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Popular >

solvefor x,4cos^3(x)=3cos(x)

Solución

resolver para

x, 4 cos^{3} (x) = 3 cos (x)

Solución

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn, x = \frac{5 π}{6} + 2 πn, x = \frac{7 π}{6} + 2 πn, x = \frac{π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

Grados

x = 9 0^{\circ} + 36 0^{\circ} n, x = 27 0^{\circ} + 36 0^{\circ} n, x = 15 0^{\circ} + 36 0^{\circ} n, x = 21 0^{\circ} + 36 0^{\circ} n, x = 3 0^{\circ} + 36 0^{\circ} n, x = 33 0^{\circ} + 36 0^{\circ} n

Pasos de solución

4 cos^{3} (x) = 3 cos (x)

Usando el método de sustitución

4 cos^{3} (x) = 3 cos (x)

Sea:

cos (x) = u

4 u^{3} = 3 u

4 u^{3} = 3 u : u = 0, u = - \frac{3}{2}, u = \frac{3}{2}

4 u^{3} = 3 u

Mover

3 u

al lado izquierdo

4 u^{3} = 3 u

Restar

3 u

de ambos lados

4 u^{3} - 3 u = 3 u - 3 u

Simplificar

4 u^{3} - 3 u = 0

4 u^{3} - 3 u = 0

Factorizar

4 u^{3} - 3 u : u (2 u + 3) (2 u - 3)

4 u^{3} - 3 u

Factorizar el termino común

u : u (4 u^{2} - 3)

4 u^{3} - 3 u

Aplicar las leyes de los exponentes:

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

u^{3} = u^{2} u

= 4 u^{2} u - 3 u

Factorizar el termino común

u

= u (4 u^{2} - 3)

= u (4 u^{2} - 3)

Factorizar

4 u^{2} - 3 : (2 u + 3) (2 u - 3)

4 u^{2} - 3

Reescribir

4 u^{2} - 3

como

(2 u)^{2} - (3)^{2}

4 u^{2} - 3

Reescribir

4

como

2^{2}

= 2^{2} u^{2} - 3

Aplicar las leyes de los exponentes:

a = (a)^{2}

3 = (3)^{2}

= 2^{2} u^{2} - (3)^{2}

Aplicar las leyes de los exponentes:

a^{m} b^{m} = (ab)^{m}

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

= (2 u)^{2} - (3)^{2}

= (2 u)^{2} - (3)^{2}

Aplicar la siguiente regla para binomios al cuadrado:

x^{2} - y^{2} = (x + y) (x - y)

(2 u)^{2} - (3)^{2} = (2 u + 3) (2 u - 3)

= (2 u + 3) (2 u - 3)

= u (2 u + 3) (2 u - 3)

u (2 u + 3) (2 u - 3) = 0

Utilizando el teorema de factor cero: si

ab = 0

entonces

a = 0

b = 0

u = 0 or 2 u + 3 = 0 or 2 u - 3 = 0

Resolver

2 u + 3 = 0 : u = - \frac{3}{2}

2 u + 3 = 0

Mover

3

al lado derecho

2 u + 3 = 0

Restar

3

de ambos lados

2 u + 3 - 3 = 0 - 3

Simplificar

2 u = - 3

2 u = - 3

Dividir ambos lados entre

2

2 u = - 3

Dividir ambos lados entre

2

\frac{2 u}{2} = \frac{- 3}{2}

Simplificar

u = - \frac{3}{2}

u = - \frac{3}{2}

Resolver

2 u - 3 = 0 : u = \frac{3}{2}

2 u - 3 = 0

Mover

3

al lado derecho

2 u - 3 = 0

Sumar

3

a ambos lados

2 u - 3 + 3 = 0 + 3

Simplificar

2 u = 3

2 u = 3

Dividir ambos lados entre

2

2 u = 3

Dividir ambos lados entre

2

\frac{2 u}{2} = \frac{3}{2}

Simplificar

u = \frac{3}{2}

u = \frac{3}{2}

Las soluciones son

u = 0, u = - \frac{3}{2}, u = \frac{3}{2}

Sustituir en la ecuación

u = cos (x)

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

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

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

cos (x) = - \frac{3}{2} : x = \frac{5 π}{6} + 2 πn, x = \frac{7 π}{6} + 2 πn

cos (x) = - \frac{3}{2}

Soluciones generales para

cos (x) = - \frac{3}{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{5 π}{6} + 2 πn, x = \frac{7 π}{6} + 2 πn

x = \frac{5 π}{6} + 2 πn, x = \frac{7 π}{6} + 2 πn

cos (x) = \frac{3}{2} : x = \frac{π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

cos (x) = \frac{3}{2}

Soluciones generales para

cos (x) = \frac{3}{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{π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

x = \frac{π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

Combinar toda las soluciones

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn, x = \frac{5 π}{6} + 2 πn, x = \frac{7 π}{6} + 2 πn, x = \frac{π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

Ejemplos populares

cos^2(x)=2cos(x)-1

cos^{2} (x) = 2 cos (x) - 1

sqrt(11)sin(x+37.09)=1

11 sin (x + 37.0 9^{\circ}) = 1

cot^2(x)+3csc(x)+3=0

cot^{2} (x) + 3 csc (x) + 3 = 0

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

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

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

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