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Popular Trigonometria >

1-8sin^2(x)cos^2(x)=0

Solução

1 - 8 sin^{2} (x) cos^{2} (x) = 0

Solução

x = \frac{5 π}{8} + πn, x = \frac{7 π}{8} + πn, x = \frac{π}{8} + πn, x = \frac{3 π}{8} + πn

Graus

x = 112. 5^{\circ} + 18 0^{\circ} n, x = 157. 5^{\circ} + 18 0^{\circ} n, x = 22. 5^{\circ} + 18 0^{\circ} n, x = 67. 5^{\circ} + 18 0^{\circ} n

Passos da solução

1 - 8 sin^{2} (x) cos^{2} (x) = 0

Fatorar

1 - 8 sin^{2} (x) cos^{2} (x) : (22 sin (x) cos (x) + 1) (- 22 sin (x) cos (x) + 1)

1 - 8 sin^{2} (x) cos^{2} (x)

Reescrever

1 - 8 sin^{2} (x) cos^{2} (x)

como

1 - (8 sin (x) cos (x))^{2}

1 - 8 sin^{2} (x) cos^{2} (x)

Aplicar as propriedades dos radicais:

a = (a)^{2}

8 = (8)^{2}

= 1 - (8)^{2} sin^{2} (x) cos^{2} (x)

Aplicar as propriedades dos expoentes:

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

(8)^{2} sin^{2} (x) cos^{2} (x) = (8 sin (x) cos (x))^{2}

= 1 - (8 sin (x) cos (x))^{2}

= 1 - (8 sin (x) cos (x))^{2}

Aplicar a regra da diferença de quadrados:

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

1 - (8 sin (x) cos (x))^{2} = (8 sin (x) cos (x) + 1) (- 8 sin (x) cos (x) + 1)

= (8 sin (x) cos (x) + 1) (- 8 sin (x) cos (x) + 1)

Simplificar

= (22 sin (x) cos (x) + 1) (- 22 sin (x) cos (x) + 1)

(22 sin (x) cos (x) + 1) (- 22 sin (x) cos (x) + 1) = 0

Resolver cada parte separadamente

22 sin (x) cos (x) + 1 = 0 or - 22 sin (x) cos (x) + 1 = 0

22 sin (x) cos (x) + 1 = 0 : x = \frac{5 π}{8} + πn, x = \frac{7 π}{8} + πn

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

Reeecreva usando identidades trigonométricas

22 sin (x) cos (x) + 1

Utilizar a identidade trigonométrica do arco duplo:

2 sin (x) cos (x) = sin (2 x)

= 1 + sin (2 x) 2

1 + sin (2 x) 2 = 0

Mova

1

para o lado direito

1 + sin (2 x) 2 = 0

Subtrair

1

de ambos os lados

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

Simplificar

sin (2 x) 2 = - 1

sin (2 x) 2 = - 1

Dividir ambos os lados por

2

sin (2 x) 2 = - 1

Dividir ambos os lados por

2

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

Simplificar

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

Simplificar

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

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

Eliminar o fator comum:

2

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

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

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

Soluções gerais para

sin (2 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}

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

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

Resolver

2 x = \frac{5 π}{4} + 2 πn : x = \frac{5 π}{8} + πn

2 x = \frac{5 π}{4} + 2 πn

Dividir ambos os lados por

2

2 x = \frac{5 π}{4} + 2 πn

Dividir ambos os lados por

2

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

Simplificar

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

Simplificar

\frac{2 x}{2} : x

\frac{2 x}{2}

Dividir:

\frac{2}{2} = 1

= x

Simplificar

\frac{\frac{5 π}{4}}{2} + \frac{2 πn}{2} : \frac{5 π}{8} + πn

\frac{\frac{5 π}{4}}{2} + \frac{2 πn}{2}

\frac{\frac{5 π}{4}}{2} = \frac{5 π}{8}

\frac{\frac{5 π}{4}}{2}

Aplicar as propriedades das frações:

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

= \frac{5 π}{4 \cdot 2}

Multiplicar os números:

4 \cdot 2 = 8

= \frac{5 π}{8}

\frac{2 πn}{2} = πn

\frac{2 πn}{2}

Dividir:

\frac{2}{2} = 1

= πn

= \frac{5 π}{8} + πn

x = \frac{5 π}{8} + πn

x = \frac{5 π}{8} + πn

x = \frac{5 π}{8} + πn

Resolver

2 x = \frac{7 π}{4} + 2 πn : x = \frac{7 π}{8} + πn

2 x = \frac{7 π}{4} + 2 πn

Dividir ambos os lados por

2

2 x = \frac{7 π}{4} + 2 πn

Dividir ambos os lados por

2

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

Simplificar

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

Simplificar

\frac{2 x}{2} : x

\frac{2 x}{2}

Dividir:

