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Popular >

4sin(x)cos(x)+2sin(x)-2cos(x)=1

Solución

4 sin (x) cos (x) + 2 sin (x) - 2 cos (x) = 1

Solución

x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn, x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

Grados

x = 12 0^{\circ} + 36 0^{\circ} n, x = 24 0^{\circ} + 36 0^{\circ} n, x = 3 0^{\circ} + 36 0^{\circ} n, x = 15 0^{\circ} + 36 0^{\circ} n

Pasos de solución

4 sin (x) cos (x) + 2 sin (x) - 2 cos (x) = 1

Restar

1

de ambos lados

4 sin (x) cos (x) + 2 sin (x) - 2 cos (x) - 1 = 0

Factorizar

4 sin (x) cos (x) + 2 sin (x) - 2 cos (x) - 1 : (2 cos (x) + 1) (2 sin (x) - 1)

4 sin (x) cos (x) + 2 sin (x) - 2 cos (x) - 1

= (4 sin (x) cos (x) + 2 sin (x)) + (- 2 cos (x) - 1)

Factorizar

- 1

- 2 cos (x) - 1

- (2 cos (x) + 1)

- 2 cos (x) - 1

Factorizar el termino común

- 1

= - (2 cos (x) + 1)

Factorizar

2 sin (x)

4 sin (x) cos (x) + 2 sin (x)

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

4 sin (x) cos (x) + 2 sin (x)

Reescribir

4

como

2 \cdot 2

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

Factorizar el termino común

2 sin (x)

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

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

Factorizar el termino común

2 cos (x) + 1

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

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

Resolver cada parte por separado

2 cos (x) + 1 = 0 or 2 sin (x) - 1 = 0

2 cos (x) + 1 = 0 : x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn

2 cos (x) + 1 = 0

Mover

1

al lado derecho

2 cos (x) + 1 = 0

Restar

1

de ambos lados

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

Simplificar

2 cos (x) = - 1

2 cos (x) = - 1

Dividir ambos lados entre

2

2 cos (x) = - 1

Dividir ambos lados entre

2

\frac{2 cos ( x )}{2} = \frac{- 1}{2}

Simplificar

cos (x) = - \frac{1}{2}

cos (x) = - \frac{1}{2}

Soluciones generales para

cos (x) = - \frac{1}{2}

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{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn

x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn

2 sin (x) - 1 = 0 : x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

2 sin (x) - 1 = 0

Mover

1

al lado derecho

2 sin (x) - 1 = 0

Sumar

1

a ambos lados

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

Simplificar

2 sin (x) = 1

2 sin (x) = 1

Dividir ambos lados entre

2

2 sin (x) = 1

Dividir ambos lados entre

2

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

Simplificar

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

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

Soluciones generales para

sin (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}

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

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

Combinar toda las soluciones

x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn, x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

Ejemplos populares

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

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

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

3 sin (x) cos (x) = cos^{2} (x)

2cos^2(4x)+1=0

2 cos^{2} (4 x) + 1 = 0

1+cos(2θ)=-3sin(θ)

1 + cos (2 θ) = - 3 sin (θ)

2cos(θ)+sin(2θ)=0

2 cos (θ) + sin (2 θ) = 0