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Popular Trigonometria >

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

Solução

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

Solução

x = \frac{0.33347 \dots}{2} + \frac{18 0 ^{\circ} n}{2}

Radianos

x = \frac{0.33347 \dots}{2} + \frac{π}{2} n

Passos da solução

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

Reeecreva usando identidades trigonométricas

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

Reeecreva usando identidades trigonométricas

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

Use a identidade de soma de ângulos:

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

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

Simplificar

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

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

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

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

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

Subtrair

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

de ambos os lados

\frac{5}{2} sin (2 x) - \frac{3}{2} cos (2 x) = 0

Simplificar

\frac{5}{2} sin (2 x) - \frac{3}{2} cos (2 x) : \frac{5 sin ( 2 x ) - 3 cos ( 2 x )}{2}

\frac{5}{2} sin (2 x) - \frac{3}{2} cos (2 x)

Multiplicar

\frac{5}{2} sin (2 x) : \frac{5 sin ( 2 x )}{2}

\frac{5}{2} sin (2 x)

Multiplicar frações:

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

= \frac{5 sin ( 2 x )}{2}

= \frac{5 sin ( 2 x )}{2} - \frac{3}{2} cos (2 x)

Multiplicar

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

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

Multiplicar frações:

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

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

= \frac{5 sin ( 2 x )}{2} - \frac{3 cos ( 2 x )}{2}

Aplicar a regra

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

= \frac{5 sin ( 2 x ) - 3 cos ( 2 x )}{2}

\frac{5 sin ( 2 x ) - 3 cos ( 2 x )}{2} = 0

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

5 sin (2 x) - 3 cos (2 x) = 0

Reeecreva usando identidades trigonométricas

5 sin (2 x) - 3 cos (2 x) = 0

Dividir ambos os lados por

cos (2 x), cos (2 x) \neq = 0

\frac{5 sin ( 2 x ) - 3 cos ( 2 x )}{cos ( 2 x )} = \frac{0}{cos ( 2 x )}

Simplificar

\frac{5 sin ( 2 x )}{cos ( 2 x )} - 3 = 0

Utilizar a Identidade Básica da Trigonometria:

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

5 tan (2 x) - 3 = 0

5 tan (2 x) - 3 = 0

Mova

3

para o lado direito

5 tan (2 x) - 3 = 0

Adicionar

3

a ambos os lados

5 tan (2 x) - 3 + 3 = 0 + 3

Simplificar

5 tan (2 x) = 3

5 tan (2 x) = 3

Dividir ambos os lados por

5

5 tan (2 x) = 3

Dividir ambos os lados por

5

\frac{5 tan ( 2 x )}{5} = \frac{3}{5}

Simplificar

tan (2 x) = \frac{3}{5}

tan (2 x) = \frac{3}{5}

Aplique as propriedades trigonométricas inversas

tan (2 x) = \frac{3}{5}

Soluções gerais para

tan (2 x) = \frac{3}{5}

tan (x) = a \Rightarrow x = arctan (a) + 18 0^{\circ} n

2 x = arctan (\frac{3}{5}) + 18 0^{\circ} n

2 x = arctan (\frac{3}{5}) + 18 0^{\circ} n

Resolver

2 x = arctan (\frac{3}{5}) + 18 0^{\circ} n : x = \frac{arctan ( \frac{3}{5} )}{2} + \frac{18 0 ^{\circ} n}{2}

2 x = arctan (\frac{3}{5}) + 18 0^{\circ} n

Dividir ambos os lados por

2

2 x = arctan (\frac{3}{5}) + 18 0^{\circ} n

Dividir ambos os lados por

2

\frac{2 x}{2} = \frac{arctan ( \frac{3}{5} )}{2} + \frac{18 0 ^{\circ} n}{2}

Simplificar

x = \frac{arctan ( \frac{3}{5} )}{2} + \frac{18 0 ^{\circ} n}{2}

x = \frac{arctan ( \frac{3}{5} )}{2} + \frac{18 0 ^{\circ} n}{2}

x = \frac{arctan ( \frac{3}{5} )}{2} + \frac{18 0 ^{\circ} n}{2}

Mostrar soluções na forma decimal

x = \frac{0.33347 \dots}{2} + \frac{18 0 ^{\circ} n}{2}

Gráfico

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