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3sin(2x)-cos(x)=0

Solución

3 sin (2 x) - cos (x) = 0

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn, x = 0.16744 \dots + 2 πn, x = π - 0.16744 \dots + 2 πn

Grados

x = 9 0^{\circ} + 36 0^{\circ} n, x = 27 0^{\circ} + 36 0^{\circ} n, x = 9.59406 \dots^{\circ} + 36 0^{\circ} n, x = 170.40593 \dots^{\circ} + 36 0^{\circ} n

Pasos de solución

3 sin (2 x) - cos (x) = 0

Re-escribir usando identidades trigonométricas

- cos (x) + 3 sin (2 x)

Utilizar la identidad trigonométrica del ángulo doble:

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

= - cos (x) + 3 \cdot 2 sin (x) cos (x)

Simplificar

= - cos (x) + 6 sin (x) cos (x)

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

Factorizar

- cos (x) + 6 cos (x) sin (x) : cos (x) (6 sin (x) - 1)

- cos (x) + 6 cos (x) sin (x)

Factorizar el termino común

cos (x)

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

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

Resolver cada parte por separado

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

cos (x) = 0 : x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

cos (x) = 0

Soluciones generales para

cos (x) = 0

cos (x)

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} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{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} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

6 sin (x) - 1 = 0 : x = arcsin (\frac{1}{6}) + 2 πn, x = π - arcsin (\frac{1}{6}) + 2 πn

6 sin (x) - 1 = 0

Mover

1

al lado derecho

6 sin (x) - 1 = 0

Sumar

1

a ambos lados

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

Simplificar

6 sin (x) = 1

6 sin (x) = 1

Dividir ambos lados entre

6

6 sin (x) = 1

Dividir ambos lados entre

6

\frac{6 sin ( x )}{6} = \frac{1}{6}

Simplificar

sin (x) = \frac{1}{6}

sin (x) = \frac{1}{6}

Aplicar propiedades trigonométricas inversas

sin (x) = \frac{1}{6}

Soluciones generales para

sin (x) = \frac{1}{6}

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

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

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

Combinar toda las soluciones

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn, x = arcsin (\frac{1}{6}) + 2 πn, x = π - arcsin (\frac{1}{6}) + 2 πn

Mostrar soluciones en forma decimal

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn, x = 0.16744 \dots + 2 πn, x = π - 0.16744 \dots + 2 πn

2 cos^{2} (x) + tan^{2} (x) = 2

sin^{3} (o) = 4 sin (o) sin^{2} (o) sin^{4} (o)

cos^{2} (x) = - 4

8 sin^{4} (x) - 6 sin^{2} (x) + 1 = 0

4 sin^{2} (x) + 2 cos (x) + a = 3