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Popular Trigonometria >

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

Solução

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

Solução

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

Graus

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

Passos da solução

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

Subtrair

2

de ambos os lados

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

Reeecreva usando identidades trigonométricas

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

Utilizar a identidade trigonométrica pitagórica:

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

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

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

Expandir

1 + 2 (1 - sin^{2} (x)) : - 2 sin^{2} (x) + 3

1 + 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)

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

Somar:

1 + 2 = 3

= - 2 sin^{2} (x) + 3

= - 2 + \frac{- 2 sin ^{2} ( x ) + 3}{2} + \frac{1 + 2 sin ^{2} ( x )}{2}

- 2 + \frac{1 + 2 sin ^{2} ( x )}{2} + \frac{3 - 2 sin ^{2} ( x )}{2} = 0

Usando o método de substituição

- 2 + \frac{1 + 2 sin ^{2} ( x )}{2} + \frac{3 - 2 sin ^{2} ( x )}{2} = 0

Sea:

sin (x) = u

- 2 + \frac{1 + 2 u ^{2}}{2} + \frac{3 - 2 u ^{2}}{2} = 0

- 2 + \frac{1 + 2 u ^{2}}{2} + \frac{3 - 2 u ^{2}}{2} = 0 : u = \frac{1}{2}, u = - \frac{1}{2}

- 2 + \frac{1 + 2 u ^{2}}{2} + \frac{3 - 2 u ^{2}}{2} = 0

Remova as raízes quadradas

- 2 + \frac{1 + 2 u ^{2}}{2} + \frac{3 - 2 u ^{2}}{2} = 0

Adicionar

2

a ambos os lados

- 2 + \frac{1 + 2 u ^{2}}{2} + \frac{3 - 2 u ^{2}}{2} + 2 = 0 + 2

Simplificar

\frac{1 + 2 u ^{2}}{2} + \frac{3 - 2 u ^{2}}{2} = 2

Subtrair

\frac{3 - 2 u ^{2}}{2}

de ambos os lados

\frac{1 + 2 u ^{2}}{2} + \frac{3 - 2 u ^{2}}{2} - \frac{3 - 2 u ^{2}}{2} = 2 - \frac{3 - 2 u ^{2}}{2}

Simplificar

\frac{1 + 2 u ^{2}}{2} = 2 - \frac{3 - 2 u ^{2}}{2}

Elevar ambos os lados ao quadrado :

\frac{1 + 2 u ^{2}}{2} = 4 - 4 \frac{3 - 2 u ^{2}}{2} + \frac{3 - 2 u ^{2}}{2}

- 2 + \frac{1 + 2 u ^{2}}{2} + \frac{3 - 2 u ^{2}}{2} = 0

(\frac{1 + 2 u ^{2}}{2})^{2} = (2 - \frac{3 - 2 u ^{2}}{2})^{2}

Expandir

(\frac{1 + 2 u ^{2}}{2})^{2} : \frac{1 + 2 u ^{2}}{2}

(\frac{1 + 2 u ^{2}}{2})^{2}

Aplicar as propriedades dos radicais:

a = a^{\frac{1}{2}}

= ((\frac{1 + 2 u ^{2}}{2})^{\frac{1}{2}})^{2}

Aplicar as propriedades dos expoentes:

(a^{b})^{c} = a^{b c}

= (\frac{1 + 2 u ^{2}}{2})^{\frac{1}{2} \cdot 2}

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

\frac{1}{2} \cdot 2

Multiplicar frações:

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

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

Eliminar o fator comum:

2

= 1

= \frac{1 + 2 u ^{2}}{2}

Expandir

(2 - \frac{3 - 2 u ^{2}}{2})^{2} : 4 - 4 \frac{3 - 2 u ^{2}}{2} + \frac{3 - 2 u ^{2}}{2}

(2 - \frac{3 - 2 u ^{2}}{2})^{2}

Aplique a fórmula do quadrado perfeito:

