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Popular Trigonometría >

(sin^2(x))/(1-cos^2(x))=cot(x)

Solución

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

Solución

x = \frac{π}{4} + πn

Grados

x = 4 5^{\circ} + 18 0^{\circ} n

Pasos de solución

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

Restar

cot (x)

de ambos lados

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

Simplificar

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

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

Convertir a fracción:

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

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

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

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

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

Re-escribir usando identidades trigonométricas

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

Utilizar la identidad pitagórica:

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

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

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

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

Factorizar

sin^{2} (x) - cot (x) sin^{2} (x) : - sin^{2} (x) (cot (x) - 1)

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

Factorizar el termino común

- sin^{2} (x)

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

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

Resolver cada parte por separado

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

sin^{2} (x) = 0 : x = 2 πn, x = π + 2 πn

sin^{2} (x) = 0

Aplicar la regla

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

sin (x) = 0

Soluciones generales para

sin (x) = 0

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} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{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} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

x = 0 + 2 πn, x = π + 2 πn

x = 0 + 2 πn, x = π + 2 πn

Resolver

x = 0 + 2 πn : x = 2 πn

x = 0 + 2 πn

0 + 2 πn = 2 πn

x = 2 πn

x = 2 πn, x = π + 2 πn

cot (x) - 1 = 0 : x = \frac{π}{4} + πn

cot (x) - 1 = 0

Desplace

1

a la derecha

cot (x) - 1 = 0

Sumar

1

a ambos lados

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

Simplificar

cot (x) = 1

cot (x) = 1

Soluciones generales para

cot (x) = 1

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}{4} + πn

x = \frac{π}{4} + πn

Combinar toda las soluciones

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

Siendo que la ecuación esta indefinida para:

2 πn, π + 2 πn

x = \frac{π}{4} + πn

Gráfica

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