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Popular Trigonometria >

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

Solução

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

Solução

x = 2 πn, x = π + 2 πn

+1

Graus

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

Passos da solução

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

Usando o método de substituição

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

Sea:

cos (x) = u

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

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

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

Mova

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

para o lado esquerdo

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

Subtrair

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

de ambos os lados

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

Simplificar

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

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

Multiplicar ambos os lados por

2

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

Multiplicar ambos os lados por

2

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

Simplificar

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

Simplificar

2 \cdot \frac{1}{2} u^{2} : u^{2}

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

Multiplicar frações:

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

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

Eliminar o fator comum:

2

= u^{2} \cdot 1

Multiplicar:

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

= u^{2}

Simplificar

\frac{1 \cdot 2}{2} : 1

\frac{1 \cdot 2}{2}

Multiplicar os números:

1 \cdot 2 = 2

= \frac{2}{2}

Aplicar a regra

\frac{a}{a} = 1

= 1

u^{2} = 1

u^{2} = 1

u^{2} = 1

Para

x^{2} = f (a)

as soluções são

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

u = 1, u = - 1

1 = 1

1

Aplicar a regra

1 = 1

= 1

- 1 = - 1

- 1

Aplicar a regra

1 = 1

= - 1

u = 1, u = - 1

Substituir na equação

u = cos (x)

cos (x) = 1, cos (x) = - 1

cos (x) = 1, cos (x) = - 1

cos (x) = 1 : x = 2 πn

cos (x) = 1

Soluções gerais para

cos (x) = 1

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 = 0 + 2 πn

x = 0 + 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

cos (x) = - 1 : x = π + 2 πn

cos (x) = - 1

Soluções gerais para

cos (x) = - 1

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 = π + 2 πn

x = π + 2 πn

Combinar toda as soluções

x = 2 πn, x = π + 2 πn

Gráfico

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