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2cos(a)*sin(2a)-1+2sin(2a)-cos(a)=0

Solución

2 cos (a) \cdot sin (2 a) - 1 + 2 sin (2 a) - cos (a) = 0

Solución

a = π + 2 πn, a = \frac{π}{12} + πn, a = \frac{5 π}{12} + πn

Grados

a = 18 0^{\circ} + 36 0^{\circ} n, a = 1 5^{\circ} + 18 0^{\circ} n, a = 7 5^{\circ} + 18 0^{\circ} n

Pasos de solución

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

Re-escribir usando identidades trigonométricas

- 1 - cos (a) + 2 sin (2 a) + 2 cos (a) sin (2 a)

Utilizar la identidad trigonométrica del ángulo doble:

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

= - 1 - cos (a) + 2 \cdot 2 sin (a) cos (a) + 2 cos (a) \cdot 2 sin (a) cos (a)

Simplificar

- 1 - cos (a) + 2 \cdot 2 sin (a) cos (a) + 2 cos (a) \cdot 2 sin (a) cos (a) : - 1 - cos (a) + 4 sin (a) cos (a) + 4 cos^{2} (a) sin (a)

- 1 - cos (a) + 2 \cdot 2 sin (a) cos (a) + 2 cos (a) \cdot 2 sin (a) cos (a)

2 \cdot 2 sin (a) cos (a) = 4 sin (a) cos (a)

2 \cdot 2 sin (a) cos (a)

Multiplicar los numeros:

2 \cdot 2 = 4

= 4 sin (a) cos (a)

2 cos (a) \cdot 2 sin (a) cos (a) = 4 cos^{2} (a) sin (a)

2 cos (a) \cdot 2 sin (a) cos (a)

Multiplicar los numeros:

2 \cdot 2 = 4

= 4 cos (a) sin (a) cos (a)

Aplicar las leyes de los exponentes:

a^{b} \cdot a^{c} = a^{b + c}

cos (a) cos (a) = cos^{1 + 1} (a)

= 4 sin (a) cos^{1 + 1} (a)

Sumar:

1 + 1 = 2

= 4 sin (a) cos^{2} (a)

= - 1 - cos (a) + 4 sin (a) cos (a) + 4 cos^{2} (a) sin (a)

= - 1 - cos (a) + 4 sin (a) cos (a) + 4 cos^{2} (a) sin (a)

- 1 - cos (a) + 4 cos (a) sin (a) + 4 cos^{2} (a) sin (a) = 0

Factorizar

- 1 - cos (a) + 4 cos (a) sin (a) + 4 cos^{2} (a) sin (a) : (cos (a) + 1) (4 sin (a) cos (a) - 1)

- 1 - cos (a) + 4 cos (a) sin (a) + 4 cos^{2} (a) sin (a)

= (- cos (a) - 1) + (4 sin (a) cos^{2} (a) + 4 sin (a) cos (a))

Factorizar

4 sin (a) cos (a)

4 sin (a) cos^{2} (a) + 4 sin (a) cos (a)

4 sin (a) cos (a) (cos (a) + 1)

4 sin (a) cos^{2} (a) + 4 sin (a) cos (a)

Aplicar las leyes de los exponentes:

a^{b + c} = a^{b} a^{c}

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

= 4 sin (a) cos (a) cos (a) + 4 sin (a) cos (a)

Factorizar el termino común

4 sin (a) cos (a)

= 4 sin (a) cos (a) (cos (a) + 1)

Factorizar

- 1

- cos (a) - 1

- (cos (a) + 1)

- cos (a) - 1

Factorizar el termino común

- 1

= - (cos (a) + 1)

= 4 sin (a) cos (a) (cos (a) + 1) - (cos (a) + 1)

Factorizar el termino común

cos (a) + 1

= (cos (a) + 1) (4 sin (a) cos (a) - 1)

(cos (a) + 1) (4 sin (a) cos (a) - 1) = 0

Resolver cada parte por separado

cos (a) + 1 = 0 or 4 sin (a) cos (a) - 1 = 0

cos (a) + 1 = 0 : a = π + 2 πn

cos (a) + 1 = 0

Mover

1

al lado derecho

cos (a) + 1 = 0

Restar

1

de ambos lados

cos (a) + 1 - 1 = 0 - 1

Simplificar

cos (a) = - 1

cos (a) = - 1

Soluciones generales para

cos (a) = - 1

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}

a = π + 2 πn

a = π + 2 πn

4 sin (a) cos (a) - 1 = 0 : a = \frac{π}{12} + πn, a = \frac{5 π}{12} + πn

4 sin (a) cos (a) - 1 = 0

Re-escribir usando identidades trigonométricas

4 sin (a) cos (a) - 1

Utilizar la identidad trigonométrica del ángulo doble:

