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

Solução

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

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn, x = - 0.25268 \dots + 2 πn, x = π + 0.25268 \dots + 2 πn

Graus

x = 9 0^{\circ} + 36 0^{\circ} n, x = 27 0^{\circ} + 36 0^{\circ} n, x = - 14.47751 \dots^{\circ} + 36 0^{\circ} n, x = 194.47751 \dots^{\circ} + 36 0^{\circ} n

Passos da solução

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

Reeecreva usando identidades trigonométricas

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

Utilizar a identidade trigonométrica do arco duplo:

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

= - 2 cos (x) - 4 \cdot 2 sin (x) cos (x)

Simplificar

= - 2 cos (x) - 8 sin (x) cos (x)

- 2 cos (x) - 8 cos (x) sin (x) = 0

Fatorar

- 2 cos (x) - 8 cos (x) sin (x) : - 2 cos (x) (4 sin (x) + 1)

- 2 cos (x) - 8 cos (x) sin (x)

Reescrever

- 8

como

4 \cdot 2

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

Fatorar o termo comum

- 2 cos (x)

= - 2 cos (x) (1 + 4 sin (x))

- 2 cos (x) (4 sin (x) + 1) = 0

Resolver cada parte separadamente

cos (x) = 0 or 4 sin (x) + 1 = 0

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

cos (x) = 0

Soluções gerais para

cos (x) = 0

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}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

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

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

4 sin (x) + 1 = 0

Mova

1

para o lado direito

4 sin (x) + 1 = 0

Subtrair

1

de ambos os lados

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

Simplificar

4 sin (x) = - 1

4 sin (x) = - 1

Dividir ambos os lados por

4

4 sin (x) = - 1

Dividir ambos os lados por

4

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

Simplificar

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

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

Aplique as propriedades trigonométricas inversas

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

Soluções gerais para

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

sin (x) = - a \Rightarrow x = arcsin (- a) + 2 πn, x = π + arcsin (a) + 2 πn

x = arcsin (- \frac{1}{4}) + 2 πn, x = π + arcsin (\frac{1}{4}) + 2 πn

x = arcsin (- \frac{1}{4}) + 2 πn, x = π + arcsin (\frac{1}{4}) + 2 πn

Combinar toda as soluções

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

Mostrar soluções na forma decimal

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn, x = - 0.25268 \dots + 2 πn, x = π + 0.25268 \dots + 2 πn

2 sin (x) cos (x) = 3 cos (x)

33 cos (x) - 6 sin (x) cos (x) = 0

2 tan^{2} (x) + 4 sec (x) + sec^{2} (x) = 2

2 + cos (2 x) = - 3 cos (x), 0 \leq x \leq 2 π

sec (x + 1 0^{\circ}) = 5.759