frações:

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

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

Eliminar o fator comum:

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

Fatorar

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

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

Fatorar o termo comum

sin (x)

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

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

Resolver cada parte separadamente

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

sin (x) = 0 : x = 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n

sin (x) = 0

Soluções gerais para

sin (x) = 0

sin (x)

tabela de periodicidade com ciclo de

36 0^{\circ} n

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}

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

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

Resolver

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

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

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

x = 36 0^{\circ} n

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

tan (x) - 1 = 0 : x = 4 5^{\circ} + 18 0^{\circ} n

tan (x) - 1 = 0

Mova

1

para o lado direito

tan (x) - 1 = 0

Adicionar

1

a ambos os lados

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

Simplificar

tan (x) = 1

tan (x) = 1

Soluções gerais para

tan (x) = 1

tan (x)

tabela de periodicidade com ciclo de

18 0^{\circ} n

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

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

Combinar toda as soluções

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

Gráfico

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