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Popular Trigonometría >

3/(cos^2(x))=(7+4)/(cot(x))

Solución

\frac{3}{cos ^{2} ( x )} = \frac{7 + 4}{cot ( x )}

Solución

x = \frac{0.57693 \dots}{2} + πn, x = \frac{π}{2} - \frac{0.57693 \dots}{2} + πn

Grados

x = 16.52786 \dots^{\circ} + 18 0^{\circ} n, x = 73.47213 \dots^{\circ} + 18 0^{\circ} n

Pasos de solución

\frac{3}{cos ^{2} ( x )} = \frac{7 + 4}{cot ( x )}

Restar

\frac{7 + 4}{cot ( x )}

de ambos lados

\frac{3}{cos ^{2} ( x )} - \frac{11}{cot ( x )} = 0

Simplificar

\frac{3}{cos ^{2} ( x )} - \frac{11}{cot ( x )} : \frac{3 cot ( x ) - 11 cos ^{2} ( x )}{cos ^{2} ( x ) cot ( x )}

\frac{3}{cos ^{2} ( x )} - \frac{11}{cot ( x )}

Mínimo común múltiplo de

cos^{2} (x), cot (x) : cos^{2} (x) cot (x)

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

Mínimo común múltiplo (MCM)

Calcular una expresión que este compuesta de factores que aparezcan tanto en

cos^{2} (x)

cot (x)

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

Reescribir las fracciones basandose en el mínimo común denominador

Multiplicar cada numerador por la misma cantidad necesaria para multiplicar el denominador correspondiente y convertirlo en el mínimo común denominador

Para

\frac{3}{cos ^{2} ( x )} :

multiplicar el denominador y el numerador por

cot (x)

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

Para

\frac{11}{cot ( x )} :

multiplicar el denominador y el numerador por

cos^{2} (x)

\frac{11}{cot ( x )} = \frac{11 cos ^{2} ( x )}{cot ( x ) cos ^{2} ( x )}

= \frac{3 cot ( x )}{cos ^{2} ( x ) cot ( x )} - \frac{11 cos ^{2} ( x )}{cot ( x ) cos ^{2} ( x )}

Ya que los denominadores son iguales, combinar las fracciones:

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

= \frac{3 cot ( x ) - 11 cos ^{2} ( x )}{cos ^{2} ( x ) cot ( x )}

\frac{3 cot ( x ) - 11 cos ^{2} ( x )}{cos ^{2} ( x ) cot ( x )} = 0

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

3 cot (x) - 11 cos^{2} (x) = 0

Expresar con seno, coseno

- 11 cos^{2} (x) + 3 cot (x)

Utilizar la identidad trigonométrica básica:

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

= - 11 cos^{2} (x) + 3 \cdot \frac{cos ( x )}{sin ( x )}

Simplificar

- 11 cos^{2} (x) + 3 \cdot \frac{cos ( x )}{sin ( x )} : \frac{- 11 cos ^{2} ( x ) sin ( x ) + 3 cos ( x )}{sin ( x )}

- 11 cos^{2} (x) + 3 \cdot \frac{cos ( x )}{sin ( x )}

Multiplicar

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

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

Multiplicar fracciones:

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

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

= - 11 cos^{2} (x) + \frac{3 cos ( x )}{sin ( x )}

Convertir a fracción:

11 cos^{2} (x) = \frac{11 cos ^{2} ( x ) sin ( x )}{sin ( x )}

= - \frac{11 cos ^{2} ( x ) sin ( x )}{sin ( x )} + \frac{cos ( x ) \cdot 3}{sin ( x )}

Ya que los denominadores son iguales, combinar las fracciones:

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

= \frac{- 11 cos ^{2} ( x ) sin ( x ) + cos ( x ) \cdot 3}{sin ( x )}

= \frac{- 11 cos ^{2} ( x ) sin ( x ) + 3 cos ( x )}{sin ( x )}

\frac{3 cos ( x ) - 11 cos ^{2} ( x ) sin ( x )}{sin ( x )} = 0

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

3 cos (x) - 11 cos^{2} (x) sin (x) = 0

Factorizar

3 cos (x) - 11 cos^{2} (x) sin (x) : cos (x) (3 - 11 sin (x) cos (x))

3 cos (x) - 11 cos^{2} (x) sin (x)

Aplicar las leyes de los exponentes:

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

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

= 3 cos (x) - 11 cos (x) cos (x)

Factorizar el termino común

cos (x)

= cos (x) (3 - 11 sin (x) cos (x))

cos (x) (3 - 11 sin (x) cos (x)) = 0

Resolver cada parte por separado

cos (x) = 0 or 3 - 11 sin (x) cos (x) = 0

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

3 - 11 sin (x) cos (x) = 0 : x = \frac{arcsin ( \frac{6}{11} )}{2} + πn, x = \frac{π}{2} - \frac{arcsin ( \frac{6}{11} )}{2} + πn

3 - 11 sin (x) cos (x) = 0

Re-escribir usando identidades trigonométricas

3 - 11 sin (x) cos (x)

Utilizar la identidad trigonométrica del ángulo doble:

