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10sin^2(2u)+6cos^2(2u)=8

Solución

10 sin^{2} (2 u) + 6 cos^{2} (2 u) = 8

Solución

u = \frac{π}{8} + πn, u = \frac{3 π}{8} + πn, u = \frac{5 π}{8} + πn, u = \frac{7 π}{8} + πn

Grados

u = 22. 5^{\circ} + 18 0^{\circ} n, u = 67. 5^{\circ} + 18 0^{\circ} n, u = 112. 5^{\circ} + 18 0^{\circ} n, u = 157. 5^{\circ} + 18 0^{\circ} n

Pasos de solución

10 sin^{2} (2 u) + 6 cos^{2} (2 u) = 8

Restar

8

de ambos lados

10 sin^{2} (2 u) + 6 cos^{2} (2 u) - 8 = 0

Re-escribir usando identidades trigonométricas

- 8 + 10 sin^{2} (2 u) + 6 cos^{2} (2 u)

Utilizar la identidad pitagórica:

cos^{2} (x) + sin^{2} (x) = 1

cos^{2} (x) = 1 - sin^{2} (x)

= - 8 + 10 sin^{2} (2 u) + 6 (1 - sin^{2} (2 u))

Simplificar

- 8 + 10 sin^{2} (2 u) + 6 (1 - sin^{2} (2 u)) : 4 sin^{2} (2 u) - 2

- 8 + 10 sin^{2} (2 u) + 6 (1 - sin^{2} (2 u))

Expandir

6 (1 - sin^{2} (2 u)) : 6 - 6 sin^{2} (2 u)

6 (1 - sin^{2} (2 u))

Poner los parentesis utilizando:

a (b - c) = ab - a c

a = 6, b = 1, c = sin^{2} (2 u)

= 6 \cdot 1 - 6 sin^{2} (2 u)

Multiplicar los numeros:

6 \cdot 1 = 6

= 6 - 6 sin^{2} (2 u)

= - 8 + 10 sin^{2} (2 u) + 6 - 6 sin^{2} (2 u)

Simplificar

- 8 + 10 sin^{2} (2 u) + 6 - 6 sin^{2} (2 u) : 4 sin^{2} (2 u) - 2

- 8 + 10 sin^{2} (2 u) + 6 - 6 sin^{2} (2 u)

Agrupar términos semejantes

= 10 sin^{2} (2 u) - 6 sin^{2} (2 u) - 8 + 6

Sumar elementos similares:

10 sin^{2} (2 u) - 6 sin^{2} (2 u) = 4 sin^{2} (2 u)

= 4 sin^{2} (2 u) - 8 + 6

Sumar/restar lo siguiente:

- 8 + 6 = - 2

= 4 sin^{2} (2 u) - 2

= 4 sin^{2} (2 u) - 2

= 4 sin^{2} (2 u) - 2

- 2 + 4 sin^{2} (2 u) = 0

Usando el método de sustitución

- 2 + 4 sin^{2} (2 u) = 0

Sea:

sin (2 u) = v

- 2 + 4 v^{2} = 0

- 2 + 4 v^{2} = 0 : v = \frac{1}{2}, v = - \frac{1}{2}

- 2 + 4 v^{2} = 0

Mover

2

al lado derecho

- 2 + 4 v^{2} = 0

Sumar

2

a ambos lados

- 2 + 4 v^{2} + 2 = 0 + 2

Simplificar

4 v^{2} = 2

4 v^{2} = 2

Dividir ambos lados entre

4

4 v^{2} = 2

Dividir ambos lados entre

4

\frac{4 v ^{2}}{4} = \frac{2}{4}

Simplificar

v^{2} = \frac{1}{2}

v^{2} = \frac{1}{2}

Para

x^{2} = f (a)

las soluciones son

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

v = \frac{1}{2}, v = - \frac{1}{2}

Sustituir en la ecuación

v = sin (2 u)

sin (2 u) = \frac{1}{2}, sin (2 u) = - \frac{1}{2}

sin (2 u) = \frac{1}{2}, sin (2 u) = - \frac{1}{2}

sin (2 u) = \frac{1}{2} : u = \frac{π}{8} + πn, u = \frac{3 π}{8} + πn

sin (2 u) = \frac{1}{2}

Soluciones generales para

sin (2 u) = \frac{1}{2}

tabla de valores periódicos con

2 πn

intervalos:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

2 u = \frac{π}{4} + 2 πn, 2 u = \frac{3 π}{4} + 2 πn

2 u = \frac{π}{4} + 2 πn, 2 u = \frac{3 π}{4} + 2 πn

Resolver

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

2 u = \frac{π}{4} + 2 πn

Dividir ambos lados entre

2

2 u = \frac{π}{4} + 2 πn

Dividir ambos lados entre

2

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

Simplificar

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

Simplificar

\frac{2 u}{2} : u

\frac{2 u}{2}

Dividir:

