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

Solução

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

Solução

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

Graus

x = 18 0^{\circ} + 36 0^{\circ} n, x = 27 0^{\circ} + 36 0^{\circ} n, x = 0^{\circ} + 36 0^{\circ} n, x = 9 0^{\circ} + 36 0^{\circ} n

Passos da solução

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

Subtrair

1^{2}

de ambos os lados

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

Fatorar

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

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

Reescrever

1

como

1^{2}

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

Aplicar a regra da diferença de quadrados:

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

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

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

Simplificar

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

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

Resolver cada parte separadamente

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

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

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

Reeecreva usando identidades trigonométricas

sin (x) + cos (x) + 1

sin (x) + cos (x) = 2 sin (x + \frac{π}{4})

sin (x) + cos (x)

Reescrever como

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

Utilizar a seguinte identidade trivial:

cos (\frac{π}{4}) = \frac{1}{2}

Utilizar a seguinte identidade trivial:

sin (\frac{π}{4}) = \frac{1}{2}

= 2 (cos (\frac{π}{4}) sin (x) + sin (\frac{π}{4}) cos (x))

Use a identidade de soma de ângulos:

sin (s + t) = sin (s) cos (t) + cos (s) sin (t)

= 2 sin (x + \frac{π}{4})

= 1 + 2 sin (x + \frac{π}{4})

1 + 2 sin (x + \frac{π}{4}) = 0

Mova

1

para o lado direito

1 + 2 sin (x + \frac{π}{4}) = 0

Subtrair

1

de ambos os lados

1 + 2 sin (x + \frac{π}{4}) - 1 = 0 - 1

Simplificar

2 sin (x + \frac{π}{4}) = - 1

2 sin (x + \frac{π}{4}) = - 1

Dividir ambos os lados por

2

2 sin (x + \frac{π}{4}) = - 1

Dividir ambos os lados por

2

\frac{2 sin ( x + \frac{π}{4} )}{2} = \frac{- 1}{2}

Simplificar

\frac{2 sin ( x + \frac{π}{4} )}{2} = \frac{- 1}{2}

Simplificar

\frac{2 sin ( x + \frac{π}{4} )}{2} : sin (x + \frac{π}{4})

\frac{2 sin ( x + \frac{π}{4} )}{2}

Eliminar o fator comum:

2

= sin (x + \frac{π}{4})

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{π}{4}) = - \frac{2}{2}

sin (x + \frac{π}{4}) = - \frac{2}{2}

sin (x + \frac{π}{4}) = - \frac{2}{2}

Soluções gerais para

sin (x + \frac{π}{4}) = - \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} = \frac{5 π}{4} + 2 πn, x + \frac{π}{4} = \frac{7 π}{4} + 2 πn

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

Resolver

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

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

Mova

\frac{π}{4}

para o lado direito

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

Subtrair

\frac{π}{4}

de ambos os lados

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

Simplificar

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

Simplificar

x + \frac{π}{4} - \frac{π}{4} : x

x + \frac{π}{4} - \frac{π}{4}

Somar elementos similares:

\frac{π}{4} - \frac{π}{4} = 0

= x

Simplificar

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

\frac{5 π}{4} + 2 πn - \frac{π}{4}

Agrupar termos semelhantes

= 2 πn - \frac{π}{4} + \frac{5 π}{4}

Combinar as frações usando o mínimo múltiplo comum:

π

Aplicar a regra

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

= \frac{- π + 5 π}{4}

Somar elementos similares:

- π + 5 π = 4 π

= \frac{4 π}{4}

Dividir:

\frac{4}{4} = 1

= π

= 2 πn + π

x = 2 πn + π

x = 2 πn + π

x = 2 πn + π

Resolver

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

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

Mova

\frac{π}{4}

para o lado direito

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

Subtrair

\frac{π}{4}

de ambos os lados

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

Simplificar

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

Simplificar

x + \frac{π}{4} - \frac{π}{4} : x

x + \frac{π}{4} - \frac{π}{4}

Somar elementos similares:

\frac{π}{4} - \frac{π}{4} = 0

= x

Simplificar

\frac{7 π}{4} + 2 πn - \frac{π}{4} : 2 πn + \frac{3 π}{2}

\frac{7 π}{4} + 2 πn - \frac{π}{4}

Agrupar termos semelhantes

= 2 πn - \frac{π}{4} + \frac{7 π}{4}

Combinar as frações usando o mínimo múltiplo comum:

