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

Solución

2 cos^{3} (x) - cos^{2} (x) = 0

Solución

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

+1

Grados

x = 9 0^{\circ} + 36 0^{\circ} n, x = 27 0^{\circ} + 36 0^{\circ} n, x = 6 0^{\circ} + 36 0^{\circ} n, x = 30 0^{\circ} + 36 0^{\circ} n

Pasos de solución

2 cos^{3} (x) - cos^{2} (x) = 0

Usando el método de sustitución

2 cos^{3} (x) - cos^{2} (x) = 0

Sea:

cos (x) = u

2 u^{3} - u^{2} = 0

2 u^{3} - u^{2} = 0 : u = 0, u = \frac{1}{2}

2 u^{3} - u^{2} = 0

Factorizar

2 u^{3} - u^{2} : u^{2} (2 u - 1)

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

Aplicar las leyes de los exponentes:

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

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

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

Factorizar el termino común

u^{2}

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

u^{2} (2 u - 1) = 0

Utilizando el teorema de factor cero: si

ab = 0

entonces

a = 0

o

b = 0

u = 0 or 2 u - 1 = 0

Resolver

2 u - 1 = 0 : u = \frac{1}{2}

2 u - 1 = 0

Mover

1

al lado derecho

2 u - 1 = 0

Sumar

1

a ambos lados

2 u - 1 + 1 = 0 + 1

Simplificar

2 u = 1

2 u = 1

Dividir ambos lados entre

2

2 u = 1

Dividir ambos lados entre

2

\frac{2 u}{2} = \frac{1}{2}

Simplificar

u = \frac{1}{2}

u = \frac{1}{2}

Las soluciones son

u = 0, u = \frac{1}{2}

Sustituir en la ecuación

u = cos (x)

cos (x) = 0, cos (x) = \frac{1}{2}

cos (x) = 0, cos (x) = \frac{1}{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{1}{2} : x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

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

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

Combinar toda las soluciones

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

Ejemplos populares

sin^2(x)-2=sin(x)

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

arccosh(x)=(2pi)/7

arccosh (x) = \frac{2 π}{7}

arccosh(x)=(2pi)/5

arccosh (x) = \frac{2 π}{5}

-6sin(x)-4sin(2x)=0

- 6 sin (x) - 4 sin (2 x) = 0

solvefor t,x=2cos(t)resolver para

t, x = 2 cos (t)