\frac{2}{2} = 1

= u

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

u = \frac{π}{8} + πn

u = \frac{π}{8} + πn

u = \frac{π}{8} + πn

Resolver

2 u = \frac{3 π}{4} + 2 πn : u = \frac{3 π}{8} + πn

2 u = \frac{3 π}{4} + 2 πn

Dividir ambos lados entre

2

2 u = \frac{3 π}{4} + 2 πn

Dividir ambos lados entre

2

\frac{2 u}{2} = \frac{\frac{3 π}{4}}{2} + \frac{2 πn}{2}

Simplificar

\frac{2 u}{2} = \frac{\frac{3 π}{4}}{2} + \frac{2 πn}{2}

Simplificar

\frac{2 u}{2} : u

\frac{2 u}{2}

Dividir:

\frac{2}{2} = 1

= u

Simplificar

\frac{\frac{3 π}{4}}{2} + \frac{2 πn}{2} : \frac{3 π}{8} + πn

\frac{\frac{3 π}{4}}{2} + \frac{2 πn}{2}

\frac{\frac{3 π}{4}}{2} = \frac{3 π}{8}

\frac{\frac{3 π}{4}}{2}

Aplicar las propiedades de las fracciones:

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

= \frac{3 π}{4 \cdot 2}

Multiplicar los numeros:

4 \cdot 2 = 8

= \frac{3 π}{8}

\frac{2 πn}{2} = πn

\frac{2 πn}{2}

Dividir:

\frac{2}{2} = 1

= πn

= \frac{3 π}{8} + πn

u = \frac{3 π}{8} + πn

u = \frac{3 π}{8} + πn

u = \frac{3 π}{8} + πn

u = \frac{π}{8} + πn, u = \frac{3 π}{8} + πn

sin (2 u) = - \frac{1}{2} : u = \frac{5 π}{8} + πn, u = \frac{7 π}{8} + πn

sin (2 u) = - \frac{1}{2}

Soluciones generales para

sin (2 u) = - \frac{1}{2}

tabla de valores periódicos con

2 πn

intervalos:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

2 u = \frac{5 π}{4} + 2 πn, 2 u = \frac{7 π}{4} + 2 πn

2 u = \frac{5 π}{4} + 2 πn, 2 u = \frac{7 π}{4} + 2 πn

Resolver

2 u = \frac{5 π}{4} + 2 πn : u = \frac{5 π}{8} + πn

2 u = \frac{5 π}{4} + 2 πn

Dividir ambos lados entre

2

2 u = \frac{5 π}{4} + 2 πn

Dividir ambos lados entre

2

\frac{2 u}{2} = \frac{\frac{5 π}{4}}{2} + \frac{2 πn}{2}

Simplificar

\frac{2 u}{2} = \frac{\frac{5 π}{4}}{2} + \frac{2 πn}{2}

Simplificar

\frac{2 u}{2} : u

\frac{2 u}{2}

Dividir:

\frac{2}{2} = 1

= u

Simplificar

\frac{\frac{5 π}{4}}{2} + \frac{2 πn}{2} : \frac{5 π}{8} + πn

\frac{\frac{5 π}{4}}{2} + \frac{2 πn}{2}

\frac{\frac{5 π}{4}}{2} = \frac{5 π}{8}

\frac{\frac{5 π}{4}}{2}

Aplicar las propiedades de las fracciones:

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

= \frac{5 π}{4 \cdot 2}

Multiplicar los numeros:

4 \cdot 2 = 8

= \frac{5 π}{8}

\frac{2 πn}{2} = πn

\frac{2 πn}{2}

Dividir:

\frac{2}{2} = 1

= πn

= \frac{5 π}{8} + πn

u = \frac{5 π}{8} + πn

u = \frac{5 π}{8} + πn

u = \frac{5 π}{8} + πn

Resolver

2 u = \frac{7 π}{4} + 2 πn : u = \frac{7 π}{8} + πn

2 u = \frac{7 π}{4} + 2 πn

Dividir ambos lados entre

2

2 u = \frac{7 π}{4} + 2 πn

Dividir ambos lados entre

2

\frac{2 u}{2} = \frac{\frac{7 π}{4}}{2} + \frac{2 πn}{2}

Simplificar

\frac{2 u}{2} = \frac{\frac{7 π}{4}}{2} + \frac{2 πn}{2}

Simplificar

\frac{2 u}{2} : u

\frac{2 u}{2}

Dividir:

\frac{2}{2} = 1

= u

Simplificar

\frac{\frac{7 π}{4}}{2} + \frac{2 πn}{2} : \frac{7 π}{8} + πn

\frac{\frac{7 π}{4}}{2} + \frac{2 πn}{2}

\frac{\frac{7 π}{4}}{2} = \frac{7 π}{8}

\frac{\frac{7 π}{4}}{2}

Aplicar las propiedades de las fracciones:

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

= \frac{7 π}{4 \cdot 2}

Multiplicar los numeros:

4 \cdot 2 = 8

= \frac{7 π}{8}

\frac{2 πn}{2} = πn

\frac{2 πn}{2}

Dividir:

\frac{2}{2} = 1

= πn

= \frac{7 π}{8} + πn

u = \frac{7 π}{8} + πn

u = \frac{7 π}{8} + πn

u = \frac{7 π}{8} + πn

u = \frac{5 π}{8} + πn, u = \frac{7 π}{8} + πn

Combinar toda las soluciones

u = \frac{π}{8} + πn, u = \frac{3 π}{8} + πn, u = \frac{5 π}{8} + πn, u = \frac{7 π}{8} + πn

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