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solvefor α,sin^2(3α)=0.56

Solución

resolver para

α, sin^{2} (3 α) = 0.56

Solución

α = \frac{0.84554 \dots}{3} + \frac{2 πn}{3}, α = \frac{π}{3} - \frac{0.84554 \dots}{3} + \frac{2 πn}{3}, α = - \frac{0.84554 \dots}{3} + \frac{2 πn}{3}, α = \frac{π}{3} + \frac{0.84554 \dots}{3} + \frac{2 πn}{3}

Grados

α = 16.14868 \dots^{\circ} + 12 0^{\circ} n, α = 43.85131 \dots^{\circ} + 12 0^{\circ} n, α = - 16.14868 \dots^{\circ} + 12 0^{\circ} n, α = 76.14868 \dots^{\circ} + 12 0^{\circ} n

Pasos de solución

sin^{2} (3 α) = 0.56

Usando el método de sustitución

sin^{2} (3 α) = 0.56

Sea:

sin (3 α) = u

u^{2} = 0.56

u^{2} = 0.56 : u = 0.56, u = - 0.56

u^{2} = 0.56

Para

x^{2} = f (a)

las soluciones son

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

u = 0.56, u = - 0.56

Sustituir en la ecuación

u = sin (3 α)

sin (3 α) = 0.56, sin (3 α) = - 0.56

sin (3 α) = 0.56, sin (3 α) = - 0.56

sin (3 α) = 0.56 : α = \frac{arcsin ( 0.56 )}{3} + \frac{2 πn}{3}, α = \frac{π}{3} - \frac{arcsin ( 0.56 )}{3} + \frac{2 πn}{3}

sin (3 α) = 0.56

Aplicar propiedades trigonométricas inversas

sin (3 α) = 0.56

Soluciones generales para

sin (3 α) = 0.56

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

3 α = arcsin (0.56) + 2 πn, 3 α = π - arcsin (0.56) + 2 πn

3 α = arcsin (0.56) + 2 πn, 3 α = π - arcsin (0.56) + 2 πn

Resolver

3 α = arcsin (0.56) + 2 πn : α = \frac{arcsin ( 0.56 )}{3} + \frac{2 πn}{3}

3 α = arcsin (0.56) + 2 πn

Dividir ambos lados entre

3

3 α = arcsin (0.56) + 2 πn

Dividir ambos lados entre

3

\frac{3 α}{3} = \frac{arcsin ( 0.56 )}{3} + \frac{2 πn}{3}

Simplificar

α = \frac{arcsin ( 0.56 )}{3} + \frac{2 πn}{3}

α = \frac{arcsin ( 0.56 )}{3} + \frac{2 πn}{3}

Resolver

3 α = π - arcsin (0.56) + 2 πn : α = \frac{π}{3} - \frac{arcsin ( 0.56 )}{3} + \frac{2 πn}{3}

3 α = π - arcsin (0.56) + 2 πn

Dividir ambos lados entre

3

3 α = π - arcsin (0.56) + 2 πn

Dividir ambos lados entre

3

\frac{3 α}{3} = \frac{π}{3} - \frac{arcsin ( 0.56 )}{3} + \frac{2 πn}{3}

Simplificar

α = \frac{π}{3} - \frac{arcsin ( 0.56 )}{3} + \frac{2 πn}{3}

α = \frac{π}{3} - \frac{arcsin ( 0.56 )}{3} + \frac{2 πn}{3}

α = \frac{arcsin ( 0.56 )}{3} + \frac{2 πn}{3}, α = \frac{π}{3} - \frac{arcsin ( 0.56 )}{3} + \frac{2 πn}{3}

sin (3 α) = - 0.56 : α = - \frac{arcsin ( \frac{14}{5} )}{3} + \frac{2 πn}{3}, α = \frac{π}{3} + \frac{arcsin ( 0.56 )}{3} + \frac{2 πn}{3}

sin (3 α) = - 0.56

Aplicar propiedades trigonométricas inversas

sin (3 α) = - 0.56

Soluciones generales para

sin (3 α) = - 0.56

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

3 α = arcsin (- 0.56) + 2 πn, 3 α = π + arcsin (0.56) + 2 πn

3 α = arcsin (- 0.56) + 2 πn, 3 α = π + arcsin (0.56) + 2 πn

Resolver

3 α = arcsin (- 0.56) + 2 πn : α = - \frac{arcsin ( \frac{14}{5} )}{3} + \frac{2 πn}{3}

3 α = arcsin (- 0.56) + 2 πn

Simplificar

arcsin (- 0.56) + 2 πn : - arcsin (\frac{14}{5}) + 2 πn

arcsin (- 0.56) + 2 πn

arcsin (- 0.56) = - arcsin (\frac{14}{5})

arcsin (- 0.56)

= arcsin (- \frac{14}{25})

Utilizar la siguiente propiedad:

arcsin (- x) = - arcsin (x)

arcsin (- \frac{14}{25}) = - arcsin (\frac{14}{25})

= - arcsin (\frac{14}{25})

= - arcsin (\frac{14}{5})

= - arcsin (\frac{14}{5}) + 2 πn

3 α = - arcsin (\frac{14}{5}) + 2 πn

Dividir ambos lados entre

3

3 α = - arcsin (\frac{14}{5}) + 2 πn

Dividir ambos lados entre

3

\frac{3 α}{3} = - \frac{arcsin ( \frac{14}{5} )}{3} + \frac{2 πn}{3}

Simplificar

α = - \frac{arcsin ( \frac{14}{5} )}{3} + \frac{2 πn}{3}

α = - \frac{arcsin ( \frac{14}{5} )}{3} + \frac{2 πn}{3}

Resolver

3 α = π + arcsin (0.56) + 2 πn : α = \frac{π}{3} + \frac{arcsin ( 0.56 )}{3} + \frac{2 πn}{3}

3 α = π + arcsin (0.56) + 2 πn

Dividir ambos lados entre

3

3 α = π + arcsin (0.56) + 2 πn

Dividir ambos lados entre

3

\frac{3 α}{3} = \frac{π}{3} + \frac{arcsin ( 0.56 )}{3} + \frac{2 πn}{3}

Simplificar

α = \frac{π}{3} + \frac{arcsin ( 0.56 )}{3} + \frac{2 πn}{3}

α = \frac{π}{3} + \frac{arcsin ( 0.56 )}{3} + \frac{2 πn}{3}

α = - \frac{arcsin ( \frac{14}{5} )}{3} + \frac{2 πn}{3}, α = \frac{π}{3} + \frac{arcsin ( 0.56 )}{3} + \frac{2 πn}{3}

Combinar toda las soluciones

α = \frac{arcsin ( 0.56 )}{3} + \frac{2 πn}{3}, α = \frac{π}{3} - \frac{arcsin ( 0.56 )}{3} + \frac{2 πn}{3}, α = - \frac{arcsin ( \frac{14}{5} )}{3} + \frac{2 πn}{3}, α = \frac{π}{3} + \frac{arcsin ( 0.56 )}{3} + \frac{2 πn}{3}

Mostrar soluciones en forma decimal

α = \frac{0.84554 \dots}{3} + \frac{2 πn}{3}, α = \frac{π}{3} - \frac{0.84554 \dots}{3} + \frac{2 πn}{3}, α = - \frac{0.84554 \dots}{3} + \frac{2 πn}{3}, α = \frac{π}{3} + \frac{0.84554 \dots}{3} + \frac{2 πn}{3}

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