(a - b)^{2} = a^{2} - 2 ab + b^{2}

a = 2, b = \frac{3 - 2 u ^{2}}{2}

= 2^{2} - 2 \cdot 2 \frac{3 - 2 u ^{2}}{2} + (\frac{3 - 2 u ^{2}}{2})^{2}

Simplificar

2^{2} - 2 \cdot 2 \frac{3 - 2 u ^{2}}{2} + (\frac{3 - 2 u ^{2}}{2})^{2} : 4 - 4 \frac{3 - 2 u ^{2}}{2} + \frac{3 - 2 u ^{2}}{2}

2^{2} - 2 \cdot 2 \frac{3 - 2 u ^{2}}{2} + (\frac{3 - 2 u ^{2}}{2})^{2}

2^{2} = 4

2^{2}

2^{2} = 4

= 4

2 \cdot 2 \frac{3 - 2 u ^{2}}{2} = 4 \frac{3 - 2 u ^{2}}{2}

2 \cdot 2 \frac{3 - 2 u ^{2}}{2}

Multiplicar os números:

2 \cdot 2 = 4

= 4 \frac{3 - 2 u ^{2}}{2}

(\frac{3 - 2 u ^{2}}{2})^{2} = \frac{3 - 2 u ^{2}}{2}

(\frac{3 - 2 u ^{2}}{2})^{2}

Aplicar as propriedades dos radicais:

a = a^{\frac{1}{2}}

= ((\frac{3 - 2 u ^{2}}{2})^{\frac{1}{2}})^{2}

Aplicar as propriedades dos expoentes:

(a^{b})^{c} = a^{b c}

= (\frac{3 - 2 u ^{2}}{2})^{\frac{1}{2} \cdot 2}

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

\frac{1}{2} \cdot 2

Multiplicar frações:

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

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

Eliminar o fator comum:

2

= 1

= \frac{3 - 2 u ^{2}}{2}

= 4 - 4 \frac{3 - 2 u ^{2}}{2} + \frac{3 - 2 u ^{2}}{2}

= 4 - 4 \frac{3 - 2 u ^{2}}{2} + \frac{3 - 2 u ^{2}}{2}

\frac{1 + 2 u ^{2}}{2} = 4 - 4 \frac{3 - 2 u ^{2}}{2} + \frac{3 - 2 u ^{2}}{2}

\frac{1 + 2 u ^{2}}{2} = 4 - 4 \frac{3 - 2 u ^{2}}{2} + \frac{3 - 2 u ^{2}}{2}

Subtrair

\frac{3 - 2 u ^{2}}{2}

de ambos os lados

\frac{1 + 2 u ^{2}}{2} - \frac{3 - 2 u ^{2}}{2} = 4 - 4 \frac{3 - 2 u ^{2}}{2} + \frac{3 - 2 u ^{2}}{2} - \frac{3 - 2 u ^{2}}{2}

Simplificar

\frac{1 + 2 u ^{2}}{2} - \frac{3 - 2 u ^{2}}{2} = 4 - 4 \frac{3 - 2 u ^{2}}{2}

Subtrair

4

de ambos os lados

\frac{1 + 2 u ^{2}}{2} - \frac{3 - 2 u ^{2}}{2} - 4 = 4 - 4 \frac{3 - 2 u ^{2}}{2} - 4

Simplificar

\frac{1 + 2 u ^{2}}{2} - \frac{3 - 2 u ^{2}}{2} - 4 = - 4 \frac{3 - 2 u ^{2}}{2}

Elevar ambos os lados ao quadrado :

4 u^{4} - 20 u^{2} + 25 = 8 (- 2 u^{2} + 3)

\frac{1 + 2 u ^{2}}{2} = 4 - 4 \frac{3 - 2 u ^{2}}{2} + \frac{3 - 2 u ^{2}}{2}

(\frac{1 + 2 u ^{2}}{2} - \frac{3 - 2 u ^{2}}{2} - 4)^{2} = (- 4 \frac{3 - 2 u ^{2}}{2})^{2}

