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6sin(2x)+2sin(x)=0

Solución

6 sin (2 x) + 2 sin (x) = 0

x = 2 πn, x = π + 2 πn, x = 1.73824 \dots + 2 πn, x = - 1.73824 \dots + 2 πn

Grados

x = 0^{\circ} + 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n, x = 99.59406 \dots^{\circ} + 36 0^{\circ} n, x = - 99.59406 \dots^{\circ} + 36 0^{\circ} n

Pasos de solución

6 sin (2 x) + 2 sin (x) = 0

Re-escribir usando identidades trigonométricas

2 sin (x) + 6 sin (2 x)

Utilizar la identidad trigonométrica del ángulo doble:

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

= 2 sin (x) + 6 \cdot 2 sin (x) cos (x)

Simplificar

= 2 sin (x) + 12 sin (x) cos (x)

2 sin (x) + 12 cos (x) sin (x) = 0

Factorizar

2 sin (x) + 12 cos (x) sin (x) : 2 sin (x) (6 cos (x) + 1)

2 sin (x) + 12 cos (x) sin (x)

Reescribir

12

como

6 \cdot 2

= 2 sin (x) + 6 \cdot 2 sin (x) cos (x)

Factorizar el termino común

2 sin (x)

= 2 sin (x) (1 + 6 cos (x))

2 sin (x) (6 cos (x) + 1) = 0

Resolver cada parte por separado

sin (x) = 0 or 6 cos (x) + 1 = 0

sin (x) = 0 : x = 2 πn, x = π + 2 πn

sin (x) = 0

Soluciones generales para

sin (x) = 0

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}

x = 0 + 2 πn, x = π + 2 πn

x = 0 + 2 πn, x = π + 2 πn

Resolver

x = 0 + 2 πn : x = 2 πn

x = 0 + 2 πn

0 + 2 πn = 2 πn

x = 2 πn

x = 2 πn, x = π + 2 πn

6 cos (x) + 1 = 0 : x = arccos (- \frac{1}{6}) + 2 πn, x = - arccos (- \frac{1}{6}) + 2 πn

6 cos (x) + 1 = 0

Mover

1

al lado derecho

6 cos (x) + 1 = 0

Restar

1

de ambos lados

6 cos (x) + 1 - 1 = 0 - 1

Simplificar

6 cos (x) = - 1

6 cos (x) = - 1

Dividir ambos lados entre

6

6 cos (x) = - 1

Dividir ambos lados entre

6

\frac{6 cos ( x )}{6} = \frac{- 1}{6}

Simplificar

cos (x) = - \frac{1}{6}

cos (x) = - \frac{1}{6}

Aplicar propiedades trigonométricas inversas

cos (x) = - \frac{1}{6}

Soluciones generales para

cos (x) = - \frac{1}{6}

cos (x) = - a \Rightarrow x = arccos (- a) + 2 πn, x = - arccos (- a) + 2 πn

x = arccos (- \frac{1}{6}) + 2 πn, x = - arccos (- \frac{1}{6}) + 2 πn

x = arccos (- \frac{1}{6}) + 2 πn, x = - arccos (- \frac{1}{6}) + 2 πn

Combinar toda las soluciones

x = 2 πn, x = π + 2 πn, x = arccos (- \frac{1}{6}) + 2 πn, x = - arccos (- \frac{1}{6}) + 2 πn

Mostrar soluciones en forma decimal

x = 2 πn, x = π + 2 πn, x = 1.73824 \dots + 2 πn, x = - 1.73824 \dots + 2 πn

3 sin^{2} (w) - 4 sin (w) - 4 = 0

sin (θ) = \frac{9}{10}

2 sec^{2} (x) - tan^{2} (x) = 3

tan (x) - 2 sin (x) = 0

4 sin (2 t) = 4 cos (t)