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

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

= - 1 + 4 \cdot \frac{sin ( 2 a )}{2}

- 1 + 4 \cdot \frac{sin ( 2 a )}{2} = 0

4 \cdot \frac{sin ( 2 a )}{2} = 2 sin (2 a)

4 \cdot \frac{sin ( 2 a )}{2}

Multiplicar fracciones:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{sin ( 2 a ) \cdot 4}{2}

Dividir:

\frac{4}{2} = 2

= 2 sin (2 a)

- 1 + 2 sin (2 a) = 0

Mover

1

al lado derecho

- 1 + 2 sin (2 a) = 0

Sumar

1

a ambos lados

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

Simplificar

2 sin (2 a) = 1

2 sin (2 a) = 1

Dividir ambos lados entre

2

2 sin (2 a) = 1

Dividir ambos lados entre

2

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

Simplificar

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

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

Soluciones generales para

sin (2 a) = \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}

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

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

Resolver

2 a = \frac{π}{6} + 2 πn : a = \frac{π}{12} + πn

2 a = \frac{π}{6} + 2 πn

Dividir ambos lados entre

2

2 a = \frac{π}{6} + 2 πn

Dividir ambos lados entre

2

\frac{2 a}{2} = \frac{\frac{π}{6}}{2} + \frac{2 πn}{2}

Simplificar

\frac{2 a}{2} = \frac{\frac{π}{6}}{2} + \frac{2 πn}{2}

Simplificar

\frac{2 a}{2} : a

\frac{2 a}{2}

Dividir:

\frac{2}{2} = 1

= a

Simplificar

\frac{\frac{π}{6}}{2} + \frac{2 πn}{2} : \frac{π}{12} + πn

\frac{\frac{π}{6}}{2} + \frac{2 πn}{2}

\frac{\frac{π}{6}}{2} = \frac{π}{12}

\frac{\frac{π}{6}}{2}

Aplicar las propiedades de las fracciones:

\frac{\frac{b}{c}}{a} = \frac{b}{c \cdot a}

= \frac{π}{6 \cdot 2}

Multiplicar los numeros:

6 \cdot 2 = 12

= \frac{π}{12}

\frac{2 πn}{2} = πn

\frac{2 πn}{2}

Dividir:

\frac{2}{2} = 1

= πn

= \frac{π}{12} + πn

a = \frac{π}{12} + πn

a = \frac{π}{12} + πn

a = \frac{π}{12} + πn

Resolver

2 a = \frac{5 π}{6} + 2 πn : a = \frac{5 π}{12} + πn

2 a = \frac{5 π}{6} + 2 πn

Dividir ambos lados entre

2

2 a = \frac{5 π}{6} + 2 πn

Dividir ambos lados entre

2

\frac{2 a}{2} = \frac{\frac{5 π}{6}}{2} + \frac{2 πn}{2}

Simplificar

\frac{2 a}{2} = \frac{\frac{5 π}{6}}{2} + \frac{2 πn}{2}

Simplificar

\frac{2 a}{2} : a

\frac{2 a}{2}

Dividir:

\frac{2}{2} = 1

= a

Simplificar

\frac{\frac{5 π}{6}}{2} + \frac{2 πn}{2} : \frac{5 π}{12} + πn

\frac{\frac{5 π}{6}}{2} + \frac{2 πn}{2}

\frac{\frac{5 π}{6}}{2} = \frac{5 π}{12}

\frac{\frac{5 π}{6}}{2}

Aplicar las propiedades de las fracciones:

\frac{\frac{b}{c}}{a} = \frac{b}{c \cdot a}

= \frac{5 π}{6 \cdot 2}

Multiplicar los numeros:

6 \cdot 2 = 12

= \frac{5 π}{12}

\frac{2 πn}{2} = πn

\frac{2 πn}{2}

Dividir:

\frac{2}{2} = 1

= πn

= \frac{5 π}{12} + πn

a = \frac{5 π}{12} + πn

a = \frac{5 π}{12} + πn

a = \frac{5 π}{12} + πn

a = \frac{π}{12} + πn, a = \frac{5 π}{12} + πn

Combinar toda las soluciones

a = π + 2 πn, a = \frac{π}{12} + πn, a = \frac{5 π}{12} + πn

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