Expandir

(\frac{1 + 2 u ^{2}}{2} - \frac{3 - 2 u ^{2}}{2} - 4)^{2} : 4 u^{4} - 20 u^{2} + 25

(\frac{1 + 2 u ^{2}}{2} - \frac{3 - 2 u ^{2}}{2} - 4)^{2}

Combinar as frações usando o mínimo múltiplo comum:

\frac{1 + 2 u ^{2} - ( 3 - 2 u ^{2} )}{2}

Aplicar a regra

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

= \frac{1 + 2 u ^{2} - ( - 2 u ^{2} + 3 )}{2}

= (\frac{2 u ^{2} - ( - 2 u ^{2} + 3 ) + 1}{2} - 4)^{2}

\frac{1 + 2 u ^{2} - ( 3 - 2 u ^{2} )}{2} = 2 u^{2} - 1

\frac{1 + 2 u ^{2} - ( 3 - 2 u ^{2} )}{2}

Expandir

1 + 2 u^{2} - (3 - 2 u^{2}) : 4 u^{2} - 2

1 + 2 u^{2} - (3 - 2 u^{2})

- (3 - 2 u^{2}) : - 3 + 2 u^{2}

- (3 - 2 u^{2})

Colocar os parênteses

= - (3) - (- 2 u^{2})

Aplicar as regras dos sinais

- (- a) = a, - (a) = - a

= - 3 + 2 u^{2}

= 1 + 2 u^{2} - 3 + 2 u^{2}

Simplificar

1 + 2 u^{2} - 3 + 2 u^{2} : 4 u^{2} - 2

1 + 2 u^{2} - 3 + 2 u^{2}

Agrupar termos semelhantes

= 2 u^{2} + 2 u^{2} + 1 - 3

Somar elementos similares:

2 u^{2} + 2 u^{2} = 4 u^{2}

= 4 u^{2} + 1 - 3

Somar/subtrair:

1 - 3 = - 2

= 4 u^{2} - 2

= 4 u^{2} - 2

= \frac{4 u ^{2} - 2}{2}

Fatorar

4 u^{2} - 2 : 2 (2 u^{2} - 1)

4 u^{2} - 2

Reescrever como

= 2 \cdot 2 u^{2} - 2 \cdot 1

Fatorar o termo comum

2

= 2 (2 u^{2} - 1)

= \frac{2 ( 2 u ^{2} - 1 )}{2}

Dividir:

\frac{2}{2} = 1

= 2 u^{2} - 1

= (2 u^{2} - 1 - 4)^{2}

Subtrair:

- 1 - 4 = - 5

= (2 u^{2} - 5)^{2}

Aplique a fórmula do quadrado perfeito:

(a - b)^{2} = a^{2} - 2 ab + b^{2}

a = 2 u^{2}, b = 5

= (2 u^{2})^{2} - 2 \cdot 2 u^{2} \cdot 5 + 5^{2}

Simplificar

(2 u^{2})^{2} - 2 \cdot 2 u^{2} \cdot 5 + 5^{2} : 4 u^{4} - 20 u^{2} + 25

(2 u^{2})^{2} - 2 \cdot 2 u^{2} \cdot 5 + 5^{2}

(2 u^{2})^{2} = 4 u^{4}

(2 u^{2})^{2}

Aplicar as propriedades dos expoentes:

(a \cdot b)^{n} = a^{n} b^{n}

= 2^{2} (u^{2})^{2}

(u^{2})^{2} : u^{4}

Aplicar as propriedades dos expoentes:

(a^{b})^{c} = a^{b c}

= u^{2 \cdot 2}

Multiplicar os números:

2 \cdot 2 = 4

= u^{4}

= 2^{2} u^{4}

2^{2} = 4

= 4 u^{4}

2 \cdot 2 u^{2} \cdot 5 = 20 u^{2}

2 \cdot 2 u^{2} \cdot 5

Multiplicar os números:

2 \cdot 2 \cdot 5 = 20

= 20 u^{2}

5^{2} = 25

5^{2}

5^{2} = 25

= 25

= 4 u^{4} - 20 u^{2} + 25

= 4 u^{4} - 20 u^{2} + 25

Expandir

(- 4 \frac{3 - 2 u ^{2}}{2})^{2} : 8 (- 2 u^{2} + 3)

(- 4 \frac{3 - 2 u ^{2}}{2})^{2}

Aplicar as propriedades dos expoentes:

(- a)^{n} = a^{n},

n

é par

(- 4 \frac{3 - 2 u ^{2}}{2})^{2} = (4 \frac{3 - 2 u ^{2}}{2})^{2}

= (4 \frac{3 - 2 u ^{2}}{2})^{2}

Aplicar as propriedades dos expoentes:

(a \cdot b)^{n} = a^{n} b^{n}

= 4^{2} (\frac{3 - 2 u ^{2}}{2})^{2}

(\frac{3 - 2 u ^{2}}{2})^{2} : \frac{3 - 2 u ^{2}}{2}

Aplicar as propriedades dos radicais:

a = a^{\frac{1}{2}}

= ((\frac{3 - 2 u ^{2}}{2})^{\frac{1}{2}})^{2}

Aplicar as propriedades dos expoentes:

(a^{b})^{c} = a^{b c}

= (\frac{3 - 2 u ^{2}}{2})^{\frac{1}{2} \cdot 2}

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

\frac{1}{2} \cdot 2

Multiplicar frações:

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

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

Eliminar o fator comum:

2

= 1

= \frac{3 - 2 u ^{2}}{2}

= 4^{2} \frac{3 - 2 u ^{2}}{2}

4^{2} = 16

= 16 \frac{3 - 2 u ^{2}}{2}

Simplificar

= 8 (- 2 u^{2} + 3)

4 u^{4} - 20 u^{2} + 25 = 8 (- 2 u^{2} + 3)

4 u^{4} - 20 u^{2} + 25 = 8 (- 2 u^{2} + 3)

4 u^{4} - 20 u^{2} + 25 = 8 (- 2 u^{2} + 3)

Resolver

4 u^{4} - 20 u^{2} + 25 = 8 (- 2 u^{2} + 3) : u = \frac{1}{2}, u = - \frac{1}{2}

4 u^{4} - 20 u^{2} + 25 = 8 (- 2 u^{2} + 3)

Expandir

8 (- 2 u^{2} + 3) : - 16 u^{2} + 24

8 (- 2 u^{2} + 3)

Colocar os parênteses utilizando:

a (b + c) = ab + a c

a = 8, b = - 2 u^{2}, c = 3

= 8 (- 2 u^{2}) + 8 \cdot 3

Aplicar as regras dos sinais

+ (- a) = - a

= - 8 \cdot 2 u^{2} + 8 \cdot 3

Simplificar

- 8 \cdot 2 u^{2} + 8 \cdot 3 : - 16 u^{2} + 24

- 8 \cdot 2 u^{2} + 8 \cdot 3

Multiplicar os números:

8 \cdot 2 = 16

= - 16 u^{2} + 8 \cdot 3

Multiplicar os números:

