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Popular Trigonometria >

6sin^2(4x)-4=0

Solução

6 sin^{2} (4 x) - 4 = 0

Solução

x = \frac{0.95531 \dots}{4} + \frac{πn}{2}, x = \frac{π}{4} - \frac{0.95531 \dots}{4} + \frac{πn}{2}, x = - \frac{0.95531 \dots}{4} + \frac{πn}{2}, x = \frac{π}{4} + \frac{0.95531 \dots}{4} + \frac{πn}{2}

Graus

x = 13.68390 \dots^{\circ} + 9 0^{\circ} n, x = 31.31609 \dots^{\circ} + 9 0^{\circ} n, x = - 13.68390 \dots^{\circ} + 9 0^{\circ} n, x = 58.68390 \dots^{\circ} + 9 0^{\circ} n

Passos da solução

6 sin^{2} (4 x) - 4 = 0

Usando o método de substituição

6 sin^{2} (4 x) - 4 = 0

Sea:

sin (4 x) = u

6 u^{2} - 4 = 0

6 u^{2} - 4 = 0 : u = \frac{2}{3}, u = - \frac{2}{3}

6 u^{2} - 4 = 0

Mova

4

para o lado direito

6 u^{2} - 4 = 0

Adicionar

4

a ambos os lados

6 u^{2} - 4 + 4 = 0 + 4

Simplificar

6 u^{2} = 4

6 u^{2} = 4

Dividir ambos os lados por

6

6 u^{2} = 4

Dividir ambos os lados por

6

\frac{6 u ^{2}}{6} = \frac{4}{6}

Simplificar

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

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

Para

x^{2} = f (a)

as soluções são

x = f (a), - f (a)

u = \frac{2}{3}, u = - \frac{2}{3}

Substituir na equação

u = sin (4 x)

sin (4 x) = \frac{2}{3}, sin (4 x) = - \frac{2}{3}

sin (4 x) = \frac{2}{3}, sin (4 x) = - \frac{2}{3}

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

sin (4 x) = \frac{2}{3}

Aplique as propriedades trigonométricas inversas

sin (4 x) = \frac{2}{3}

Soluções gerais para

sin (4 x) = \frac{2}{3}

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

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

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

Resolver

4 x = arcsin (\frac{2}{3}) + 2 πn : x = \frac{arcsin ( \frac{2}{3} )}{4} + \frac{πn}{2}

4 x = arcsin (\frac{2}{3}) + 2 πn

Dividir ambos os lados por

4

4 x = arcsin (\frac{2}{3}) + 2 πn

Dividir ambos os lados por

4

\frac{4 x}{4} = \frac{arcsin ( \frac{2}{3} )}{4} + \frac{2 πn}{4}

Simplificar

x = \frac{arcsin ( \frac{2}{3} )}{4} + \frac{πn}{2}

x = \frac{arcsin ( \frac{2}{3} )}{4} + \frac{πn}{2}

Resolver

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

4 x = π - arcsin (\frac{2}{3}) + 2 πn

Dividir ambos os lados por

4

4 x = π - arcsin (\frac{2}{3}) + 2 πn

Dividir ambos os lados por

4

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

Simplificar

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

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

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

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

sin (4 x) = - \frac{2}{3}

Aplique as propriedades trigonométricas inversas

sin (4 x) = - \frac{2}{3}

Soluções gerais para

sin (4 x) = - \frac{2}{3}

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

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

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

Resolver

4 x = arcsin (- \frac{2}{3}) + 2 πn : x = - \frac{arcsin ( \frac{2}{3} )}{4} + \frac{πn}{2}

4 x = arcsin (- \frac{2}{3}) + 2 πn

Simplificar

arcsin (- \frac{2}{3}) + 2 πn : - arcsin (\frac{2}{3}) + 2 πn

arcsin (- \frac{2}{3}) + 2 πn

Utilizar a seguinte propriedade:

arcsin (- x) = - arcsin (x)

arcsin (- \frac{2}{3}) = - arcsin (\frac{2}{3})

= - arcsin (\frac{2}{3}) + 2 πn

4 x = - arcsin (\frac{2}{3}) + 2 πn

Dividir ambos os lados por

4

4 x = - arcsin (\frac{2}{3}) + 2 πn

Dividir ambos os lados por

4

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

Simplificar

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

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

Resolver

4 x = π + arcsin (\frac{2}{3}) + 2 πn : x = \frac{π}{4} + \frac{arcsin ( \frac{2}{3} )}{4} + \frac{πn}{2}

4 x = π + arcsin (\frac{2}{3}) + 2 πn

Dividir ambos os lados por

4

4 x = π + arcsin (\frac{2}{3}) + 2 πn

Dividir ambos os lados por

4

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

Simplificar

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

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

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

Combinar toda as soluções

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

Mostrar soluções na forma decimal

x = \frac{0.95531 \dots}{4} + \frac{πn}{2}, x = \frac{π}{4} - \frac{0.95531 \dots}{4} + \frac{πn}{2}, x = - \frac{0.95531 \dots}{4} + \frac{πn}{2}, x = \frac{π}{4} + \frac{0.95531 \dots}{4} + \frac{πn}{2}

Gráfico

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