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sin(2x)=2sin(x)

Solução

sin (2 x) = 2 sin (x)

x = 2 πn, x = π + 2 πn

Graus

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

Passos da solução

sin (2 x) = 2 sin (x)

Subtrair

2 sin (x)

de ambos os lados

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

Reeecreva usando identidades trigonométricas

sin (2 x) - 2 sin (x)

Utilizar a identidade trigonométrica do arco duplo:

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

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

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

Fatorar

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

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

Fatorar o termo comum

2 sin (x)

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

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

Resolver cada parte separadamente

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

cos (x) - 1 = 0 : x = 2 πn

cos (x) - 1 = 0

Mova

1

para o lado direito

cos (x) - 1 = 0

Adicionar

1

a ambos os lados

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

Simplificar

cos (x) = 1

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

Combinar toda as soluções

x = 2 πn, x = π + 2 πn

sin (x) - 1 = 0

cos (x) = \frac{1}{3}

tan (x) + 1 = 0

sin (x) = - cos (x)

tan (x) = \frac{1}{3}