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Popular Trigonometria >

4sec^2(x)-3tan^2(x)=5tan^2(x)

Solução

4 sec^{2} (x) - 3 tan^{2} (x) = 5 tan^{2} (x)

Solução

x = \frac{5 π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn, x = \frac{π}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn

Graus

x = 22 5^{\circ} + 36 0^{\circ} n, x = 31 5^{\circ} + 36 0^{\circ} n, x = 4 5^{\circ} + 36 0^{\circ} n, x = 13 5^{\circ} + 36 0^{\circ} n

Passos da solução

4 sec^{2} (x) - 3 tan^{2} (x) = 5 tan^{2} (x)

Subtrair

5 tan^{2} (x)

de ambos os lados

4 sec^{2} (x) - 8 tan^{2} (x) = 0

Fatorar

4 sec^{2} (x) - 8 tan^{2} (x) : 4 (sec (x) + 2 tan (x)) (sec (x) - 2 tan (x))

4 sec^{2} (x) - 8 tan^{2} (x)

Reescrever

- 8

como

2 \cdot 4

= 4 sec^{2} (x) + 2 \cdot 4 tan^{2} (x)

Fatorar o termo comum

4

= 4 (sec^{2} (x) - 2 tan^{2} (x))

Fatorar

sec^{2} (x) - 2 tan^{2} (x) : (sec (x) + 2 tan (x)) (sec (x) - 2 tan (x))

sec^{2} (x) - 2 tan^{2} (x)

Reescrever

sec^{2} (x) - 2 tan^{2} (x)

como

sec^{2} (x) - (2 tan (x))^{2}

sec^{2} (x) - 2 tan^{2} (x)

Aplicar as propriedades dos radicais:

a = (a)^{2}

2 = (2)^{2}

= sec^{2} (x) - (2)^{2} tan^{2} (x)

Aplicar as propriedades dos expoentes:

a^{m} b^{m} = (ab)^{m}

(2)^{2} tan^{2} (x) = (2 tan (x))^{2}

= sec^{2} (x) - (2 tan (x))^{2}

= sec^{2} (x) - (2 tan (x))^{2}

Aplicar a regra da diferença de quadrados:

x^{2} - y^{2} = (x + y) (x - y)

sec^{2} (x) - (2 tan (x))^{2} = (sec (x) + 2 tan (x)) (sec (x) - 2 tan (x))

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

= 4 (sec (x) + 2 tan (x)) (sec (x) - 2 tan (x))

4 (sec (x) + 2 tan (x)) (sec (x) - 2 tan (x)) = 0

Resolver cada parte separadamente

sec (x) + 2 tan (x) = 0 or sec (x) - 2 tan (x) = 0

sec (x) + 2 tan (x) = 0 : x = \frac{5 π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

sec (x) + 2 tan (x) = 0

Expresar com seno, cosseno

sec (x) + 2 tan (x)

Utilizar a Identidade Básica da Trigonometria:

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

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

Utilizar a Identidade Básica da Trigonometria:

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

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

Simplificar

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

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

Multiplicar

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

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

Multiplicar frações:

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

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

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

Aplicar a regra

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

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

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

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

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

1 + sin (x) 2 = 0

Mova

1

para o lado direito

1 + sin (x) 2 = 0

Subtrair

1

de ambos os lados

1 + sin (x) 2 - 1 = 0 - 1

Simplificar

sin (x) 2 = - 1

sin (x) 2 = - 1

Dividir ambos os lados por

2

sin (x) 2 = - 1

Dividir ambos os lados por

2

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

Simplificar

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

Simplificar

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

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

Eliminar o fator comum:

2

= sin (x)

Simplificar

\frac{- 1}{2} : - \frac{2}{2}

\frac{- 1}{2}

Aplicar as propriedades das frações:

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

= - \frac{1}{2}

Racionalizar

- \frac{1}{2} : - \frac{2}{2}

- \frac{1}{2}

Multiplicar pelo conjugado

\frac{2}{2}

= - \frac{1 \cdot 2}{2 2}

1 \cdot 2 = 2

22 = 2

22

Aplicar as propriedades dos radicais:

a a = a

22 = 2

= 2

= - \frac{2}{2}

= - \frac{2}{2}

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

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

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

Soluções gerais para

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

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}

x = \frac{5 π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

x = \frac{5 π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

sec (x) - 2 tan (x) = 0 : x = \frac{π}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn

sec (x) - 2 tan (x) = 0

Expresar com seno, cosseno

sec (x) - 2 tan (x)

Utilizar a Identidade Básica da Trigonometria:

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

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

Utilizar a Identidade Básica da Trigonometria:

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

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

Simplificar

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

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

Multiplicar

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

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

Multiplicar frações:

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

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

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

Aplicar a regra

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

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

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

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

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

1 - sin (x) 2 = 0

Mova

1

para o lado direito

1 - sin (x) 2 = 0

Subtrair

1

de ambos os lados

1 - sin (x) 2 - 1 = 0 - 1

Simplificar

- sin (x) 2 = - 1

- sin (x) 2 = - 1

Dividir ambos os lados por

- 2

- sin (x) 2 = - 1

Dividir ambos os lados por

- 2

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

Simplificar

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

Simplificar

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

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

Aplicar as propriedades das frações:

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

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

Eliminar o fator comum:

2

= sin (x)

Simplificar

\frac{- 1}{- 2} : \frac{2}{2}

\frac{- 1}{- 2}

Aplicar as propriedades das frações:

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

= \frac{1}{2}

Racionalizar

\frac{1}{2} : \frac{2}{2}

\frac{1}{2}

Multiplicar pelo conjugado

\frac{2}{2}

= \frac{1 \cdot 2}{2 2}

1 \cdot 2 = 2

22 = 2

22

Aplicar as propriedades dos radicais:

a a = a

22 = 2

= 2

= \frac{2}{2}

= \frac{2}{2}

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

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

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

Soluções gerais para

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

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}

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

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

Combinar toda as soluções

x = \frac{5 π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn, x = \frac{π}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn

Gráfico

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