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sin(x)-2cos(x)cot(x)=csc(x)

Solución

sin (x) - 2 cos (x) cot (x) = csc (x)

Solución

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

Grados

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

Pasos de solución

sin (x) - 2 cos (x) cot (x) = csc (x)

Restar

csc (x)

de ambos lados

sin (x) - 2 cos (x) cot (x) - csc (x) = 0

Expresar con seno, coseno

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

Simplificar

sin (x) - 2 cos (x) \frac{cos ( x )}{sin ( x )} - \frac{1}{sin ( x )} : \frac{sin ^{2} ( x ) - 2 cos ^{2} ( x ) - 1}{sin ( x )}

sin (x) - 2 cos (x) \frac{cos ( x )}{sin ( x )} - \frac{1}{sin ( x )}

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

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

Multiplicar fracciones:

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

= \frac{cos ( x ) \cdot 2 cos ( x )}{sin ( x )}

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

cos (x) \cdot 2 cos (x)

Aplicar las leyes de los exponentes:

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

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

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

Sumar:

1 + 1 = 2

= 2 cos^{2} (x)

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

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

Combinar las fracciones usando el mínimo común denominador:

\frac{- 2 cos ^{2} ( x ) - 1}{sin ( x )}

Aplicar la regla

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

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

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

Convertir a fracción:

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

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

Ya que los denominadores son iguales, combinar las fracciones:

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

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

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

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

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

sin (x) sin (x)

Aplicar las leyes de los exponentes:

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

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

= sin^{1 + 1} (x)

Sumar:

1 + 1 = 2

= sin^{2} (x)

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

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

\frac{sin ^{2} ( x ) - 2 cos ^{2} ( x ) - 1}{sin ( x )} = 0

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

sin^{2} (x) - 2 cos^{2} (x) - 1 = 0

Sumar

2 cos^{2} (x)

a ambos lados

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

Elevar al cuadrado ambos lados

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

Restar

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

de ambos lados

(sin^{2} (x) - 1)^{2} - 4 cos^{4} (x) = 0

Factorizar

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

(sin^{2} (x) - 1)^{2} - 4 cos^{4} (x)

Reescribir

(sin^{2} (x) - 1)^{2} - 4 cos^{4} (x)

como

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

(sin^{2} (x) - 1)^{2} - 4 cos^{4} (x)

Reescribir

4

como

2^{2}

= (sin^{2} (x) - 1)^{2} - 2^{2} cos^{4} (x)

Aplicar las leyes de los exponentes:

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

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

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

Aplicar las leyes de los exponentes:

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

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

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

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

Aplicar la siguiente regla para binomios al cuadrado:

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

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

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

Simplificar

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

(sin^{2} (x) - 1 + 2 cos^{2} (x)) (sin^{2} (x) - 1 - 2 cos^{2} (x)) = 0

Resolver cada parte por separado

sin^{2} (x) - 1 + 2 cos^{2} (x) = 0 or sin^{2} (x) - 1 - 2 cos^{2} (x) = 0

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

sin^{2} (x) - 1 + 2 cos^{2} (x) = 0

Re-escribir usando identidades trigonométricas

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

Utilizar la identidad pitagórica:

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

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

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

Simplificar

= cos^{2} (x)

cos^{2} (x) = 0

Aplicar la regla

x^{n} = 0 \Rightarrow x = 0

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

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

sin^{2} (x) - 1 - 2 cos^{2} (x) = 0

Re-escribir usando identidades trigonométricas

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

Utilizar la identidad pitagórica:

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

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

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

Simplificar

= - 3 cos^{2} (x)

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

Dividir ambos lados entre

- 3

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

Dividir ambos lados entre

- 3

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

Dividir ambos lados entre

- 3

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

Simplificar

cos^{2} (x) = 0

cos^{2} (x) = 0

Aplicar la regla

x^{n} = 0 \Rightarrow x = 0

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

Combinar toda las soluciones

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

Verificar las soluciones sustituyendo en la ecuación original

Verificar las soluciones sustituyéndolas en

sin (x) - 2 cos (x) cot (x) = csc (x)

Quitar las que no concuerden con la ecuación.

Verificar la solución

\frac{π}{2} + 2 πn :

Verdadero

\frac{π}{2} + 2 πn

Sustituir

n = 1

\frac{π}{2} + 2 π 1

Multiplicar

sin (x) - 2 cos (x) cot (x) = csc (x)

por

x = \frac{π}{2} + 2 π 1

sin (\frac{π}{2} + 2 π 1) - 2 cos (\frac{π}{2} + 2 π 1) cot (\frac{π}{2} + 2 π 1) = csc (\frac{π}{2} + 2 π 1)

Simplificar

1 = 1

\Rightarrow Verdadero

Verificar la solución

\frac{3 π}{2} + 2 πn :

Verdadero

\frac{3 π}{2} + 2 πn

Sustituir

n = 1

\frac{3 π}{2} + 2 π 1

Multiplicar

sin (x) - 2 cos (x) cot (x) = csc (x)

por

x = \frac{3 π}{2} + 2 π 1

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

Simplificar

- 1 = - 1

\Rightarrow Verdadero

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

Ejemplos populares

sin(2x)=cos(x-60)

sin (2 x) = cos (x - 6 0^{\circ})

tan(180-θ)=3sqrt(3)

tan (18 0^{\circ} - θ) = 33

tan(x)= 1/(cos^2(x))

tan (x) = \frac{1}{cos ^{2} ( x )}

arccsc(2x)= pi/6

arccsc (2 x) = \frac{π}{6}

2csc(θ)=sqrt(2)cot(θ)+3csc(θ)

2 csc (θ) = 2 cot (θ) + 3 csc (θ)