8 \cdot 3 = 24

= - 16 u^{2} + 24

= - 16 u^{2} + 24

4 u^{4} - 20 u^{2} + 25 = - 16 u^{2} + 24

Mova

24

para o lado esquerdo

4 u^{4} - 20 u^{2} + 25 = - 16 u^{2} + 24

Subtrair

24

de ambos os lados

4 u^{4} - 20 u^{2} + 25 - 24 = - 16 u^{2} + 24 - 24

Simplificar

4 u^{4} - 20 u^{2} + 1 = - 16 u^{2}

4 u^{4} - 20 u^{2} + 1 = - 16 u^{2}

Mova

16 u^{2}

para o lado esquerdo

4 u^{4} - 20 u^{2} + 1 = - 16 u^{2}

Adicionar

16 u^{2}

a ambos os lados

4 u^{4} - 20 u^{2} + 1 + 16 u^{2} = - 16 u^{2} + 16 u^{2}

Simplificar

4 u^{4} - 4 u^{2} + 1 = 0

4 u^{4} - 4 u^{2} + 1 = 0

Reescrever a equação com

v = u^{2}

v^{2} = u^{4}

4 v^{2} - 4 v + 1 = 0

Resolver

4 v^{2} - 4 v + 1 = 0 : v = \frac{1}{2}

4 v^{2} - 4 v + 1 = 0

Resolver com a fórmula quadrática

4 v^{2} - 4 v + 1 = 0

Fórmula geral para equações de segundo grau:

Para

a = 4, b = - 4, c = 1

v_{1, 2} = \frac{- ( - 4 ) \pm ( - 4 ) ^{2} - 4 \cdot 4 \cdot 1}{2 \cdot 4}

v_{1, 2} = \frac{- ( - 4 ) \pm ( - 4 ) ^{2} - 4 \cdot 4 \cdot 1}{2 \cdot 4}

(- 4)^{2} - 4 \cdot 4 \cdot 1 = 0

(- 4)^{2} - 4 \cdot 4 \cdot 1

Aplicar as propriedades dos expoentes:

(- a)^{n} = a^{n},

n

é par

(- 4)^{2} = 4^{2}

= 4^{2} - 4 \cdot 4 \cdot 1

Multiplicar os números:

4 \cdot 4 \cdot 1 = 16

= 4^{2} - 16

4^{2} = 16

= 16 - 16

Subtrair:

16 - 16 = 0

= 0

v_{1, 2} = \frac{- ( - 4 ) \pm 0}{2 \cdot 4}

v = \frac{- ( - 4 )}{2 \cdot 4}

\frac{- ( - 4 )}{2 \cdot 4} = \frac{1}{2}

\frac{- ( - 4 )}{2 \cdot 4}

Aplicar a regra

- (- a) = a

= \frac{4}{2 \cdot 4}

Multiplicar os números:

2 \cdot 4 = 8

= \frac{4}{8}

Eliminar o fator comum:

4

= \frac{1}{2}

v = \frac{1}{2}

A solução para a equação de segundo grau é:

v = \frac{1}{2}

v = \frac{1}{2}

Substitua

v = u^{2},

solucione para

u

Resolver

u^{2} = \frac{1}{2} : u = \frac{1}{2}, u = - \frac{1}{2}

u^{2} = \frac{1}{2}

Para

x^{2} = f (a)

as soluções são

x = f (a), - f (a)

u = \frac{1}{2}, u = - \frac{1}{2}

As soluções são

u = \frac{1}{2}, u = - \frac{1}{2}

u = \frac{1}{2}, u = - \frac{1}{2}

Verifique soluções:

u = \frac{1}{2}

Verdadeiro

, u = - \frac{1}{2}

Verdadeiro

Verificar as soluções inserindo-as em

- 2 + \frac{1 + 2 u ^{2}}{2} + \frac{3 - 2 u ^{2}}{2} = 0

Eliminar aquelas que não estejam de acordo com a equação.

Inserir

u = \frac{1}{2} :

Verdadeiro

- 2 + \frac{1 + 2 ( \frac{1}{2} ) ^{2}}{2} + \frac{3 - 2 ( \frac{1}{2} ) ^{2}}{2} = 0

- 2 + \frac{1 + 2 ( \frac{1}{2} ) ^{2}}{2} + \frac{3 - 2 ( \frac{1}{2} ) ^{2}}{2} = 0

- 2 + \frac{1 + 2 ( \frac{1}{2} ) ^{2}}{2} + \frac{3 - 2 ( \frac{1}{2} ) ^{2}}{2}

