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tan(pi/2-x)+2tan(pi/2+x)=1

Solução

tan (\frac{π}{2} - x) + 2 tan (\frac{π}{2} + x) = 1

Solução

x = \frac{3 π}{4} + πn

Graus

x = 13 5^{\circ} + 18 0^{\circ} n

Passos da solução

tan (\frac{π}{2} - x) + 2 tan (\frac{π}{2} + x) = 1

Reeecreva usando identidades trigonométricas

tan (\frac{π}{2} - x) + 2 tan (\frac{π}{2} + x) = 1

Reeecreva usando identidades trigonométricas

tan (\frac{π}{2} - x)

Utilizar a Identidade Básica da Trigonometria:

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

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

Use a identidade de diferença de ângulos:

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

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

Use a identidade de diferença de ângulos:

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

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

Simplificar

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

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

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

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

sin (\frac{π}{2}) cos (x) = cos (x)

sin (\frac{π}{2}) cos (x)

Simplificar

sin (\frac{π}{2}) : 1

sin (\frac{π}{2})

Utilizar a seguinte identidade trivial:

sin (\frac{π}{2}) = 1

sin (x)

tabela de periodicidade com ciclo de

2 πn

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}

= 1

= 1 \cdot cos (x)

Multiplicar:

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

= cos (x)

cos (\frac{π}{2}) sin (x) = 0

cos (\frac{π}{2}) sin (x)

Simplificar

cos (\frac{π}{2}) : 0

cos (\frac{π}{2})

Utilizar a seguinte identidade trivial:

cos (\frac{π}{2}) = 0

cos (x)

tabela de periodicidade com ciclo de

2 πn

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}

= 0

= 0 \cdot sin (x)

Aplicar a regra

0 \cdot a = 0

= 0

= cos (x) - 0

cos (x) - 0 = cos (x)

= cos (x)

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

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

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

cos (\frac{π}{2}) cos (x) = 0

cos (\frac{π}{2}) cos (x)

Simplificar

cos (\frac{π}{2}) : 0

cos (\frac{π}{2})

Utilizar a seguinte identidade trivial:

cos (\frac{π}{2}) = 0

cos (x)

tabela de periodicidade com ciclo de

2 πn

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}

= 0

= 0 \cdot cos (x)

Aplicar a regra

0 \cdot a = 0

= 0

sin (\frac{π}{2}) sin (x) = sin (x)

sin (\frac{π}{2}) sin (x)

Simplificar

sin (\frac{π}{2}) : 1

sin (\frac{π}{2})

Utilizar a seguinte identidade trivial:

sin (\frac{π}{2}) = 1

sin (x)

tabela de periodicidade com ciclo de

2 πn

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}

= 1

= 1 \cdot sin (x)

Multiplicar:

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

= sin (x)

= 0 + sin (x)

0 + sin (x) = sin (x)

= sin (x)

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

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

Utilizar a Identidade Básica da Trigonometria:

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

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

Use a identidade de soma de ângulos:

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

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

Use a identidade de soma de ângulos:

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

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

Simplificar

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

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

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

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

sin (\frac{π}{2}) cos (x) = cos (x)

sin (\frac{π}{2}) cos (x)

Simplificar

sin (\frac{π}{2}) : 1

sin (\frac{π}{2})

Utilizar a seguinte identidade trivial:

sin (\frac{π}{2}) = 1

sin (x)

tabela de periodicidade com ciclo de

2 πn

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}

= 1

= 1 \cdot cos (x)

Multiplicar:

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

= cos (x)

cos (\frac{π}{2}) sin (x) = 0

cos (\frac{π}{2}) sin (x)

Simplificar

cos (\frac{π}{2}) : 0

cos (\frac{π}{2})

Utilizar a seguinte identidade trivial:

cos (\frac{π}{2}) = 0

cos (x)

tabela de periodicidade com ciclo de

2 πn

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}

= 0

= 0 \cdot sin (x)

Aplicar a regra

0 \cdot a = 0

= 0

= cos (x) + 0

cos (x) + 0 = cos (x)

= cos (x)

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

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

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

cos (\frac{π}{2}) cos (x) = 0

cos (\frac{π}{2}) cos (x)

Simplificar

cos (\frac{π}{2}) : 0

cos (\frac{π}{2})

Utilizar a seguinte identidade trivial:

cos (\frac{π}{2}) = 0

cos (x)

tabela de periodicidade com ciclo de

2 πn

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}

= 0

= 0 \cdot cos (x)

Aplicar a regra

0 \cdot a = 0

= 0

sin (\frac{π}{2}) sin (x) = sin (x)

sin (\frac{π}{2}) sin (x)

Simplificar

sin (\frac{π}{2}) : 1

sin (\frac{π}{2})

Utilizar a seguinte identidade trivial:

sin (\frac{π}{2}) = 1

sin (x)

tabela de periodicidade com ciclo de

2 πn

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}

= 1

= 1 \cdot sin (x)

Multiplicar:

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

= sin (x)

= 0 - sin (x)

0 - sin (x) = - sin (x)

= - sin (x)

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

Aplicar as propriedades das frações:

\frac{a}{- b} = - \frac{a}{b}

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

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

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

Simplificar

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

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

Remover os parênteses:

(- a) = - a

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

Somar elementos similares:

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

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

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

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

Subtrair

1

de ambos os lados

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

Simplificar

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

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

Converter para fração:

1 = \frac{1 sin ( x )}{sin ( x )}

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

Multiplicar:

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

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

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

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

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

Reeecreva usando identidades trigonométricas

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

Dividir ambos os lados por

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

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

Simplificar

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

Utilizar a Identidade Básica da Trigonometria:

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

- 1 - tan (x) = 0

- 1 - tan (x) = 0

Mova

1

para o lado direito

- 1 - tan (x) = 0

Adicionar

1

a ambos os lados

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

Simplificar

- tan (x) = 1

- tan (x) = 1

Dividir ambos os lados por

- 1

- tan (x) = 1

Dividir ambos os lados por

- 1

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

Simplificar

tan (x) = - 1

tan (x) = - 1

Soluções gerais para

tan (x) = - 1

tan (x)

tabela de periodicidade com ciclo de

πn

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{3 π}{4} + πn

x = \frac{3 π}{4} + πn

Exemplos populares

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(\frac{sec ( x )}{cos ( x )})^{2} = tan^{2} (x)

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

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

2sin^2(x/3)=1,-pi<= ,x<= pi

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