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

Solução

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

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

Graus

x = 0^{\circ} + 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n, x = 6 0^{\circ} + 36 0^{\circ} n, x = 30 0^{\circ} + 36 0^{\circ} n

Passos da solução

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

Subtrair

sin (2 x)

de ambos os lados

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

Reeecreva usando identidades trigonométricas

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

Utilizar a identidade trigonométrica do arco duplo:

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

= - 2 \cdot 2 sin (x) cos (x) + 2 sin (x)

Simplificar

= - 4 sin (x) cos (x) + 2 sin (x)

2 sin (x) - 4 cos (x) sin (x) = 0

Fatorar

2 sin (x) - 4 cos (x) sin (x) : - 2 sin (x) (2 cos (x) - 1)

2 sin (x) - 4 cos (x) sin (x)

Reescrever

- 4

como

2 \cdot 2

= 2 sin (x) + 2 \cdot 2 sin (x) cos (x)

Fatorar o termo comum

- 2 sin (x)

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

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

Resolver cada parte separadamente

sin (x) = 0 or 2 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

2 cos (x) - 1 = 0 : x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

2 cos (x) - 1 = 0

Mova

1

para o lado direito

2 cos (x) - 1 = 0

Adicionar

1

a ambos os lados

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

Simplificar

2 cos (x) = 1

2 cos (x) = 1

Dividir ambos os lados por

2

2 cos (x) = 1

Dividir ambos os lados por

2

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

Simplificar

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

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

Soluções gerais para

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

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

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

Combinar toda as soluções

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

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

3 cos (x) + 5 = 7, 0^{\circ} \leq x \leq 36 0^{\circ}

cot (3 x - 6^{\circ}) = \frac{3}{3}

sec (x) = 4

\frac{182.5}{sin ( 36. 6 ^{\circ} )} = \frac{248.6}{sin ( x )}