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csc(x)+cot(x)=(sqrt(3))/3

Solución

csc (x) + cot (x) = \frac{3}{3}

Solución

x = \frac{2 π}{3} + 2 πn

Grados

x = 12 0^{\circ} + 36 0^{\circ} n

Pasos de solución

csc (x) + cot (x) = \frac{3}{3}

Restar

\frac{3}{3}

de ambos lados

csc (x) + cot (x) - \frac{1}{3} = 0

Simplificar

csc (x) + cot (x) - \frac{1}{3} : \frac{3 csc ( x ) + 3 cot ( x ) - 1}{3}

csc (x) + cot (x) - \frac{1}{3}

Convertir a fracción:

csc (x) = \frac{csc ( x ) 3}{3}, cot (x) = \frac{cot ( x ) 3}{3}

= \frac{csc ( x ) 3}{3} + \frac{cot ( x ) 3}{3} - \frac{1}{3}

Ya que los denominadores son iguales, combinar las fracciones:

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

= \frac{csc ( x ) 3 + cot ( x ) 3 - 1}{3}

\frac{3 csc ( x ) + 3 cot ( x ) - 1}{3} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

3 csc (x) + 3 cot (x) - 1 = 0

Expresar con seno, coseno

3 \frac{1}{sin ( x )} + 3 \frac{cos ( x )}{sin ( x )} - 1 = 0

Simplificar

3 \frac{1}{sin ( x )} + 3 \frac{cos ( x )}{sin ( x )} - 1 : \frac{3 + 3 cos ( x ) - sin ( x )}{sin ( x )}

3 \frac{1}{sin ( x )} + 3 \frac{cos ( x )}{sin ( x )} - 1

3 \frac{1}{sin ( x )} = \frac{3}{sin ( x )}

3 \frac{1}{sin ( x )}

Multiplicar fracciones:

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

= \frac{1 \cdot 3}{sin ( x )}

Multiplicar:

1 \cdot 3 = 3

= \frac{3}{sin ( x )}

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

3 \frac{cos ( x )}{sin ( x )}

Multiplicar fracciones:

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

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

= \frac{3}{sin ( x )} + \frac{3 cos ( x )}{sin ( x )} - 1

Combinar las fracciones usando el mínimo común denominador:

\frac{3 + 3 cos ( x )}{sin ( x )}

Aplicar la regla

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

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

= \frac{3 cos ( x ) + 3}{sin ( x )} - 1

Convertir a fracción:

1 = \frac{1 sin ( x )}{sin ( x )}

= \frac{3 + cos ( x ) 3}{sin ( x )} - \frac{1 \cdot sin ( x )}{sin ( x )}

Ya que los denominadores son iguales, combinar las fracciones:

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

= \frac{3 + cos ( x ) 3 - 1 \cdot sin ( x )}{sin ( x )}

Multiplicar:

1 \cdot sin (x) = sin (x)

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

\frac{3 + 3 cos ( x ) - sin ( x )}{sin ( x )} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

3 + 3 cos (x) - sin (x) = 0

Sumar

sin (x)

a ambos lados

3 + 3 cos (x) = sin (x)

Elevar al cuadrado ambos lados

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

Restar

sin^{2} (x)

de ambos lados

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

Re-escribir usando identidades trigonométricas

(3 + cos (x) 3)^{2} - sin^{2} (x)

Utilizar la identidad pitagórica:

cos^{2} (x) + sin^{2} (x) = 1

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

= (3 + cos (x) 3)^{2} - (1 - cos^{2} (x))

Simplificar

(3 + cos (x) 3)^{2} - (1 - cos^{2} (x)) : 4 cos^{2} (x) + 6 cos (x) + 2

(3 + cos (x) 3)^{2} - (1 - cos^{2} (x))

= (3 + 3 cos (x))^{2} - (1 - cos^{2} (x))

(3 + cos (x) 3)^{2} : 3 + 6 cos (x) + 3 cos^{2} (x)

Aplicar la formula del binomio al cuadrado:

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

a = 3, b = cos (x) 3

= (3)^{2} + 23 cos (x) 3 + (cos (x) 3)^{2}

Simplificar

(3)^{2} + 23 cos (x) 3 + (cos (x) 3)^{2} : 3 + 6 cos (x) + 3 cos^{2} (x)

