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3+sin(2x)=1-6sin(2x)

Solución

3 + sin (2 x) = 1 - 6 sin (2 x)

Solución

x = - \frac{0.28975 \dots}{2} + πn, x = \frac{π}{2} + \frac{0.28975 \dots}{2} + πn

+1

Grados

x = - 8.30077 \dots^{\circ} + 18 0^{\circ} n, x = 98.30077 \dots^{\circ} + 18 0^{\circ} n

Pasos de solución

3 + sin (2 x) = 1 - 6 sin (2 x)

Usando el método de sustitución

3 + sin (2 x) = 1 - 6 sin (2 x)

Sea:

sin (2 x) = u

3 + u = 1 - 6 u

3 + u = 1 - 6 u : u = - \frac{2}{7}

3 + u = 1 - 6 u

Mover

3

al lado derecho

3 + u = 1 - 6 u

Restar

3

de ambos lados

3 + u - 3 = 1 - 6 u - 3

Simplificar

u = - 6 u - 2

u = - 6 u - 2

Mover

6 u

al lado izquierdo

u = - 6 u - 2

Sumar

6 u

a ambos lados

u + 6 u = - 6 u - 2 + 6 u

Simplificar

7 u = - 2

7 u = - 2

Dividir ambos lados entre

7

7 u = - 2

Dividir ambos lados entre

7

\frac{7 u}{7} = \frac{- 2}{7}

Simplificar

u = - \frac{2}{7}

u = - \frac{2}{7}

Sustituir en la ecuación

u = sin (2 x)

sin (2 x) = - \frac{2}{7}

sin (2 x) = - \frac{2}{7}

sin (2 x) = - \frac{2}{7} : x = - \frac{arcsin ( \frac{2}{7} )}{2} + πn, x = \frac{π}{2} + \frac{arcsin ( \frac{2}{7} )}{2} + πn

sin (2 x) = - \frac{2}{7}

Aplicar propiedades trigonométricas inversas

sin (2 x) = - \frac{2}{7}

Soluciones generales para

sin (2 x) = - \frac{2}{7}

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

2 x = arcsin (- \frac{2}{7}) + 2 πn, 2 x = π + arcsin (\frac{2}{7}) + 2 πn

2 x = arcsin (- \frac{2}{7}) + 2 πn, 2 x = π + arcsin (\frac{2}{7}) + 2 πn

Resolver

2 x = arcsin (- \frac{2}{7}) + 2 πn : x = - \frac{arcsin ( \frac{2}{7} )}{2} + πn

2 x = arcsin (- \frac{2}{7}) + 2 πn

Simplificar

arcsin (- \frac{2}{7}) + 2 πn : - arcsin (\frac{2}{7}) + 2 πn

arcsin (- \frac{2}{7}) + 2 πn

Utilizar la siguiente propiedad:

arcsin (- x) = - arcsin (x)

arcsin (- \frac{2}{7}) = - arcsin (\frac{2}{7})

= - arcsin (\frac{2}{7}) + 2 πn

2 x = - arcsin (\frac{2}{7}) + 2 πn

Dividir ambos lados entre

2

2 x = - arcsin (\frac{2}{7}) + 2 πn

Dividir ambos lados entre

2

\frac{2 x}{2} = - \frac{arcsin ( \frac{2}{7} )}{2} + \frac{2 πn}{2}

Simplificar

x = - \frac{arcsin ( \frac{2}{7} )}{2} + πn

x = - \frac{arcsin ( \frac{2}{7} )}{2} + πn

Resolver

2 x = π + arcsin (\frac{2}{7}) + 2 πn : x = \frac{π}{2} + \frac{arcsin ( \frac{2}{7} )}{2} + πn

2 x = π + arcsin (\frac{2}{7}) + 2 πn

Dividir ambos lados entre

2

2 x = π + arcsin (\frac{2}{7}) + 2 πn

Dividir ambos lados entre

2

\frac{2 x}{2} = \frac{π}{2} + \frac{arcsin ( \frac{2}{7} )}{2} + \frac{2 πn}{2}

Simplificar

x = \frac{π}{2} + \frac{arcsin ( \frac{2}{7} )}{2} + πn

x = \frac{π}{2} + \frac{arcsin ( \frac{2}{7} )}{2} + πn

x = - \frac{arcsin ( \frac{2}{7} )}{2} + πn, x = \frac{π}{2} + \frac{arcsin ( \frac{2}{7} )}{2} + πn

Combinar toda las soluciones

x = - \frac{arcsin ( \frac{2}{7} )}{2} + πn, x = \frac{π}{2} + \frac{arcsin ( \frac{2}{7} )}{2} + πn

Mostrar soluciones en forma decimal

x = - \frac{0.28975 \dots}{2} + πn, x = \frac{π}{2} + \frac{0.28975 \dots}{2} + πn

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