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5sin(2x)+9sin(x)=0

Solución

5 sin (2 x) + 9 sin (x) = 0

x = 2 πn, x = π + 2 πn, x = 2.69056 \dots + 2 πn, x = - 2.69056 \dots + 2 πn

Grados

x = 0^{\circ} + 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n, x = 154.15806 \dots^{\circ} + 36 0^{\circ} n, x = - 154.15806 \dots^{\circ} + 36 0^{\circ} n

Pasos de solución

5 sin (2 x) + 9 sin (x) = 0

Re-escribir usando identidades trigonométricas

5 sin (2 x) + 9 sin (x)

Utilizar la identidad trigonométrica del ángulo doble:

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

= 5 \cdot 2 sin (x) cos (x) + 9 sin (x)

Simplificar

= 10 sin (x) cos (x) + 9 sin (x)

9 sin (x) + 10 cos (x) sin (x) = 0

Factorizar

9 sin (x) + 10 cos (x) sin (x) : sin (x) (10 cos (x) + 9)

9 sin (x) + 10 cos (x) sin (x)

Factorizar el termino común

sin (x)

= sin (x) (9 + 10 cos (x))

sin (x) (10 cos (x) + 9) = 0

Resolver cada parte por separado

sin (x) = 0 or 10 cos (x) + 9 = 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

10 cos (x) + 9 = 0 : x = arccos (- \frac{9}{10}) + 2 πn, x = - arccos (- \frac{9}{10}) + 2 πn

10 cos (x) + 9 = 0

Mover

9

al lado derecho

10 cos (x) + 9 = 0

Restar

9

de ambos lados

10 cos (x) + 9 - 9 = 0 - 9

Simplificar

10 cos (x) = - 9

10 cos (x) = - 9

Dividir ambos lados entre

10

10 cos (x) = - 9

Dividir ambos lados entre

10

\frac{10 cos ( x )}{10} = \frac{- 9}{10}

Simplificar

cos (x) = - \frac{9}{10}

cos (x) = - \frac{9}{10}

Aplicar propiedades trigonométricas inversas

cos (x) = - \frac{9}{10}

Soluciones generales para

cos (x) = - \frac{9}{10}

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

x = arccos (- \frac{9}{10}) + 2 πn, x = - arccos (- \frac{9}{10}) + 2 πn

x = arccos (- \frac{9}{10}) + 2 πn, x = - arccos (- \frac{9}{10}) + 2 πn

Combinar toda las soluciones

x = 2 πn, x = π + 2 πn, x = arccos (- \frac{9}{10}) + 2 πn, x = - arccos (- \frac{9}{10}) + 2 πn

Mostrar soluciones en forma decimal

x = 2 πn, x = π + 2 πn, x = 2.69056 \dots + 2 πn, x = - 2.69056 \dots + 2 πn

4 sin (x) - 6 cos (x) = 3

4 cos (x) = cos^{3} (x)

cos^{2} (a) = \frac{3}{5 sin ( a )}

3 cos (x) = 2 sec (x) - 5

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