(3)^{2} + 23 cos (x) 3 + (cos (x) 3)^{2}

(3)^{2} = 3

(3)^{2}

Aplicar las leyes de los exponentes:

a = a^{\frac{1}{2}}

= (3^{\frac{1}{2}})^{2}

Aplicar las leyes de los exponentes:

(a^{b})^{c} = a^{b c}

= 3^{\frac{1}{2} \cdot 2}

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

Multiplicar fracciones:

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

= \frac{1 \cdot 2}{2}

Eliminar los terminos comunes:

2

= 1

= 3

23 cos (x) 3 = 6 cos (x)

23 cos (x) 3

Aplicar las leyes de los exponentes:

a a = a

33 = 3

= 2 \cdot 3 cos (x)

Multiplicar los numeros:

2 \cdot 3 = 6

= 6 cos (x)

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

(cos (x) 3)^{2}

Aplicar las leyes de los exponentes:

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

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

(3)^{2} : 3

Aplicar las leyes de los exponentes:

a = a^{\frac{1}{2}}

= (3^{\frac{1}{2}})^{2}

Aplicar las leyes de los exponentes:

(a^{b})^{c} = a^{b c}

= 3^{\frac{1}{2} \cdot 2}

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

Multiplicar fracciones:

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

= \frac{1 \cdot 2}{2}

Eliminar los terminos comunes:

2

= 1

= 3

= cos^{2} (x) \cdot 3

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

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

= 3 + 6 cos (x) + 3 cos^{2} (x) - (1 - cos^{2} (x))

- (1 - cos^{2} (x)) : - 1 + cos^{2} (x)

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

Poner los parentesis

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

Aplicar las reglas de los signos

- (- a) = a, - (a) = - a

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

= 3 + 6 cos (x) + 3 cos^{2} (x) - 1 + cos^{2} (x)

Simplificar

3 + 6 cos (x) + 3 cos^{2} (x) - 1 + cos^{2} (x) : 4 cos^{2} (x) + 6 cos (x) + 2

3 + 6 cos (x) + 3 cos^{2} (x) - 1 + cos^{2} (x)

Agrupar términos semejantes

= 6 cos (x) + 3 cos^{2} (x) + cos^{2} (x) + 3 - 1

Sumar elementos similares:

3 cos^{2} (x) + cos^{2} (x) = 4 cos^{2} (x)

= 6 cos (x) + 4 cos^{2} (x) + 3 - 1

Sumar/restar lo siguiente:

3 - 1 = 2

= 4 cos^{2} (x) + 6 cos (x) + 2

= 4 cos^{2} (x) + 6 cos (x) + 2

= 4 cos^{2} (x) + 6 cos (x) + 2

2 + 4 cos^{2} (x) + 6 cos (x) = 0

Usando el método de sustitución

2 + 4 cos^{2} (x) + 6 cos (x) = 0

Sea:

cos (x) = u

2 + 4 u^{2} + 6 u = 0

2 + 4 u^{2} + 6 u = 0 : u = - \frac{1}{2}, u = - 1

2 + 4 u^{2} + 6 u = 0

Escribir en la forma binómica

a x^{2} + b x + c = 0

4 u^{2} + 6 u + 2 = 0

Resolver con la fórmula general para ecuaciones de segundo grado:

4 u^{2} + 6 u + 2 = 0

Formula general para ecuaciones de segundo grado:

Para

a = 4, b = 6, c = 2

u_{1, 2} = \frac{- 6 \pm 6 ^{2} - 4 \cdot 4 \cdot 2}{2 \cdot 4}

u_{1, 2} = \frac{- 6 \pm 6 ^{2} - 4 \cdot 4 \cdot 2}{2 \cdot 4}

6^{2} - 4 \cdot 4 \cdot 2 = 2

6^{2} - 4 \cdot 4 \cdot 2

Multiplicar los numeros:

4 \cdot 4 \cdot 2 = 32

= 6^{2} - 32

6^{2} = 36

= 36 - 32

Restar:

36 - 32 = 4

= 4

Descomponer el número en factores primos:

