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Popular Trigonometria >

sin(x+30)=2cos(x)

Solução

sin (x + 3 0^{\circ}) = 2 cos (x)

Solução

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

Radianos

x = \frac{π}{3} + πn

Passos da solução

sin (x + 3 0^{\circ}) = 2 cos (x)

Reeecreva usando identidades trigonométricas

sin (x + 3 0^{\circ}) = 2 cos (x)

Reeecreva usando identidades trigonométricas

sin (x + 3 0^{\circ})

Use a identidade de soma de ângulos:

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

= sin (x) cos (3 0^{\circ}) + cos (x) sin (3 0^{\circ})

Simplificar

sin (x) cos (3 0^{\circ}) + cos (x) sin (3 0^{\circ}) : \frac{3}{2} sin (x) + \frac{1}{2} cos (x)

sin (x) cos (3 0^{\circ}) + cos (x) sin (3 0^{\circ})

Simplificar

cos (3 0^{\circ}) : \frac{3}{2}

cos (3 0^{\circ})

Utilizar a seguinte identidade trivial:

cos (3 0^{\circ}) = \frac{3}{2}

cos (x)

tabela de periodicidade com ciclo de

36 0^{\circ} n

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

= \frac{3}{2}

= \frac{3}{2} sin (x) + sin (3 0^{\circ}) cos (x)

Simplificar

sin (3 0^{\circ}) : \frac{1}{2}

sin (3 0^{\circ})

Utilizar a seguinte identidade trivial:

sin (3 0^{\circ}) = \frac{1}{2}

sin (x)

tabela de periodicidade com ciclo de

36 0^{\circ} n

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

= \frac{1}{2}

= \frac{3}{2} sin (x) + \frac{1}{2} cos (x)

= \frac{3}{2} sin (x) + \frac{1}{2} cos (x)

\frac{3}{2} sin (x) + \frac{1}{2} cos (x) = 2 cos (x)

\frac{3}{2} sin (x) + \frac{1}{2} cos (x) = 2 cos (x)

Subtrair

2 cos (x)

de ambos os lados

\frac{3}{2} sin (x) - \frac{3}{2} cos (x) = 0

Simplificar

\frac{3}{2} sin (x) - \frac{3}{2} cos (x) : \frac{3 sin ( x ) - 3 cos ( x )}{2}

\frac{3}{2} sin (x) - \frac{3}{2} cos (x)

Multiplicar

\frac{3}{2} sin (x) : \frac{3 sin ( x )}{2}

\frac{3}{2} sin (x)

Multiplicar frações:

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

= \frac{3 sin ( x )}{2}

= \frac{3 sin ( x )}{2} - \frac{3}{2} cos (x)

Multiplicar

\frac{3}{2} cos (x) : \frac{3 cos ( x )}{2}

\frac{3}{2} cos (x)

Multiplicar frações:

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

= \frac{3 cos ( x )}{2}

= \frac{3 sin ( x )}{2} - \frac{3 cos ( x )}{2}

Aplicar a regra

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

= \frac{3 sin ( x ) - 3 cos ( x )}{2}

\frac{3 sin ( x ) - 3 cos ( x )}{2} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

3 sin (x) - 3 cos (x) = 0

Reeecreva usando identidades trigonométricas

3 sin (x) - 3 cos (x) = 0

Dividir ambos os lados por

cos (x), cos (x) \neq = 0

\frac{3 sin ( x ) - 3 cos ( x )}{cos ( x )} = \frac{0}{cos ( x )}

Simplificar

\frac{3 sin ( x )}{cos ( x )} - 3 = 0

Utilizar a Identidade Básica da Trigonometria:

\frac{sin ( x )}{cos ( x )} = tan (x)

3 tan (x) - 3 = 0

3 tan (x) - 3 = 0

Mova

3

para o lado direito

3 tan (x) - 3 = 0

Adicionar

3

a ambos os lados

3 tan (x) - 3 + 3 = 0 + 3

Simplificar

3 tan (x) = 3

3 tan (x) = 3

Dividir ambos os lados por

3

3 tan (x) = 3

Dividir ambos os lados por

3

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

Simplificar

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

Simplificar

\frac{3 tan ( x )}{3} : tan (x)

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

Eliminar o fator comum:

3

= tan (x)

Simplificar

\frac{3}{3} : 3

\frac{3}{3}

Aplicar as propriedades dos radicais:

n a = a^{\frac{1}{n}}

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

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

Aplicar as propriedades dos expoentes:

\frac{x ^{a}}{x ^{b}} = x^{a - b}

\frac{3 ^{1}}{3 ^{\frac{1}{2}}} = 3^{1 - \frac{1}{2}}

= 3^{1 - \frac{1}{2}}

Subtrair:

1 - \frac{1}{2} = \frac{1}{2}

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

Aplicar as propriedades dos radicais:

a^{\frac{1}{n}} = n a

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

= 3

tan (x) = 3

tan (x) = 3

tan (x) = 3

Soluções gerais para

tan (x) = 3

tan (x)

tabela de periodicidade com ciclo de

18 0^{\circ} n

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

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

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

Gráfico

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(sin(8))/(30)=(sin(θ))/(120)

\frac{sin ( 8 )}{30} = \frac{sin ( θ )}{120}