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Popular >

(2cos(a)+1)(2cos(a)-1)=2cos^2(a)+1

Solución

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

Solución

a = 2 πn, a = π + 2 πn

Grados

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

Pasos de solución

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

Usando el método de sustitución

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

Sea:

cos (a) = u

(2 u + 1) (2 u - 1) = 2 u^{2} + 1

(2 u + 1) (2 u - 1) = 2 u^{2} + 1 : u = 1, u = - 1

(2 u + 1) (2 u - 1) = 2 u^{2} + 1

Desarrollar

(2 u + 1) (2 u - 1) : 4 u^{2} - 1

(2 u + 1) (2 u - 1)

Aplicar la siguiente regla para binomios al cuadrado:

(a + b) (a - b) = a^{2} - b^{2}

a = 2 u, b = 1

= (2 u)^{2} - 1^{2}

Simplificar

(2 u)^{2} - 1^{2} : 4 u^{2} - 1

(2 u)^{2} - 1^{2}

Aplicar la regla

1^{a} = 1

1^{2} = 1

= (2 u)^{2} - 1

(2 u)^{2} = 4 u^{2}

(2 u)^{2}

Aplicar las leyes de los exponentes:

(a \cdot b)^{n} = a^{n} b^{n}

= 2^{2} u^{2}

2^{2} = 4

= 4 u^{2}

= 4 u^{2} - 1

= 4 u^{2} - 1

4 u^{2} - 1 = 2 u^{2} + 1

Mover

1

al lado derecho

4 u^{2} - 1 = 2 u^{2} + 1

Sumar

1

a ambos lados

4 u^{2} - 1 + 1 = 2 u^{2} + 1 + 1

Simplificar

4 u^{2} = 2 u^{2} + 2

4 u^{2} = 2 u^{2} + 2

Mover

2 u^{2}

al lado izquierdo

4 u^{2} = 2 u^{2} + 2

Restar

2 u^{2}

de ambos lados

4 u^{2} - 2 u^{2} = 2 u^{2} + 2 - 2 u^{2}

Simplificar

2 u^{2} = 2

2 u^{2} = 2

Dividir ambos lados entre

2

2 u^{2} = 2

Dividir ambos lados entre

2

\frac{2 u ^{2}}{2} = \frac{2}{2}

Simplificar

u^{2} = 1

u^{2} = 1

Para

x^{2} = f (a)

las soluciones son

x = f (a), - f (a)

u = 1, u = - 1

1 = 1

1

Aplicar la regla

1 = 1

= 1

- 1 = - 1

- 1

Aplicar la regla

1 = 1

= - 1

u = 1, u = - 1

Sustituir en la ecuación

u = cos (a)

cos (a) = 1, cos (a) = - 1

cos (a) = 1, cos (a) = - 1

cos (a) = 1 : a = 2 πn

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 = 0 + 2 πn

a = 0 + 2 πn

Resolver

a = 0 + 2 πn : a = 2 πn

a = 0 + 2 πn

0 + 2 πn = 2 πn

a = 2 πn

a = 2 πn

cos (a) = - 1 : a = π + 2 πn

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

Combinar toda las soluciones

a = 2 πn, a = π + 2 πn

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