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Popular Trigonometria >

((sin(x)+tan(x)))/(1+cos(x))=m*sec(x)

Solução

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

Solução

x = arcsin (m) + 2 πn, x = π + arcsin (- m) + 2 πn

Passos da solução

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

Subtrair

m sec (x)

de ambos os lados

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

Simplificar

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

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

Converter para fração:

m sec (x) = \frac{m sec ( x ) ( 1 + cos ( x ) )}{1 + cos ( x )}

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

Já que os denominadores são iguais, combinar as frações:

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

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

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

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

sin (x) + tan (x) - m sec (x) (1 + cos (x)) = 0

Expresar com seno, cosseno

sin (x) + tan (x) - (1 + cos (x)) sec (x) m

Utilizar a Identidade Básica da Trigonometria:

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

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

Utilizar a Identidade Básica da Trigonometria:

sec (x) = \frac{1}{cos ( x )}

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

Simplificar

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

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

(1 + cos (x)) \frac{1}{cos ( x )} m = \frac{m ( 1 + cos ( x ) )}{cos ( x )}

(1 + cos (x)) \frac{1}{cos ( x )} m

Multiplicar frações:

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

= \frac{1 \cdot ( 1 + cos ( x ) ) m}{cos ( x )}

Multiplicar:

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

= \frac{m ( cos ( x ) + 1 )}{cos ( x )}

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

Combinar as frações usando o mínimo múltiplo comum:

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

Aplicar a regra

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

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

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

Converter para fração:

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

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

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 ( x ) + sin ( x ) - ( 1 + cos ( x ) ) m}{cos ( x )}

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

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

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

sin (x) - (1 + cos (x)) m + cos (x) sin (x) = 0

Fatorar

sin (x) - (1 + cos (x)) m + cos (x) sin (x) : (1 + cos (x)) (sin (x) - m)

sin (x) - (1 + cos (x)) m + cos (x) sin (x)

Fatorar o termo comum

sin (x)

= sin (x) (1 + cos (x)) - (1 + cos (x)) m

Fatorar o termo comum

(1 + cos (x))

= (1 + cos (x)) (sin (x) - m)

(1 + cos (x)) (sin (x) - m) = 0

Resolver cada parte separadamente

1 + cos (x) = 0 or sin (x) - m = 0

1 + cos (x) = 0 : x = π + 2 πn

1 + cos (x) = 0

Mova

1

para o lado direito

1 + cos (x) = 0

Subtrair

1

de ambos os lados

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

Simplificar

cos (x) = - 1

cos (x) = - 1

Soluções gerais para

cos (x) = - 1

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 = π + 2 πn

x = π + 2 πn

sin (x) - m = 0 : x = arcsin (m) + 2 πn, x = π + arcsin (- m) + 2 πn

sin (x) - m = 0

Mova

m

para o lado direito

sin (x) - m = 0

Adicionar

m

a ambos os lados

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

Simplificar

sin (x) = m

sin (x) = m

Aplique as propriedades trigonométricas inversas

sin (x) = m

Soluções gerais para

sin (x) = m

sin (x) = a \Rightarrow x = arcsin (a) + 2 πn, x = π + arcsin (a) + 2 πn

x = arcsin (m) + 2 πn, x = π + arcsin (- m) + 2 πn

x = arcsin (m) + 2 πn, x = π + arcsin (- m) + 2 πn

Combinar toda as soluções

x = π + 2 πn, x = arcsin (m) + 2 πn, x = π + arcsin (- m) + 2 πn

Dado que a equação é indefinida para:

π + 2 πn

x = arcsin (m) + 2 πn, x = π + arcsin (- m) + 2 πn

Gráfico

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