\frac{1 + 2 ( \frac{1}{2} ) ^{2}}{2} = 1

\frac{1 + 2 ( \frac{1}{2} ) ^{2}}{2}

\frac{1 + 2 ( \frac{1}{2} ) ^{2}}{2} = 1

\frac{1 + 2 ( \frac{1}{2} ) ^{2}}{2}

2 (\frac{1}{2})^{2} = 1

2 (\frac{1}{2})^{2}

(\frac{1}{2})^{2} = \frac{1}{2}

(\frac{1}{2})^{2}

Aplicar as propriedades dos radicais:

a = a^{\frac{1}{2}}

= ((\frac{1}{2})^{\frac{1}{2}})^{2}

Aplicar as propriedades dos expoentes:

(a^{b})^{c} = a^{b c}

= (\frac{1}{2})^{\frac{1}{2} \cdot 2}

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

\frac{1}{2} \cdot 2

Multiplicar frações:

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

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

Eliminar o fator comum:

2

= 1

= \frac{1}{2}

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

Multiplicar frações:

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

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

Eliminar o fator comum:

2

= 1

= \frac{1 + 1}{2}

Somar:

1 + 1 = 2

= \frac{2}{2}

Aplicar a regra

\frac{a}{a} = 1

= 1

= 1

Aplicar a regra

1 = 1

= 1

\frac{3 - 2 ( \frac{1}{2} ) ^{2}}{2} = 1

\frac{3 - 2 ( \frac{1}{2} ) ^{2}}{2}

\frac{3 - 2 ( \frac{1}{2} ) ^{2}}{2} = 1

\frac{3 - 2 ( \frac{1}{2} ) ^{2}}{2}

2 (\frac{1}{2})^{2} = 1

2 (\frac{1}{2})^{2}

(\frac{1}{2})^{2} = \frac{1}{2}

(\frac{1}{2})^{2}

Aplicar as propriedades dos radicais:

a = a^{\frac{1}{2}}

= ((\frac{1}{2})^{\frac{1}{2}})^{2}

Aplicar as propriedades dos expoentes:

(a^{b})^{c} = a^{b c}

= (\frac{1}{2})^{\frac{1}{2} \cdot 2}

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

\frac{1}{2} \cdot 2

Multiplicar frações:

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

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

Eliminar o fator comum:

2

= 1

= \frac{1}{2}

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

Multiplicar frações:

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

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

Eliminar o fator comum:

2

= 1

= \frac{3 - 1}{2}

Subtrair:

3 - 1 = 2

= \frac{2}{2}

Aplicar a regra

\frac{a}{a} = 1

= 1

= 1

Aplicar a regra

1 = 1

= 1

= - 2 + 1 + 1

Somar/subtrair:

- 2 + 1 + 1 = 0

= 0

0 = 0

Verdadeiro

Inserir

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

Verdadeiro

- 2 + \frac{1 + 2 ( - \frac{1}{2} ) ^{2}}{2} + \frac{3 - 2 ( - \frac{1}{2} ) ^{2}}{2} = 0

- 2 + \frac{1 + 2 ( - \frac{1}{2} ) ^{2}}{2} + \frac{3 - 2 ( - \frac{1}{2} ) ^{2}}{2} = 0

- 2 + \frac{1 + 2 ( - \frac{1}{2} ) ^{2}}{2} + \frac{3 - 2 ( - \frac{1}{2} ) ^{2}}{2}

\frac{1 + 2 ( - \frac{1}{2} ) ^{2}}{2} = 1

\frac{1 + 2 ( - \frac{1}{2} ) ^{2}}{2}

\frac{1 + 2 ( - \frac{1}{2} ) ^{2}}{2} = 1

\frac{1 + 2 ( - \frac{1}{2} ) ^{2}}{2}

1 + 2 (- \frac{1}{2})^{2} = 1 + 2 (\frac{1}{2})^{2}

1 + 2 (- \frac{1}{2})^{2}

(- \frac{1}{2})^{2} = (\frac{1}{2})^{2}

(- \frac{1}{2})^{2}

Aplicar as propriedades dos expoentes:

