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

Solución

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

Solución

x = 1.10714 \dots + πn

Grados

x = 63.43494 \dots^{\circ} + 18 0^{\circ} n

Pasos de solución

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

Restar

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

de ambos lados

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

Simplificar

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

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

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

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

Multiplicar fracciones:

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

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

Multiplicar:

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

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

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

Convertir a fracción:

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

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

Ya que los denominadores son iguales, combinar las fracciones:

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

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

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

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

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

Re-escribir usando identidades trigonométricas

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

Utilizar la identidad pitagórica:

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

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

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

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

Factorizar

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

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

Factorizar el termino común

sin^{2} (x)

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

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

Resolver cada parte por separado

sin^{2} (x) = 0 or 2 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

2 cot (x) - 1 = 0 : x = arccot (\frac{1}{2}) + πn

2 cot (x) - 1 = 0

Mover

1

al lado derecho

2 cot (x) - 1 = 0

Sumar

1

a ambos lados

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

Simplificar

2 cot (x) = 1

2 cot (x) = 1

Dividir ambos lados entre

2

2 cot (x) = 1

Dividir ambos lados entre

2

\frac{2 cot ( x )}{2} = \frac{1}{2}

Simplificar

cot (x) = \frac{1}{2}

cot (x) = \frac{1}{2}

Aplicar propiedades trigonométricas inversas

cot (x) = \frac{1}{2}

Soluciones generales para

cot (x) = \frac{1}{2}

cot (x) = a \Rightarrow x = arccot (a) + πn

x = arccot (\frac{1}{2}) + πn

x = arccot (\frac{1}{2}) + πn

Combinar toda las soluciones

x = 2 πn, x = π + 2 πn, x = arccot (\frac{1}{2}) + πn

Siendo que la ecuación esta indefinida para:

2 πn, π + 2 πn

x = arccot (\frac{1}{2}) + πn

Mostrar soluciones en forma decimal

x = 1.10714 \dots + πn

Ejemplos populares

(tan(x)-1)(2sin(x)-1)=0

(tan (x) - 1) (2 sin (x) - 1) = 0

cos(x)= 20/29

cos (x) = \frac{20}{29}

2cot^4(x)-cot^2(x)-15=0

2 cot^{4} (x) - cot^{2} (x) - 15 = 0

cos(θ)=(sqrt(11))/6

cos (θ) = \frac{11}{6}

1+cos(x)=3cos(x)

1 + cos (x) = 3 cos (x)