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Popular Trigonometria >

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

Solução

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

Solução

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn, x = 4 πn, x = 2 π + 4 πn

Graus

x = 9 0^{\circ} + 36 0^{\circ} n, x = 27 0^{\circ} + 36 0^{\circ} n, x = 0^{\circ} + 72 0^{\circ} n, x = 36 0^{\circ} + 72 0^{\circ} n

Passos da solução

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

Expresar com seno, cosseno

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

Utilizar a Identidade Básica da Trigonometria:

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

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

Simplificar

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

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

Converter para fração:

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

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

Já que os denominadores são iguais, combinar as frações:

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

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

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

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

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

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

Reeecreva usando identidades trigonométricas

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

Use a identidade da transformação de produto em soma:

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

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

Simplificar

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

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

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

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

Multiplicar frações:

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

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

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

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

Multiplicar:

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

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

Remover os parênteses:

(a) = a

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

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

Simplificar

x + \frac{x}{2}

em uma fração:

\frac{3 x}{2}

x + \frac{x}{2}

Converter para fração:

x = \frac{x 2}{2}

= \frac{x}{2} + \frac{x \cdot 2}{2}

Já que os denominadores são iguais, combinar as frações:

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

= \frac{x + x \cdot 2}{2}

Somar elementos similares:

x + 2 x = 3 x

= \frac{3 x}{2}

= \frac{sin ( \frac{3 x}{2} ) + sin ( x - \frac{x}{2} )}{2}

Simplificar

x - \frac{x}{2}

em uma fração:

\frac{x}{2}

x - \frac{x}{2}

Converter para fração:

x = \frac{x 2}{2}

= - \frac{x}{2} + \frac{x \cdot 2}{2}

Já que os denominadores são iguais, combinar as frações:

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

= \frac{- x + x \cdot 2}{2}

Somar elementos similares:

- x + 2 x = x

= \frac{x}{2}

= \frac{sin ( \frac{3 x}{2} ) + sin ( \frac{x}{2} )}{2}

= sin (\frac{x}{2}) - \frac{sin ( \frac{3 x}{2} ) + sin ( \frac{x}{2} )}{2}

Converter para fração:

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

= - \frac{sin ( \frac{3 x}{2} ) + sin ( \frac{x}{2} )}{2} + \frac{sin ( \frac{x}{2} ) \cdot 2}{2}

Já que os denominadores são iguais, combinar as frações:

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

= \frac{- ( sin ( \frac{3 x}{2} ) + sin ( \frac{x}{2} ) ) + sin ( \frac{x}{2} ) \cdot 2}{2}

Expandir

- (sin (\frac{3 x}{2}) + sin (\frac{x}{2})) + sin (\frac{x}{2}) \cdot 2 : - sin (\frac{3 x}{2}) + sin (\frac{x}{2})

- (sin (\frac{3 x}{2}) + sin (\frac{x}{2})) + sin (\frac{x}{2}) \cdot 2

= - (sin (\frac{3 x}{2}) + sin (\frac{x}{2})) + 2 sin (\frac{x}{2})

- (sin (\frac{3 x}{2}) + sin (\frac{x}{2})) : - sin (\frac{3 x}{2}) - sin (\frac{x}{2})

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

Colocar os parênteses

= - (sin (\frac{3 x}{2})) - (sin (\frac{x}{2}))

Aplicar as regras dos sinais

+ (- a) = - a

= - sin (\frac{3 x}{2}) - sin (\frac{x}{2})

= - sin (\frac{3 x}{2}) - sin (\frac{x}{2}) + sin (\frac{x}{2}) \cdot 2

Somar elementos similares:

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

= - sin (\frac{3 x}{2}) + sin (\frac{x}{2})

= \frac{- sin ( \frac{3 x}{2} ) + sin ( \frac{x}{2} )}{2}

= \frac{- sin ( \frac{3 x}{2} ) + sin ( \frac{x}{2} )}{2}

Use a identidade da transformação de soma em produto:

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

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

Simplificar

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

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

Combinar as frações usando o mínimo múltiplo comum:

2 x

Aplicar a regra

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

= \frac{x + 3 x}{2}

Somar elementos similares:

x + 3 x = 4 x

= \frac{4 x}{2}

Dividir:

\frac{4}{2} = 2

= 2 x

= \frac{2 sin ( \frac{\frac{x}{2} - \frac{3 x}{2}}{2} ) cos ( \frac{2 x}{2} )}{2}

Combinar as frações usando o mínimo múltiplo comum:

- x

Aplicar a regra

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

= \frac{x - 3 x}{2}

Somar elementos similares:

x - 3 x = - 2 x

= \frac{- 2 x}{2}

Aplicar as propriedades das frações:

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

= - \frac{2 x}{2}

Dividir:

\frac{2}{2} = 1

= - x

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

Aplicar as propriedades das frações:

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

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

Dividir:

\frac{2}{2} = 1

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

Use a identidade de ângulo negativo:

sin (- x) = - sin (x)

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

Remover os parênteses:

(- a) = - a

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

Dividir:

\frac{2}{2} = 1

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

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

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

Resolver cada parte separadamente

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

cos (x) = 0 : x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

cos (x) = 0

Soluções gerais para

cos (x) = 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}

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

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

sin (\frac{x}{2}) = 0 : x = 4 πn, x = 2 π + 4 πn

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

Soluções gerais para

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

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}

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

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

Resolver

\frac{x}{2} = 0 + 2 πn : x = 4 πn

\frac{x}{2} = 0 + 2 πn

0 + 2 πn = 2 πn

\frac{x}{2} = 2 πn

Multiplicar ambos os lados por

2

\frac{x}{2} = 2 πn

Multiplicar ambos os lados por

2

\frac{2 x}{2} = 2 \cdot 2 πn

Simplificar

x = 4 πn

x = 4 πn

Resolver

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

\frac{x}{2} = π + 2 πn

Multiplicar ambos os lados por

2

\frac{x}{2} = π + 2 πn

Multiplicar ambos os lados por

2

\frac{2 x}{2} = 2 π + 2 \cdot 2 πn

Simplificar

x = 2 π + 4 πn

x = 2 π + 4 πn

x = 4 πn, x = 2 π + 4 πn

Combinar toda as soluções

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn, x = 4 πn, x = 2 π + 4 πn

Gráfico

Exemplos populares

sin(θ)= 1/3

sin (θ) = \frac{1}{3}

cos(θ)= 1/(sqrt(2))

cos (θ) = \frac{1}{2}

tan(2x)=3tan(x)

tan (2 x) = 3 tan (x)

tan(2θ)=-1

tan (2 θ) = - 1

sin(x)-cos(x)=1

sin (x) - cos (x) = 1