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Popular >

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

Solución

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

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

Grados

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

Pasos de solución

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

Restar

cos (x)

de ambos lados

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

Re-escribir usando identidades trigonométricas

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

Utilizar la identidad trigonométrica del ángulo doble:

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

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

Simplificar

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

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

Multiplicar los numeros:

2 \cdot 2 = 4

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

Sumar elementos similares:

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

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

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

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

Factorizar

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

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

Factorizar el termino común

cos (x)

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

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

Resolver cada parte por separado

cos (x) = 0 or 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

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

sin (x) - 1 = 0

Mover

1

al lado derecho

sin (x) - 1 = 0

Sumar

1

a ambos lados

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

Simplificar

sin (x) = 1

sin (x) = 1

Soluciones generales para

sin (x) = 1

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 = \frac{π}{2} + 2 πn

x = \frac{π}{2} + 2 πn

Combinar toda las soluciones

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

5 + sin (3 x) = 4

csc (θ) + 1.597 = 0

9 cos^{2} (x) = 1

csc (θ) = - 5

- 50 \cdot cos (x) - 86.6025400000000 \dots \cdot sin (x) = 50