\frac{3 π}{2}

Aplicar a regra

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

= \frac{- π + 7 π}{4}

Somar elementos similares:

- π + 7 π = 6 π

= \frac{6 π}{4}

Eliminar o fator comum:

2

= \frac{3 π}{2}

= 2 πn + \frac{3 π}{2}

x = 2 πn + \frac{3 π}{2}

x = 2 πn + \frac{3 π}{2}

x = 2 πn + \frac{3 π}{2}

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

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

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

Reeecreva usando identidades trigonométricas

sin (x) + cos (x) - 1

sin (x) + cos (x) = 2 sin (x + \frac{π}{4})

sin (x) + cos (x)

Reescrever como

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

Utilizar a seguinte identidade trivial:

cos (\frac{π}{4}) = \frac{1}{2}

Utilizar a seguinte identidade trivial:

sin (\frac{π}{4}) = \frac{1}{2}

= 2 (cos (\frac{π}{4}) sin (x) + sin (\frac{π}{4}) cos (x))

Use a identidade de soma de ângulos:

sin (s + t) = sin (s) cos (t) + cos (s) sin (t)

= 2 sin (x + \frac{π}{4})

= - 1 + 2 sin (x + \frac{π}{4})

- 1 + 2 sin (x + \frac{π}{4}) = 0

Mova

1

para o lado direito

- 1 + 2 sin (x + \frac{π}{4}) = 0

Adicionar

1

a ambos os lados

- 1 + 2 sin (x + \frac{π}{4}) + 1 = 0 + 1

Simplificar

2 sin (x + \frac{π}{4}) = 1

2 sin (x + \frac{π}{4}) = 1

Dividir ambos os lados por

2

2 sin (x + \frac{π}{4}) = 1

Dividir ambos os lados por

2

\frac{2 sin ( x + \frac{π}{4} )}{2} = \frac{1}{2}

Simplificar

\frac{2 sin ( x + \frac{π}{4} )}{2} = \frac{1}{2}

Simplificar

\frac{2 sin ( x + \frac{π}{4} )}{2} : sin (x + \frac{π}{4})

\frac{2 sin ( x + \frac{π}{4} )}{2}

Eliminar o fator comum:

2

= sin (x + \frac{π}{4})

Simplificar

\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}

sin (x + \frac{π}{4}) = \frac{2}{2}

sin (x + \frac{π}{4}) = \frac{2}{2}

sin (x + \frac{π}{4}) = \frac{2}{2}

Soluções gerais para

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

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

Resolver

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

x + \frac{π}{4} = \frac{π}{4} + 2 πn

Subtrair

\frac{π}{4}

de ambos os lados

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

Simplificar

x = 2 πn

Resolver

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

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

Mova

\frac{π}{4}

para o lado direito

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

Subtrair

\frac{π}{4}

de ambos os lados

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

Simplificar

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

Simplificar

x + \frac{π}{4} - \frac{π}{4} : x

x + \frac{π}{4} - \frac{π}{4}

Somar elementos similares:

\frac{π}{4} - \frac{π}{4} = 0

= x

Simplificar

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

\frac{3 π}{4} + 2 πn - \frac{π}{4}

Agrupar termos semelhantes

= 2 πn - \frac{π}{4} + \frac{3 π}{4}

Combinar as frações usando o mínimo múltiplo comum:

\frac{π}{2}

Aplicar a regra

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

= \frac{- π + 3 π}{4}

Somar elementos similares:

- π + 3 π = 2 π

= \frac{2 π}{4}

Eliminar o fator comum:

2

= \frac{π}{2}

= 2 πn + \frac{π}{2}

x = 2 πn + \frac{π}{2}

x = 2 πn + \frac{π}{2}

x = 2 πn + \frac{π}{2}

x = 2 πn, x = 2 πn + \frac{π}{2}

Combinar toda as soluções

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

Exemplos populares

tan(4x)=12

tan (4 x) = 12

4csc(B)+5=0

4 csc (B) + 5 = 0

2cos(pi/5 x)=sqrt(3)

2 cos (\frac{π}{5} x) = 3

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

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

solvefor x,y=arctan(2x+3y)resolver para

x, y = arctan (2 x + 3 y)