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Popular >

sin^2(2x)=0.5

Solución

sin^{2} (2 x) = 0.5

Solución

x = \frac{π}{8} + πn, x = \frac{π}{2} - \frac{π}{8} + πn, x = - \frac{π}{8} + πn, x = \frac{π}{2} + \frac{π}{8} + πn

Grados

x = 22. 5^{\circ} + 18 0^{\circ} n, x = 67. 5^{\circ} + 18 0^{\circ} n, x = - 22. 5^{\circ} + 18 0^{\circ} n, x = 112. 5^{\circ} + 18 0^{\circ} n

Pasos de solución

sin^{2} (2 x) = 0.5

Usando el método de sustitución

sin^{2} (2 x) = 0.5

Sea:

sin (2 x) = u

u^{2} = 0.5

u^{2} = 0.5 : u = 0.5, u = - 0.5

u^{2} = 0.5

Para

x^{2} = f (a)

las soluciones son

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

u = 0.5, u = - 0.5

Sustituir en la ecuación

u = sin (2 x)

sin (2 x) = 0.5, sin (2 x) = - 0.5

sin (2 x) = 0.5, sin (2 x) = - 0.5

sin (2 x) = 0.5 : x = \frac{π}{8} + πn, x = \frac{π}{2} - \frac{π}{8} + πn

sin (2 x) = 0.5

Aplicar propiedades trigonométricas inversas

sin (2 x) = 0.5

Soluciones generales para

sin (2 x) = 0.5

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

2 x = arcsin (0.5) + 2 πn, 2 x = π - arcsin (0.5) + 2 πn

2 x = arcsin (0.5) + 2 πn, 2 x = π - arcsin (0.5) + 2 πn

Resolver

2 x = arcsin (0.5) + 2 πn : x = \frac{π}{8} + πn

2 x = arcsin (0.5) + 2 πn

Simplificar

arcsin (0.5) + 2 πn : \frac{π}{4} + 2 πn

arcsin (0.5) + 2 πn

arcsin (0.5) = \frac{π}{4}

arcsin (0.5)

= arcsin (\frac{1}{2})

Utilizar la siguiente identidad trivial:

arcsin (\frac{1}{2}) = \frac{π}{4}

arcsin (\frac{1}{2})

x 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 arcsin (x) 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} arcsin (x) 0^{\circ} 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ}

= \frac{π}{4}

= \frac{π}{4}

= \frac{π}{4} + 2 πn

2 x = \frac{π}{4} + 2 πn

Dividir ambos lados entre

2

2 x = \frac{π}{4} + 2 πn

Dividir ambos lados entre

2

\frac{2 x}{2} = \frac{\frac{π}{4}}{2} + \frac{2 πn}{2}

Simplificar

\frac{2 x}{2} = \frac{\frac{π}{4}}{2} + \frac{2 πn}{2}

Simplificar

\frac{2 x}{2} : x

\frac{2 x}{2}

Dividir:

\frac{2}{2} = 1

= x

Simplificar

\frac{\frac{π}{4}}{2} + \frac{2 πn}{2} : \frac{π}{8} + πn

\frac{\frac{π}{4}}{2} + \frac{2 πn}{2}

\frac{\frac{π}{4}}{2} = \frac{π}{8}

\frac{\frac{π}{4}}{2}

Aplicar las propiedades de las fracciones:

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

= \frac{π}{4 \cdot 2}

Multiplicar los numeros:

4 \cdot 2 = 8

= \frac{π}{8}

\frac{2 πn}{2} = πn

\frac{2 πn}{2}

Dividir:

\frac{2}{2} = 1

= πn

= \frac{π}{8} + πn

x = \frac{π}{8} + πn

x = \frac{π}{8} + πn

x = \frac{π}{8} + πn

Resolver

2 x = π - arcsin (0.5) + 2 πn : x = \frac{π}{2} - \frac{π}{8} + πn

2 x = π - arcsin (0.5) + 2 πn

Simplificar

π - arcsin (0.5) + 2 πn : π - \frac{π}{4} + 2 πn

π - arcsin (0.5) + 2 πn

arcsin (0.5) = \frac{π}{4}

arcsin (0.5)

= arcsin (\frac{1}{2})

Utilizar la siguiente identidad trivial:

arcsin (\frac{1}{2}) = \frac{π}{4}

arcsin (\frac{1}{2})

x 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 arcsin (x) 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} arcsin (x) 0^{\circ} 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ}

= \frac{π}{4}

= \frac{π}{4}

= π - \frac{π}{4} + 2 πn

2 x = π - \frac{π}{4} + 2 πn

Dividir ambos lados entre

2

2 x = π - \frac{π}{4} + 2 πn

Dividir ambos lados entre

2

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

Simplificar

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

Simplificar

\frac{2 x}{2} : x

\frac{2 x}{2}

Dividir:

\frac{2}{2} = 1

= x

Simplificar

\frac{π}{2} - \frac{\frac{π}{4}}{2} + \frac{2 πn}{2} : \frac{π}{2} - \frac{π}{8} + πn

\frac{π}{2} - \frac{\frac{π}{4}}{2} + \frac{2 πn}{2}

\frac{\frac{π}{4}}{2} = \frac{π}{8}

\frac{\frac{π}{4}}{2}

Aplicar las propiedades de las fracciones:

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

= \frac{π}{4 \cdot 2}

Multiplicar los numeros:

4 \cdot 2 = 8

= \frac{π}{8}

\frac{2 πn}{2} = πn

\frac{2 πn}{2}

Dividir:

