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Popular Trigonometria >

tan^2(x)=5sin^2(x)

Solução

tan^{2} (x) = 5 sin^{2} (x)

Solução

x = 2 πn, x = π + 2 πn, x = 2.03444 \dots + 2 πn, x = - 2.03444 \dots + 2 πn, x = 1.10714 \dots + 2 πn, x = 2 π - 1.10714 \dots + 2 πn

Graus

x = 0^{\circ} + 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n, x = 116.56505 \dots^{\circ} + 36 0^{\circ} n, x = - 116.56505 \dots^{\circ} + 36 0^{\circ} n, x = 63.43494 \dots^{\circ} + 36 0^{\circ} n, x = 296.56505 \dots^{\circ} + 36 0^{\circ} n

Passos da solução

tan^{2} (x) = 5 sin^{2} (x)

Subtrair

5 sin^{2} (x)

de ambos os lados

tan^{2} (x) - 5 sin^{2} (x) = 0

Fatorar

tan^{2} (x) - 5 sin^{2} (x) : (tan (x) + 5 sin (x)) (tan (x) - 5 sin (x))

tan^{2} (x) - 5 sin^{2} (x)

Reescrever

tan^{2} (x) - 5 sin^{2} (x)

como

tan^{2} (x) - (5 sin (x))^{2}

tan^{2} (x) - 5 sin^{2} (x)

Aplicar as propriedades dos radicais:

a = (a)^{2}

5 = (5)^{2}

= tan^{2} (x) - (5)^{2} sin^{2} (x)

Aplicar as propriedades dos expoentes:

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

(5)^{2} sin^{2} (x) = (5 sin (x))^{2}

= tan^{2} (x) - (5 sin (x))^{2}

= tan^{2} (x) - (5 sin (x))^{2}

Aplicar a regra da diferença de quadrados:

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

tan^{2} (x) - (5 sin (x))^{2} = (tan (x) + 5 sin (x)) (tan (x) - 5 sin (x))

= (tan (x) + 5 sin (x)) (tan (x) - 5 sin (x))

(tan (x) + 5 sin (x)) (tan (x) - 5 sin (x)) = 0

Resolver cada parte separadamente

tan (x) + 5 sin (x) = 0 or tan (x) - 5 sin (x) = 0

tan (x) + 5 sin (x) = 0 : x = 2 πn, x = π + 2 πn, x = arccos (- \frac{5}{5}) + 2 πn, x = - arccos (- \frac{5}{5}) + 2 πn

tan (x) + 5 sin (x) = 0

Expresar com seno, cosseno

tan (x) + sin (x) 5

Utilizar a Identidade Básica da Trigonometria:

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

= \frac{sin ( x )}{cos ( x )} + sin (x) 5

Simplificar

\frac{sin ( x )}{cos ( x )} + sin (x) 5 : \frac{sin ( x ) + 5 sin ( x ) cos ( x )}{cos ( x )}

\frac{sin ( x )}{cos ( x )} + sin (x) 5

Converter para fração:

5 sin (x) = \frac{sin ( x ) 5 cos ( x )}{cos ( x )}

= \frac{sin ( x )}{cos ( x )} + \frac{sin ( x ) 5 cos ( x )}{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 ) + sin ( x ) 5 cos ( x )}{cos ( x )}

= \frac{sin ( x ) + 5 sin ( x ) cos ( x )}{cos ( x )}

\frac{sin ( x ) + cos ( x ) sin ( x ) 5}{cos ( x )} = 0

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

sin (x) + cos (x) sin (x) 5 = 0

Fatorar

sin (x) + cos (x) sin (x) 5 : sin (x) (1 + 5 cos (x))

sin (x) + cos (x) sin (x) 5

Fatorar o termo comum

sin (x)

