English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

cot^3(x)+2cot^2(x)-3cot(x)-6=0

Solución

cot^{3} (x) + 2 cot^{2} (x) - 3 cot (x) - 6 = 0

Solución

x = 2.67794 \dots + πn, x = \frac{5 π}{6} + πn, x = \frac{π}{6} + πn

Grados

x = 153.43494 \dots^{\circ} + 18 0^{\circ} n, x = 15 0^{\circ} + 18 0^{\circ} n, x = 3 0^{\circ} + 18 0^{\circ} n

Pasos de solución

cot^{3} (x) + 2 cot^{2} (x) - 3 cot (x) - 6 = 0

Usando el método de sustitución

cot^{3} (x) + 2 cot^{2} (x) - 3 cot (x) - 6 = 0

Sea:

cot (x) = u

u^{3} + 2 u^{2} - 3 u - 6 = 0

u^{3} + 2 u^{2} - 3 u - 6 = 0 : u = - 2, u = - 3, u = 3

u^{3} + 2 u^{2} - 3 u - 6 = 0

Factorizar

u^{3} + 2 u^{2} - 3 u - 6 : (u + 2) (u + 3) (u - 3)

u^{3} + 2 u^{2} - 3 u - 6

= (u^{3} + 2 u^{2}) + (- 3 u - 6)

Factorizar

- 3

- 3 u - 6

- 3 (u + 2)

- 3 u - 6

Reescribir

6

como

3 \cdot 2

= - 3 u - 3 \cdot 2

Factorizar el termino común

- 3

= - 3 (u + 2)

Factorizar

u^{2}

u^{3} + 2 u^{2}

u^{2} (u + 2)

u^{3} + 2 u^{2}

Aplicar las leyes de los exponentes:

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

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

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

Factorizar el termino común

u^{2}

= u^{2} (u + 2)

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

Factorizar el termino común

u + 2

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

Factorizar

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

u^{2} - 3

Aplicar las leyes de los exponentes:

a = (a)^{2}

3 = (3)^{2}

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

Aplicar la siguiente regla para binomios al cuadrado:

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

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

= (u + 3) (u - 3)

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

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

Utilizando el teorema de factor cero: si

ab = 0

entonces

a = 0

b = 0

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

Resolver

u + 2 = 0 : u = - 2

u + 2 = 0

Mover

2

al lado derecho

u + 2 = 0

Restar

2

de ambos lados

u + 2 - 2 = 0 - 2

Simplificar

u = - 2

u = - 2

Resolver

u + 3 = 0 : u = - 3

u + 3 = 0

Mover

3

al lado derecho

u + 3 = 0

Restar

3

de ambos lados

u + 3 - 3 = 0 - 3

Simplificar

u = - 3

u = - 3

Resolver

u - 3 = 0 : u = 3

u - 3 = 0

Mover

3

al lado derecho

u - 3 = 0

Sumar

3

a ambos lados

u - 3 + 3 = 0 + 3

Simplificar

u = 3

u = 3

Las soluciones son

u = - 2, u = - 3, u = 3

Sustituir en la ecuación

u = cot (x)

cot (x) = - 2, cot (x) = - 3, cot (x) = 3

cot (x) = - 2, cot (x) = - 3, cot (x) = 3

cot (x) = - 2 : x = arccot (- 2) + πn

cot (x) = - 2

Aplicar propiedades trigonométricas inversas

cot (x) = - 2

Soluciones generales para

cot (x) = - 2

cot (x) = - a \Rightarrow x = arccot (- a) + πn

x = arccot (- 2) + πn

x = arccot (- 2) + πn

cot (x) = - 3 : x = \frac{5 π}{6} + πn

cot (x) = - 3

Soluciones generales para

cot (x) = - 3

cot (x)

tabla de valores periódicos con

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} cot (x) \mp \infty 31 \frac{3}{3} 0 - \frac{3}{3} - 1 - 3

x = \frac{5 π}{6} + πn

x = \frac{5 π}{6} + πn

cot (x) = 3 : x = \frac{π}{6} + πn

cot (x) = 3

Soluciones generales para

cot (x) = 3

cot (x)

tabla de valores periódicos con

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} cot (x) \mp \infty 31 \frac{3}{3} 0 - \frac{3}{3} - 1 - 3

x = \frac{π}{6} + πn

x = \frac{π}{6} + πn

Combinar toda las soluciones

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

Mostrar soluciones en forma decimal

x = 2.67794 \dots + πn, x = \frac{5 π}{6} + πn, x = \frac{π}{6} + πn

Ejemplos populares

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

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

12cot^2(x)-cot(x)=1

12 cot^{2} (x) - cot (x) = 1

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

2 + cos (x) = 3 \frac{cos ^{2} ( x )}{2} + \frac{sin ^{2} ( x )}{2}

sin^2(x)+sin^6(x)=3cos^2(2x)

sin^{2} (x) + sin^{6} (x) = 3 cos^{2} (2 x)

cot^2(x)-7cot(x)+10=0

cot^{2} (x) - 7 cot (x) + 10 = 0