\frac{2}{2} = 1

= πn

= \frac{π}{2} - \frac{π}{8} + πn

x = \frac{π}{2} - \frac{π}{8} + πn

x = \frac{π}{2} - \frac{π}{8} + πn

x = \frac{π}{2} - \frac{π}{8} + πn

x = \frac{π}{8} + πn, x = \frac{π}{2} - \frac{π}{8} + πn

sin (2 x) = - 0.5 : x = - \frac{π}{8} + πn, x = \frac{π}{2} + \frac{π}{8} + πn

sin (2 x) = - 0.5

Aplicar propiedades trigonométricas inversas

sin (2 x) = - 0.5

Soluciones generales para

sin (2 x) = - 0.5

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

2 x = arcsin (- 0.5) + 2 πn, 2 x = π + arcsin (0.5) + 2 πn

2 x = arcsin (- 0.5) + 2 πn, 2 x = π + arcsin (0.5) + 2 πn

Resolver

2 x = arcsin (- 0.5) + 2 πn : x = - \frac{π}{8} + πn

2 x = arcsin (- 0.5) + 2 πn

Simplificar

arcsin (- 0.5) + 2 πn : - \frac{π}{4} + 2 πn

arcsin (- 0.5) + 2 πn

arcsin (- 0.5) = - \frac{π}{4}

arcsin (- 0.5)

= arcsin (- \frac{1}{2})

Utilizar la siguiente propiedad:

arcsin (- x) = - arcsin (x)

arcsin (- \frac{1}{2}) = - arcsin (\frac{1}{2})

= - arcsin (\frac{1}{2})

Utilizar la siguiente identidad trivial:

arcsin (\frac{1}{2}) = \frac{π}{4}

arcsin (\frac{1}{2})

x 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 arcsin (x) 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} arcsin (x) 0^{\circ} 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ}

= \frac{π}{4}

= - \frac{π}{4}

= - \frac{π}{4} + 2 πn

2 x = - \frac{π}{4} + 2 πn

Dividir ambos lados entre

2

2 x = - \frac{π}{4} + 2 πn

Dividir ambos lados entre

2

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

Simplificar

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

Simplificar

\frac{2 x}{2} : x

\frac{2 x}{2}

Dividir:

\frac{2}{2} = 1

= x

Simplificar

- \frac{\frac{π}{4}}{2} + \frac{2 πn}{2} : - \frac{π}{8} + πn

- \frac{\frac{π}{4}}{2} + \frac{2 πn}{2}

\frac{\frac{π}{4}}{2} = \frac{π}{8}

\frac{\frac{π}{4}}{2}

Aplicar las propiedades de las fracciones:

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

= \frac{π}{4 \cdot 2}

Multiplicar los numeros:

4 \cdot 2 = 8

= \frac{π}{8}

\frac{2 πn}{2} = πn

\frac{2 πn}{2}

Dividir:

\frac{2}{2} = 1

= πn

= - \frac{π}{8} + πn

x = - \frac{π}{8} + πn

x = - \frac{π}{8} + πn

x = - \frac{π}{8} + πn

Resolver

2 x = π + arcsin (0.5) + 2 πn : x = \frac{π}{2} + \frac{π}{8} + πn

2 x = π + arcsin (0.5) + 2 πn

Simplificar

π + arcsin (0.5) + 2 πn : π + \frac{π}{4} + 2 πn

π + arcsin (0.5) + 2 πn

arcsin (0.5) = \frac{π}{4}

arcsin (0.5)

= arcsin (\frac{1}{2})

Utilizar la siguiente identidad trivial:

arcsin (\frac{1}{2}) = \frac{π}{4}

arcsin (\frac{1}{2})

x 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 arcsin (x) 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} arcsin (x) 0^{\circ} 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ}

= \frac{π}{4}

= \frac{π}{4}

= π + \frac{π}{4} + 2 πn

2 x = π + \frac{π}{4} + 2 πn

Dividir ambos lados entre

2

2 x = π + \frac{π}{4} + 2 πn

Dividir ambos lados entre

2

\frac{2 x}{2} = \frac{π}{2} + \frac{\frac{π}{4}}{2} + \frac{2 πn}{2}

Simplificar

\frac{2 x}{2} = \frac{π}{2} + \frac{\frac{π}{4}}{2} + \frac{2 πn}{2}

Simplificar

\frac{2 x}{2} : x

\frac{2 x}{2}

Dividir:

\frac{2}{2} = 1

= x

Simplificar

\frac{π}{2} + \frac{\frac{π}{4}}{2} + \frac{2 πn}{2} : \frac{π}{2} + \frac{π}{8} + πn

\frac{π}{2} + \frac{\frac{π}{4}}{2} + \frac{2 πn}{2}

\frac{\frac{π}{4}}{2} = \frac{π}{8}

\frac{\frac{π}{4}}{2}

Aplicar las propiedades de las fracciones:

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

= \frac{π}{4 \cdot 2}

Multiplicar los numeros:

4 \cdot 2 = 8

= \frac{π}{8}

\frac{2 πn}{2} = πn

\frac{2 πn}{2}

Dividir:

\frac{2}{2} = 1

= πn

= \frac{π}{2} + \frac{π}{8} + πn

x = \frac{π}{2} + \frac{π}{8} + πn

x = \frac{π}{2} + \frac{π}{8} + πn

x = \frac{π}{2} + \frac{π}{8} + πn

x = - \frac{π}{8} + πn, x = \frac{π}{2} + \frac{π}{8} + πn

Combinar toda las soluciones

x = \frac{π}{8} + πn, x = \frac{π}{2} - \frac{π}{8} + πn, x = - \frac{π}{8} + πn, x = \frac{π}{2} + \frac{π}{8} + πn

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