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Popular Trigonometría >

(-6cos(x)-5sin(x))^2+11sin^2(x)=51

Solución

(- 6 cos (x) - 5 sin (x))^{2} + 11 sin^{2} (x) = 51

Solución

x = \frac{π}{12} + πn, x = \frac{5 π}{12} + πn

Grados

x = 1 5^{\circ} + 18 0^{\circ} n, x = 7 5^{\circ} + 18 0^{\circ} n

Pasos de solución

(- 6 cos (x) - 5 sin (x))^{2} + 11 sin^{2} (x) = 51

Restar

51

de ambos lados

36 cos^{2} (x) + 60 cos (x) sin (x) + 36 sin^{2} (x) - 51 = 0

Re-escribir usando identidades trigonométricas

- 51 + 36 cos^{2} (x) + 36 sin^{2} (x) + 60 cos (x) sin (x)

Utilizar la identidad pitagórica:

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

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

= - 51 + 36 (1 - sin^{2} (x)) + 36 sin^{2} (x) + 60 cos (x) sin (x)

Simplificar

- 51 + 36 (1 - sin^{2} (x)) + 36 sin^{2} (x) + 60 cos (x) sin (x) : 60 cos (x) sin (x) - 15

- 51 + 36 (1 - sin^{2} (x)) + 36 sin^{2} (x) + 60 cos (x) sin (x)

Expandir

36 (1 - sin^{2} (x)) : 36 - 36 sin^{2} (x)

36 (1 - sin^{2} (x))

Poner los parentesis utilizando:

a (b - c) = ab - a c

a = 36, b = 1, c = sin^{2} (x)

= 36 \cdot 1 - 36 sin^{2} (x)

Multiplicar los numeros:

36 \cdot 1 = 36

= 36 - 36 sin^{2} (x)

= - 51 + 36 - 36 sin^{2} (x) + 36 sin^{2} (x) + 60 cos (x) sin (x)

Simplificar

- 51 + 36 - 36 sin^{2} (x) + 36 sin^{2} (x) + 60 cos (x) sin (x) : 60 cos (x) sin (x) - 15

- 51 + 36 - 36 sin^{2} (x) + 36 sin^{2} (x) + 60 cos (x) sin (x)

Sumar elementos similares:

- 36 sin^{2} (x) + 36 sin^{2} (x) = 0

= - 51 + 36 + 60 cos (x) sin (x)

Sumar/restar lo siguiente:

- 51 + 36 = - 15

= 60 cos (x) sin (x) - 15

= 60 cos (x) sin (x) - 15

= 60 cos (x) sin (x) - 15

Utilizar la identidad trigonométrica del ángulo doble:

2 sin (x) cos (x) = sin (2 x)

sin (x) cos (x) = \frac{sin ( 2 x )}{2}

= - 15 + 60 \cdot \frac{sin ( 2 x )}{2}

- 15 + 60 \cdot \frac{sin ( 2 x )}{2} = 0

60 \cdot \frac{sin ( 2 x )}{2} = 30 sin (2 x)

60 \cdot \frac{sin ( 2 x )}{2}

Multiplicar fracciones:

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

= \frac{sin ( 2 x ) \cdot 60}{2}

Dividir:

\frac{60}{2} = 30

= 30 sin (2 x)

- 15 + 30 sin (2 x) = 0

Desplace

15

a la derecha

- 15 + 30 sin (2 x) = 0

Sumar

15

a ambos lados

- 15 + 30 sin (2 x) + 15 = 0 + 15

Simplificar

30 sin (2 x) = 15

30 sin (2 x) = 15

Dividir ambos lados entre

30

30 sin (2 x) = 15

Dividir ambos lados entre

30

\frac{30 sin ( 2 x )}{30} = \frac{15}{30}

Simplificar

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

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

Soluciones generales para

sin (2 x) = \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 x = \frac{π}{6} + 2 πn, 2 x = \frac{5 π}{6} + 2 πn

2 x = \frac{π}{6} + 2 πn, 2 x = \frac{5 π}{6} + 2 πn

Resolver

2 x = \frac{π}{6} + 2 πn : x = \frac{π}{12} + πn

2 x = \frac{π}{6} + 2 πn

Dividir ambos lados entre

2

2 x = \frac{π}{6} + 2 πn

Dividir ambos lados entre

2

\frac{2 x}{2} = \frac{\frac{π}{6}}{2} + \frac{2 πn}{2}

Simplificar

\frac{2 x}{2} = \frac{\frac{π}{6}}{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}{6}}{2} + \frac{2 πn}{2} : \frac{π}{12} + πn

\frac{\frac{π}{6}}{2} + \frac{2 πn}{2}

\frac{\frac{π}{6}}{2} = \frac{π}{12}

\frac{\frac{π}{6}}{2}

Aplicar las propiedades de las fracciones:

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

= \frac{π}{6 \cdot 2}

Multiplicar los numeros:

6 \cdot 2 = 12

= \frac{π}{12}

\frac{2 πn}{2} = πn

\frac{2 πn}{2}

Dividir:

\frac{2}{2} = 1

= πn

= \frac{π}{12} + πn

x = \frac{π}{12} + πn

x = \frac{π}{12} + πn

x = \frac{π}{12} + πn

Resolver

2 x = \frac{5 π}{6} + 2 πn : x = \frac{5 π}{12} + πn

2 x = \frac{5 π}{6} + 2 πn

Dividir ambos lados entre

2

2 x = \frac{5 π}{6} + 2 πn

Dividir ambos lados entre

2

\frac{2 x}{2} = \frac{\frac{5 π}{6}}{2} + \frac{2 πn}{2}

Simplificar

\frac{2 x}{2} = \frac{\frac{5 π}{6}}{2} + \frac{2 πn}{2}

Simplificar

\frac{2 x}{2} : x

\frac{2 x}{2}

Dividir:

\frac{2}{2} = 1

= x

Simplificar

\frac{\frac{5 π}{6}}{2} + \frac{2 πn}{2} : \frac{5 π}{12} + πn

\frac{\frac{5 π}{6}}{2} + \frac{2 πn}{2}

\frac{\frac{5 π}{6}}{2} = \frac{5 π}{12}

\frac{\frac{5 π}{6}}{2}

Aplicar las propiedades de las fracciones:

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

= \frac{5 π}{6 \cdot 2}

Multiplicar los numeros:

6 \cdot 2 = 12

= \frac{5 π}{12}

\frac{2 πn}{2} = πn

\frac{2 πn}{2}

Dividir:

\frac{2}{2} = 1

= πn

= \frac{5 π}{12} + πn

x = \frac{5 π}{12} + πn

x = \frac{5 π}{12} + πn

x = \frac{5 π}{12} + πn

x = \frac{π}{12} + πn, x = \frac{5 π}{12} + πn

Gráfica

Ejemplos populares

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