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

Solução

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

x = \frac{1}{2} + \frac{2 n}{3}

Graus

x = 28.64788 \dots^{\circ} + 38.19718 \dots^{\circ} n

Passos da solução

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

Reeecreva usando identidades trigonométricas

sin (2 π x) cos (π x) + cos (2 π x) sin (π x)

Use a identidade de soma de ângulos:

sin (s) cos (t) + cos (s) sin (t) = sin (s + t)

= sin (2 π x + π x)

sin (2 π x + π x) = - 1

Soluções gerais para

sin (2 π x + π x) = - 1

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}

2 π x + π x = \frac{3 π}{2} + 2 πn

2 π x + π x = \frac{3 π}{2} + 2 πn

Resolver

2 π x + π x = \frac{3 π}{2} + 2 πn : x = \frac{1}{2} + \frac{2 n}{3}

2 π x + π x = \frac{3 π}{2} + 2 πn

Somar elementos similares:

2 π x + π x = 3 π x

3 π x = \frac{3 π}{2} + 2 πn

Dividir ambos os lados por

3 π

3 π x = \frac{3 π}{2} + 2 πn

Dividir ambos os lados por

3 π

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

Simplificar

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

Simplificar

\frac{3 π x}{3 π} : x

\frac{3 π x}{3 π}

Dividir:

\frac{3}{3} = 1

= \frac{π x}{π}

Eliminar o fator comum:

π

= x

Simplificar

\frac{\frac{3 π}{2}}{3 π} + \frac{2 πn}{3 π} : \frac{1}{2} + \frac{2 n}{3}

\frac{\frac{3 π}{2}}{3 π} + \frac{2 πn}{3 π}

\frac{\frac{3 π}{2}}{3 π} = \frac{1}{2}

\frac{\frac{3 π}{2}}{3 π}

Aplicar as propriedades das frações:

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

= \frac{3 π}{2 \cdot 3 π}

Multiplicar os números:

2 \cdot 3 = 6

= \frac{3 π}{6 π}

Eliminar o fator comum:

3

= \frac{π}{2 π}

Eliminar o fator comum:

π

= \frac{1}{2}

\frac{2 πn}{3 π} = \frac{2 n}{3}

\frac{2 πn}{3 π}

Eliminar o fator comum:

π

= \frac{2 n}{3}

= \frac{1}{2} + \frac{2 n}{3}

x = \frac{1}{2} + \frac{2 n}{3}

x = \frac{1}{2} + \frac{2 n}{3}

x = \frac{1}{2} + \frac{2 n}{3}

x = \frac{1}{2} + \frac{2 n}{3}

- 4 π^{2} sin (2 π x) = 0

6 sin^{2} (θ) = 2

10 tan (θ) + 9 = 0

189.4 = 100 tan^{2} (4 5^{\circ} - \frac{x}{2})

\frac{20}{3} sin (10 x) + 5 cos (10 x) = 0