= sin (x) (1 + cos (x) 5)

sin (x) (1 + 5 cos (x)) = 0

Resolver cada parte separadamente

sin (x) = 0 or 1 + 5 cos (x) = 0

sin (x) = 0 : x = 2 πn, x = π + 2 πn

sin (x) = 0

Soluções gerais para

sin (x) = 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}

x = 0 + 2 πn, x = π + 2 πn

x = 0 + 2 πn, x = π + 2 πn

Resolver

x = 0 + 2 πn : x = 2 πn

x = 0 + 2 πn

0 + 2 πn = 2 πn

x = 2 πn

x = 2 πn, x = π + 2 πn

1 + 5 cos (x) = 0 : x = arccos (- \frac{5}{5}) + 2 πn, x = - arccos (- \frac{5}{5}) + 2 πn

1 + 5 cos (x) = 0

Mova

1

para o lado direito

1 + 5 cos (x) = 0

Subtrair

1

de ambos os lados

1 + 5 cos (x) - 1 = 0 - 1

Simplificar

5 cos (x) = - 1

5 cos (x) = - 1

Dividir ambos os lados por

5

5 cos (x) = - 1

Dividir ambos os lados por

5

\frac{5 cos ( x )}{5} = \frac{- 1}{5}

Simplificar

\frac{5 cos ( x )}{5} = \frac{- 1}{5}

Simplificar

\frac{5 cos ( x )}{5} : cos (x)

\frac{5 cos ( x )}{5}

Eliminar o fator comum:

5

= cos (x)

Simplificar

\frac{- 1}{5} : - \frac{5}{5}

\frac{- 1}{5}

Aplicar as propriedades das frações:

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

= - \frac{1}{5}

Racionalizar

- \frac{1}{5} : - \frac{5}{5}

- \frac{1}{5}

Multiplicar pelo conjugado

\frac{5}{5}

= - \frac{1 \cdot 5}{5 5}

1 \cdot 5 = 5

55 = 5

55

Aplicar as propriedades dos radicais:

a a = a

55 = 5

= 5

= - \frac{5}{5}

= - \frac{5}{5}

cos (x) = - \frac{5}{5}

cos (x) = - \frac{5}{5}

cos (x) = - \frac{5}{5}

Aplique as propriedades trigonométricas inversas

cos (x) = - \frac{5}{5}

Soluções gerais para

cos (x) = - \frac{5}{5}

cos (x) = - a \Rightarrow x = arccos (- a) + 2 πn, x = - arccos (- a) + 2 πn

x = arccos (- \frac{5}{5}) + 2 πn, x = - arccos (- \frac{5}{5}) + 2 πn

x = arccos (- \frac{5}{5}) + 2 πn, x = - arccos (- \frac{5}{5}) + 2 πn

Combinar toda as soluções

x = 2 πn, x = π + 2 πn, x = arccos (- \frac{5}{5}) + 2 πn, x = - arccos (- \frac{5}{5}) + 2 πn

tan (x) - 5 sin (x) = 0 : x = 2 πn, x = π + 2 πn, x = arccos (\frac{5}{5}) + 2 πn, x = 2 π - arccos (\frac{5}{5}) + 2 πn

tan (x) - 5 sin (x) = 0

Expresar com seno, cosseno

tan (x) - sin (x) 5

Utilizar a Identidade Básica da Trigonometria:

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

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

Simplificar

\frac{sin ( x )}{cos ( x )} - sin (x) 5 : \frac{sin ( x ) - 5 sin ( x ) cos ( x )}{cos ( x )}

\frac{sin ( x )}{cos ( x )} - sin (x) 5

Converter para fração:

5 sin (x) = \frac{sin ( x ) 5 cos ( x )}{cos ( x )}

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

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

\frac{sin ( x ) - cos ( x ) sin ( x ) 5}{cos ( x )} = 0

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

sin (x) - cos (x) sin (x) 5 = 0

Fatorar

sin (x) - cos (x) sin (x) 5 : sin (x) (1 - 5 cos (x))

sin (x) - cos (x) sin (x) 5

Fatorar o termo comum

sin (x)