4 = 2^{2}

= 2^{2}

Aplicar las leyes de los exponentes:

2^{2} = 2

= 2

u_{1, 2} = \frac{- 6 \pm 2}{2 \cdot 4}

Separar las soluciones

u_{1} = \frac{- 6 + 2}{2 \cdot 4}, u_{2} = \frac{- 6 - 2}{2 \cdot 4}

u = \frac{- 6 + 2}{2 \cdot 4} : - \frac{1}{2}

\frac{- 6 + 2}{2 \cdot 4}

Sumar/restar lo siguiente:

- 6 + 2 = - 4

= \frac{- 4}{2 \cdot 4}

Multiplicar los numeros:

2 \cdot 4 = 8

= \frac{- 4}{8}

Aplicar las propiedades de las fracciones:

\frac{- a}{b} = - \frac{a}{b}

= - \frac{4}{8}

Eliminar los terminos comunes:

4

= - \frac{1}{2}

u = \frac{- 6 - 2}{2 \cdot 4} : - 1

\frac{- 6 - 2}{2 \cdot 4}

Restar:

- 6 - 2 = - 8

= \frac{- 8}{2 \cdot 4}

Multiplicar los numeros:

2 \cdot 4 = 8

= \frac{- 8}{8}

Aplicar las propiedades de las fracciones:

\frac{- a}{b} = - \frac{a}{b}

= - \frac{8}{8}

Aplicar la regla

\frac{a}{a} = 1

= - 1

Las soluciones a la ecuación de segundo grado son:

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

Sustituir en la ecuación

u = cos (x)

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

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

cos (x) = - \frac{1}{2} : x = \frac{2 π}{3} + 2 πn, x = \frac{4 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{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn

x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn

cos (x) = - 1 : x = π + 2 πn

cos (x) = - 1

Soluciones generales para

cos (x) = - 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}

x = π + 2 πn

x = π + 2 πn

Combinar toda las soluciones

x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn, x = π + 2 πn

Verificar las soluciones sustituyendo en la ecuación original

Verificar las soluciones sustituyéndolas en

csc (x) + cot (x) = \frac{3}{3}

Quitar las que no concuerden con la ecuación.

Verificar la solución

\frac{2 π}{3} + 2 πn :

Verdadero

\frac{2 π}{3} + 2 πn

Sustituir

n = 1

\frac{2 π}{3} + 2 π 1

Multiplicar

csc (x) + cot (x) = \frac{3}{3}

por

x = \frac{2 π}{3} + 2 π 1

csc (\frac{2 π}{3} + 2 π 1) + cot (\frac{2 π}{3} + 2 π 1) = \frac{3}{3}

Simplificar

0.57735 \dots = 0.57735 \dots

\Rightarrow Verdadero

Verificar la solución

\frac{4 π}{3} + 2 πn :

Falso

\frac{4 π}{3} + 2 πn

Sustituir

n = 1

\frac{4 π}{3} + 2 π 1

Multiplicar

csc (x) + cot (x) = \frac{3}{3}

por

x = \frac{4 π}{3} + 2 π 1

csc (\frac{4 π}{3} + 2 π 1) + cot (\frac{4 π}{3} + 2 π 1) = \frac{3}{3}

Simplificar

- 0.57735 \dots = 0.57735 \dots

\Rightarrow Falso

Verificar la solución

π + 2 πn :

Falso

π + 2 πn

Sustituir

n = 1

π + 2 π 1

Multiplicar

csc (x) + cot (x) = \frac{3}{3}

por

x = π + 2 π 1

csc (π + 2 π 1) + cot (π + 2 π 1) = \frac{3}{3}

Sin definir

\Rightarrow Falso

x = \frac{2 π}{3} + 2 πn

Ejemplos populares

sin(θ)+1=cos(θ)

sin (θ) + 1 = cos (θ)

sqrt(2)sin^2(θ)-sin(θ)=0

2 sin^{2} (θ) - sin (θ) = 0

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

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

tan^2(x)= 3/2 sec(x)

tan^{2} (x) = \frac{3}{2} sec (x)

cot(θ)=cot^2(θ)

cot (θ) = cot^{2} (θ)