2 sin (x) cos (x) = sin (2 x)

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

= 3 - 11 \cdot \frac{sin ( 2 x )}{2}

3 - 11 \cdot \frac{sin ( 2 x )}{2} = 0

Desplace

3

a la derecha

3 - 11 \cdot \frac{sin ( 2 x )}{2} = 0

Restar

3

de ambos lados

3 - 11 \cdot \frac{sin ( 2 x )}{2} - 3 = 0 - 3

Simplificar

- 11 \cdot \frac{sin ( 2 x )}{2} = - 3

- 11 \cdot \frac{sin ( 2 x )}{2} = - 3

Simplificar

- 11 \cdot \frac{sin ( 2 x )}{2} : - \frac{11 sin ( 2 x )}{2}

- 11 \cdot \frac{sin ( 2 x )}{2}

Multiplicar fracciones:

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

= - \frac{sin ( 2 x ) \cdot 11}{2}

- \frac{11 sin ( 2 x )}{2} = - 3

Multiplicar ambos lados por

2

- \frac{11 sin ( 2 x )}{2} = - 3

Multiplicar ambos lados por

2

- \frac{11 sin ( 2 x )}{2} \cdot 2 = - 3 \cdot 2

Simplificar

- \frac{11 sin ( 2 x )}{2} \cdot 2 = - 3 \cdot 2

Simplificar

- \frac{11 sin ( 2 x )}{2} \cdot 2 : - 11 sin (2 x)

- \frac{11 sin ( 2 x )}{2} \cdot 2

Multiplicar fracciones:

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

= - \frac{11 sin ( 2 x ) \cdot 2}{2}

Eliminar los terminos comunes:

2

= - 11 sin (2 x)

Simplificar

- 3 \cdot 2 : - 6

- 3 \cdot 2

Multiplicar los numeros:

3 \cdot 2 = 6

= - 6

- 11 sin (2 x) = - 6

- 11 sin (2 x) = - 6

- 11 sin (2 x) = - 6

Dividir ambos lados entre

- 11

- 11 sin (2 x) = - 6

Dividir ambos lados entre

- 11

\frac{- 11 sin ( 2 x )}{- 11} = \frac{- 6}{- 11}

Simplificar

sin (2 x) = \frac{6}{11}

sin (2 x) = \frac{6}{11}

Aplicar propiedades trigonométricas inversas

sin (2 x) = \frac{6}{11}

Soluciones generales para

sin (2 x) = \frac{6}{11}

sin (x) = a \Rightarrow x = arcsin (a) + 2 πn, x = π - arcsin (a) + 2 πn

2 x = arcsin (\frac{6}{11}) + 2 πn, 2 x = π - arcsin (\frac{6}{11}) + 2 πn

2 x = arcsin (\frac{6}{11}) + 2 πn, 2 x = π - arcsin (\frac{6}{11}) + 2 πn

Resolver

2 x = arcsin (\frac{6}{11}) + 2 πn : x = \frac{arcsin ( \frac{6}{11} )}{2} + πn

2 x = arcsin (\frac{6}{11}) + 2 πn

Dividir ambos lados entre

2

2 x = arcsin (\frac{6}{11}) + 2 πn

Dividir ambos lados entre

2

\frac{2 x}{2} = \frac{arcsin ( \frac{6}{11} )}{2} + \frac{2 πn}{2}

Simplificar

x = \frac{arcsin ( \frac{6}{11} )}{2} + πn

x = \frac{arcsin ( \frac{6}{11} )}{2} + πn

Resolver

2 x = π - arcsin (\frac{6}{11}) + 2 πn : x = \frac{π}{2} - \frac{arcsin ( \frac{6}{11} )}{2} + πn

2 x = π - arcsin (\frac{6}{11}) + 2 πn

Dividir ambos lados entre

2

2 x = π - arcsin (\frac{6}{11}) + 2 πn

Dividir ambos lados entre

2

\frac{2 x}{2} = \frac{π}{2} - \frac{arcsin ( \frac{6}{11} )}{2} + \frac{2 πn}{2}

Simplificar

x = \frac{π}{2} - \frac{arcsin ( \frac{6}{11} )}{2} + πn

x = \frac{π}{2} - \frac{arcsin ( \frac{6}{11} )}{2} + πn

x = \frac{arcsin ( \frac{6}{11} )}{2} + πn, x = \frac{π}{2} - \frac{arcsin ( \frac{6}{11} )}{2} + πn

Combinar toda las soluciones

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn, x = \frac{arcsin ( \frac{6}{11} )}{2} + πn, x = \frac{π}{2} - \frac{arcsin ( \frac{6}{11} )}{2} + πn

Siendo que la ecuación esta indefinida para:

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

x = \frac{arcsin ( \frac{6}{11} )}{2} + πn, x = \frac{π}{2} - \frac{arcsin ( \frac{6}{11} )}{2} + πn

Mostrar soluciones en forma decimal

x = \frac{0.57693 \dots}{2} + πn, x = \frac{π}{2} - \frac{0.57693 \dots}{2} + πn

Gráfica

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