= sin (x) (1 - cos (x) 5)

sin (x) (1 - 5 cos (x)) = 0

Resolver cada parte separadamente

sin (x) = 0 or 1 - 5 cos (x) = 0

sin (x) = 0 : x = 2 πn, x = π + 2 πn

sin (x) = 0

Soluções gerais para

sin (x) = 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}

x = 0 + 2 πn, x = π + 2 πn

x = 0 + 2 πn, x = π + 2 πn

Resolver

x = 0 + 2 πn : x = 2 πn

x = 0 + 2 πn

0 + 2 πn = 2 πn

x = 2 πn

x = 2 πn, x = π + 2 πn

1 - 5 cos (x) = 0 : x = arccos (\frac{5}{5}) + 2 πn, x = 2 π - arccos (\frac{5}{5}) + 2 πn

1 - 5 cos (x) = 0

Mova

1

para o lado direito

1 - 5 cos (x) = 0

Subtrair

1

de ambos os lados

1 - 5 cos (x) - 1 = 0 - 1

Simplificar

- 5 cos (x) = - 1

- 5 cos (x) = - 1

Dividir ambos os lados por

- 5

- 5 cos (x) = - 1

Dividir ambos os lados por

- 5

\frac{- 5 cos ( x )}{- 5} = \frac{- 1}{- 5}

Simplificar

\frac{- 5 cos ( x )}{- 5} = \frac{- 1}{- 5}

Simplificar

\frac{- 5 cos ( x )}{- 5} : cos (x)

\frac{- 5 cos ( x )}{- 5}

Aplicar as propriedades das frações:

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

= \frac{5 cos ( x )}{5}

Eliminar o fator comum:

5

= cos (x)

Simplificar

\frac{- 1}{- 5} : \frac{5}{5}

\frac{- 1}{- 5}

Aplicar as propriedades das frações:

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

= \frac{1}{5}

Racionalizar

\frac{1}{5} : \frac{5}{5}

\frac{1}{5}

Multiplicar pelo conjugado

\frac{5}{5}

= \frac{1 \cdot 5}{5 5}

1 \cdot 5 = 5

55 = 5

55

Aplicar as propriedades dos radicais:

a a = a

55 = 5

= 5

= \frac{5}{5}

= \frac{5}{5}

cos (x) = \frac{5}{5}

cos (x) = \frac{5}{5}

cos (x) = \frac{5}{5}

Aplique as propriedades trigonométricas inversas

cos (x) = \frac{5}{5}

Soluções gerais para

cos (x) = \frac{5}{5}

cos (x) = a \Rightarrow x = arccos (a) + 2 πn, x = 2 π - arccos (a) + 2 πn

x = arccos (\frac{5}{5}) + 2 πn, x = 2 π - arccos (\frac{5}{5}) + 2 πn

x = arccos (\frac{5}{5}) + 2 πn, x = 2 π - arccos (\frac{5}{5}) + 2 πn

Combinar toda as soluções

x = 2 πn, x = π + 2 πn, x = arccos (\frac{5}{5}) + 2 πn, x = 2 π - arccos (\frac{5}{5}) + 2 πn

Combinar toda as soluções

x = 2 πn, x = π + 2 πn, x = arccos (- \frac{5}{5}) + 2 πn, x = - arccos (- \frac{5}{5}) + 2 πn, x = arccos (\frac{5}{5}) + 2 πn, x = 2 π - arccos (\frac{5}{5}) + 2 πn

Mostrar soluções na forma decimal

x = 2 πn, x = π + 2 πn, x = 2.03444 \dots + 2 πn, x = - 2.03444 \dots + 2 πn, x = 1.10714 \dots + 2 πn, x = 2 π - 1.10714 \dots + 2 πn

Gráfico

Exemplos populares

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

2 sin^{2} (x) - sin (2 x) = 0

cos(2x)cos(x)-sin(2x)sin(x)=0.5

cos (2 x) cos (x) - sin (2 x) sin (x) = 0.5

sin^2(θ)=0.4567

sin^{2} (θ) = 0.4567

(sin(A))/(cos(A))-2=1

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

cos(θ)= 10/15

cos (θ) = \frac{10}{15}