\frac{2}{2} = 1

= x

Simplificar

\frac{\frac{7 π}{4}}{2} + \frac{2 πn}{2} : \frac{7 π}{8} + πn

\frac{\frac{7 π}{4}}{2} + \frac{2 πn}{2}

\frac{\frac{7 π}{4}}{2} = \frac{7 π}{8}

\frac{\frac{7 π}{4}}{2}

Aplicar as propriedades das frações:

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

= \frac{7 π}{4 \cdot 2}

Multiplicar os números:

4 \cdot 2 = 8

= \frac{7 π}{8}

\frac{2 πn}{2} = πn

\frac{2 πn}{2}

Dividir:

\frac{2}{2} = 1

= πn

= \frac{7 π}{8} + πn

x = \frac{7 π}{8} + πn

x = \frac{7 π}{8} + πn

x = \frac{7 π}{8} + πn

x = \frac{5 π}{8} + πn, x = \frac{7 π}{8} + πn

- 22 sin (x) cos (x) + 1 = 0 : x = \frac{π}{8} + πn, x = \frac{3 π}{8} + πn

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

Reeecreva usando identidades trigonométricas

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

Utilizar a identidade trigonométrica do arco duplo:

2 sin (x) cos (x) = sin (2 x)

= 1 - sin (2 x) 2

1 - sin (2 x) 2 = 0

Mova

1

para o lado direito

1 - sin (2 x) 2 = 0

Subtrair

1

de ambos os lados

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

Simplificar

- sin (2 x) 2 = - 1

- sin (2 x) 2 = - 1

Dividir ambos os lados por

- 2

- sin (2 x) 2 = - 1

Dividir ambos os lados por

- 2

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

Simplificar

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

Simplificar

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

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

Aplicar as propriedades das frações:

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

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

Eliminar o fator comum:

2

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

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

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

Soluções gerais para

sin (2 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}

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

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

Resolver

2 x = \frac{π}{4} + 2 πn : x = \frac{π}{8} + πn

2 x = \frac{π}{4} + 2 πn

Dividir ambos os lados por

2

2 x = \frac{π}{4} + 2 πn

Dividir ambos os lados por

2

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

Simplificar

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

Simplificar

\frac{2 x}{2} : x

\frac{2 x}{2}

Dividir:

\frac{2}{2} = 1

= x

Simplificar

\frac{\frac{π}{4}}{2} + \frac{2 πn}{2} : \frac{π}{8} + πn

\frac{\frac{π}{4}}{2} + \frac{2 πn}{2}

\frac{\frac{π}{4}}{2} = \frac{π}{8}

\frac{\frac{π}{4}}{2}

Aplicar as propriedades das frações:

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

= \frac{π}{4 \cdot 2}

Multiplicar os números:

4 \cdot 2 = 8

= \frac{π}{8}

\frac{2 πn}{2} = πn

\frac{2 πn}{2}

Dividir:

\frac{2}{2} = 1

= πn

= \frac{π}{8} + πn

x = \frac{π}{8} + πn

x = \frac{π}{8} + πn

x = \frac{π}{8} + πn

Resolver

2 x = \frac{3 π}{4} + 2 πn : x = \frac{3 π}{8} + πn

2 x = \frac{3 π}{4} + 2 πn

Dividir ambos os lados por

2

2 x = \frac{3 π}{4} + 2 πn

Dividir ambos os lados por

2

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

Simplificar

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

Simplificar

\frac{2 x}{2} : x

\frac{2 x}{2}

Dividir:

\frac{2}{2} = 1

= x

Simplificar

\frac{\frac{3 π}{4}}{2} + \frac{2 πn}{2} : \frac{3 π}{8} + πn

\frac{\frac{3 π}{4}}{2} + \frac{2 πn}{2}

\frac{\frac{3 π}{4}}{2} = \frac{3 π}{8}

\frac{\frac{3 π}{4}}{2}

Aplicar as propriedades das frações:

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

= \frac{3 π}{4 \cdot 2}

Multiplicar os números:

4 \cdot 2 = 8

= \frac{3 π}{8}

\frac{2 πn}{2} = πn

\frac{2 πn}{2}

Dividir:

\frac{2}{2} = 1

= πn

= \frac{3 π}{8} + πn

x = \frac{3 π}{8} + πn

x = \frac{3 π}{8} + πn

x = \frac{3 π}{8} + πn

x = \frac{π}{8} + πn, x = \frac{3 π}{8} + πn

Combinar toda as soluções

x = \frac{5 π}{8} + πn, x = \frac{7 π}{8} + πn, x = \frac{π}{8} + πn, x = \frac{3 π}{8} + πn

Gráfico

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