(- a)^{n} = a^{n},

n

é par

(- \frac{1}{2})^{2} = (\frac{1}{2})^{2}

= (\frac{1}{2})^{2}

= 1 + 2 (\frac{1}{2})^{2}

= \frac{1 + 2 ( \frac{1}{2} ) ^{2}}{2}

2 (\frac{1}{2})^{2} = 1

2 (\frac{1}{2})^{2}

(\frac{1}{2})^{2} = \frac{1}{2}

(\frac{1}{2})^{2}

Aplicar as propriedades dos radicais:

a = a^{\frac{1}{2}}

= ((\frac{1}{2})^{\frac{1}{2}})^{2}

Aplicar as propriedades dos expoentes:

(a^{b})^{c} = a^{b c}

= (\frac{1}{2})^{\frac{1}{2} \cdot 2}

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

\frac{1}{2} \cdot 2

Multiplicar frações:

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

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

Eliminar o fator comum:

2

= 1

= \frac{1}{2}

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

Multiplicar frações:

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

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

Eliminar o fator comum:

2

= 1

= \frac{1 + 1}{2}

Somar:

1 + 1 = 2

= \frac{2}{2}

Aplicar a regra

\frac{a}{a} = 1

= 1

= 1

Aplicar a regra

1 = 1

= 1

\frac{3 - 2 ( - \frac{1}{2} ) ^{2}}{2} = 1

\frac{3 - 2 ( - \frac{1}{2} ) ^{2}}{2}

\frac{3 - 2 ( - \frac{1}{2} ) ^{2}}{2} = 1

\frac{3 - 2 ( - \frac{1}{2} ) ^{2}}{2}

3 - 2 (- \frac{1}{2})^{2} = 3 - 2 (\frac{1}{2})^{2}

3 - 2 (- \frac{1}{2})^{2}

(- \frac{1}{2})^{2} = (\frac{1}{2})^{2}

(- \frac{1}{2})^{2}

Aplicar as propriedades dos expoentes:

(- a)^{n} = a^{n},

n

é par

(- \frac{1}{2})^{2} = (\frac{1}{2})^{2}

= (\frac{1}{2})^{2}

= 3 - 2 (\frac{1}{2})^{2}

= \frac{3 - 2 ( \frac{1}{2} ) ^{2}}{2}

2 (\frac{1}{2})^{2} = 1

2 (\frac{1}{2})^{2}

(\frac{1}{2})^{2} = \frac{1}{2}

(\frac{1}{2})^{2}

Aplicar as propriedades dos radicais:

a = a^{\frac{1}{2}}

= ((\frac{1}{2})^{\frac{1}{2}})^{2}

Aplicar as propriedades dos expoentes:

(a^{b})^{c} = a^{b c}

= (\frac{1}{2})^{\frac{1}{2} \cdot 2}

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

\frac{1}{2} \cdot 2

Multiplicar frações:

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

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

Eliminar o fator comum:

2

= 1

= \frac{1}{2}

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

Multiplicar frações:

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

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

Eliminar o fator comum:

2

= 1

= \frac{3 - 1}{2}

Subtrair:

3 - 1 = 2

= \frac{2}{2}

Aplicar a regra

\frac{a}{a} = 1

= 1

= 1

Aplicar a regra

1 = 1

= 1

= - 2 + 1 + 1

Somar/subtrair:

- 2 + 1 + 1 = 0

= 0

0 = 0

Verdadeiro

As soluções são

u = \frac{1}{2}, u = - \frac{1}{2}

Substituir na equação

u = sin (x)

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

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

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

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

Soluções gerais para

sin (x) = \frac{1}{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

sin (x) = - \frac{1}{2} : x = \frac{5 π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

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

Soluções gerais para

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

Combinar toda as soluções

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

Gráfico

Exemplos populares

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

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

1/2 tan^2(x)=1-sec(x)

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sin(2x)=sin^2(x)

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

2sec^2(x)+3(1/(cos(x)))=2

2 sec^{2} (x) + 3 (\frac{1}{cos ( x )}) = 2

cot^2(θ)-3=0

cot^{2} (θ) - 3 = 0