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2cos^2(x)+tan^2(x)=2

Solução

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

Solução

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

Graus

x = 0^{\circ} + 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n, x = 13 5^{\circ} + 36 0^{\circ} n, x = 22 5^{\circ} + 36 0^{\circ} n, x = 4 5^{\circ} + 36 0^{\circ} n, x = 31 5^{\circ} + 36 0^{\circ} n

Passos da solução

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

Subtrair

2

de ambos os lados

2 cos^{2} (x) + tan^{2} (x) - 2 = 0

Reeecreva usando identidades trigonométricas

- 2 + tan^{2} (x) + 2 cos^{2} (x)

Utilizar a identidade trigonométrica pitagórica:

cos^{2} (x) + sin^{2} (x) = 1

cos^{2} (x) = 1 - sin^{2} (x)

= - 2 + tan^{2} (x) + 2 (1 - sin^{2} (x))

Simplificar

- 2 + tan^{2} (x) + 2 (1 - sin^{2} (x)) : tan^{2} (x) - 2 sin^{2} (x)

- 2 + tan^{2} (x) + 2 (1 - sin^{2} (x))

Expandir

2 (1 - sin^{2} (x)) : 2 - 2 sin^{2} (x)

2 (1 - sin^{2} (x))

Colocar os parênteses utilizando:

a (b - c) = ab - a c

a = 2, b = 1, c = sin^{2} (x)

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

Multiplicar os números:

2 \cdot 1 = 2

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

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

Simplificar

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

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

Agrupar termos semelhantes

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

- 2 + 2 = 0

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

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

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

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

Fatorar

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

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

Reescrever

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

como

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

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

Aplicar as propriedades dos radicais:

a = (a)^{2}

2 = (2)^{2}

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

Aplicar as propriedades dos expoentes:

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

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

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

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

Aplicar a regra da diferença de quadrados:

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

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

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

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

Resolver cada parte separadamente

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

tan (x) + sin (x) 2 = 0 : x = 2 πn, x = π + 2 πn, x = \frac{3 π}{4} + 2 πn, x = \frac{5 π}{4} + 2 πn

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

Expresar com seno, cosseno

tan (x) + sin (x) 2

Utilizar a Identidade Básica da Trigonometria:

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

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

Simplificar

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

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

Converter para fração:

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

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

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

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

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

sin (x) + cos (x) sin (x) 2 = 0

Fatorar

sin (x) + cos (x) sin (x) 2 : sin (x) (1 + 2 cos (x))

sin (x) + cos (x) sin (x) 2

Fatorar o termo comum

sin (x)

= sin (x) (1 + cos (x) 2)

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

Resolver cada parte separadamente

sin (x) = 0 or 1 + 2 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 + 2 cos (x) = 0 : x = \frac{3 π}{4} + 2 πn, x = \frac{5 π}{4} + 2 πn

1 + 2 cos (x) = 0

Mova

1

para o lado direito

1 + 2 cos (x) = 0

Subtrair

1

de ambos os lados

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

Simplificar

2 cos (x) = - 1

2 cos (x) = - 1

Dividir ambos os lados por

2

2 cos (x) = - 1

Dividir ambos os lados por

2

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

Simplificar

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

Simplificar

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

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

Eliminar o fator comum:

2

= cos (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}

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

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

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

Soluções gerais para

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

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

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

Combinar toda as soluções

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

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

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

Expresar com seno, cosseno

tan (x) - sin (x) 2

Utilizar a Identidade Básica da Trigonometria:

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

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

Simplificar

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

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

Converter para fração:

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

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

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

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

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

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

Fatorar

sin (x) - cos (x) sin (x) 2 : sin (x) (1 - 2 cos (x))

sin (x) - cos (x) sin (x) 2

Fatorar o termo comum

sin (x)

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

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

Resolver cada parte separadamente

sin (x) = 0 or 1 - 2 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 - 2 cos (x) = 0 : x = \frac{π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

1 - 2 cos (x) = 0

Mova

1

para o lado direito

1 - 2 cos (x) = 0

Subtrair

1

de ambos os lados

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

Simplificar

- 2 cos (x) = - 1

- 2 cos (x) = - 1

Dividir ambos os lados por

- 2

- 2 cos (x) = - 1

Dividir ambos os lados por

- 2

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

Simplificar

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

Simplificar

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

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

Aplicar as propriedades das frações:

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

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

Eliminar o fator comum:

2

= cos (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}

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

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

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

Soluções gerais para

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

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}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

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

Combinar toda as soluções

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

Combinar toda as soluções

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

Exemplos populares

sin^3(o)=4sin(o)sin^2(o)sin^4(o)

sin^{3} (o) = 4 sin (o) sin^{2} (o) sin^{4} (o)

cos^2(x)=-4

cos^{2} (x) = - 4

8sin^4(x)-6sin^2(x)+1=0

8 sin^{4} (x) - 6 sin^{2} (x) + 1 = 0

4sin^2(x)+2cos(x)+a=3

4 sin^{2} (x) + 2 cos (x) + a = 3

cos^5(x)+cos(x)+4cos^2(x)=2

cos^{5} (x) + cos (x) + 4 cos^{2} (x) = 2