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Popular Trigonometría >

sqrt(2)=csc(x)cot(x)

Solución

2 = csc (x) cot (x)

Solución

x = \frac{π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

Grados

x = 4 5^{\circ} + 36 0^{\circ} n, x = 31 5^{\circ} + 36 0^{\circ} n

Pasos de solución

2 = csc (x) cot (x)

Intercambiar lados

csc (x) cot (x) = 2

Restar

2

de ambos lados

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

Expresar con seno, coseno

- 2 + cot (x) csc (x)

Utilizar la identidad trigonométrica básica:

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

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

Utilizar la identidad trigonométrica básica:

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

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

Simplificar

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

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

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

\frac{cos ( x )}{sin ( x )} \cdot \frac{1}{sin ( x )}

Multiplicar fracciones:

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

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

Multiplicar:

cos (x) \cdot 1 = cos (x)

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

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)

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

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

Convertir a fracción:

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

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

Ya que los denominadores son iguales, combinar las fracciones:

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

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

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

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

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

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

Re-escribir usando identidades trigonométricas

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

Utilizar la identidad pitagórica:

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

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

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

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

Usando el método de sustitución

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

Sea:

cos (x) = u

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

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

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

Desarrollar

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

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

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

Expandir

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

- 2 (1 - u^{2})

Poner los parentesis utilizando:

a (b - c) = ab - a c

a = - 2, b = 1, c = u^{2}

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

Aplicar las reglas de los signos

- (- a) = a

= - 1 \cdot 2 + 2 u^{2}

Multiplicar:

1 \cdot 2 = 2

= - 2 + 2 u^{2}

= u - 2 + 2 u^{2}

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

Escribir en la forma binómica

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

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

Resolver con la fórmula general para ecuaciones de segundo grado:

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

Formula general para ecuaciones de segundo grado:

Para

a = 2, b = 1, c = - 2

u_{1, 2} = \frac{- 1 \pm 1 ^{2} - 4 2 ( - 2 )}{2 2}

u_{1, 2} = \frac{- 1 \pm 1 ^{2} - 4 2 ( - 2 )}{2 2}

1^{2} - 42 (- 2) = 3

1^{2} - 42 (- 2)

Aplicar la regla

1^{a} = 1

1^{2} = 1

= 1 - 42 (- 2)

Aplicar la regla

- (- a) = a

= 1 + 422

422 = 8

422

Aplicar las leyes de los exponentes:

a a = a

22 = 2

= 4 \cdot 2

Multiplicar los numeros:

4 \cdot 2 = 8

= 8

= 1 + 8

Sumar:

1 + 8 = 9

= 9

Descomponer el número en factores primos:

9 = 3^{2}

= 3^{2}

Aplicar las leyes de los exponentes:

3^{2} = 3

= 3

u_{1, 2} = \frac{- 1 \pm 3}{2 2}

Separar las soluciones

u_{1} = \frac{- 1 + 3}{2 2}, u_{2} = \frac{- 1 - 3}{2 2}

u = \frac{- 1 + 3}{2 2} : \frac{2}{2}

\frac{- 1 + 3}{2 2}

Sumar/restar lo siguiente:

- 1 + 3 = 2

= \frac{2}{2 2}

Dividir:

\frac{2}{2} = 1

= \frac{1}{2}

Racionalizar

\frac{1}{2} : \frac{2}{2}

\frac{1}{2}

Multiplicar por el conjugado

\frac{2}{2}

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

1 \cdot 2 = 2

22 = 2

22

Aplicar las leyes de los exponentes:

a a = a

22 = 2

= 2

= \frac{2}{2}

= \frac{2}{2}

u = \frac{- 1 - 3}{2 2} : - 2

\frac{- 1 - 3}{2 2}

Restar:

- 1 - 3 = - 4

= \frac{- 4}{2 2}

Aplicar las propiedades de las fracciones:

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

= - \frac{4}{2 2}

Dividir:

\frac{4}{2} = 2

= \frac{2}{2}

Aplicar las leyes de los exponentes:

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

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

Aplicar las leyes de los exponentes:

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

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

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

Restar:

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

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

Aplicar las leyes de los exponentes:

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

= - 2

Las soluciones a la ecuación de segundo grado son:

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

Sustituir en la ecuación

u = cos (x)

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

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

cos (x) = \frac{2}{2} : x = \frac{π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

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

Soluciones generales para

cos (x) = \frac{2}{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}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

x = \frac{π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

cos (x) = - 2 :

Sin solución

cos (x) = - 2

- 1 \leq cos (x) \leq 1

Sin soluci \overset{o}{ˊ} n

Combinar toda las soluciones

x = \frac{π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

Gráfica

Ejemplos populares

5cos^2(θ)+3sqrt(2)sin(θ)-11/2 =0