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Popular Trigonometria >

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

Solução

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

Solução

x = 1.10714 \dots + πn

Graus

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

Passos da solução

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

Subtrair

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

de ambos os 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 frações:

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}

Converter para fração:

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}

Já que os denominadores são iguais, combinar as frações:

\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

Reeecreva usando identidades trigonométricas

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

Utilizar a identidade trigonométrica 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

Fatorar

- 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)

Fatorar o termo comum

sin^{2} (x)

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

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

Resolver cada parte separadamente

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 a regra

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

sin (x) = 0

Soluções gerais para

sin (x) = 0

sin (x)

tabela de periodicidade com ciclo de

2 πn

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

Mova

1

para o lado direito

2 cot (x) - 1 = 0

Adicionar

1

a ambos os lados

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

Simplificar

2 cot (x) = 1

2 cot (x) = 1

Dividir ambos os lados por

2

2 cot (x) = 1

Dividir ambos os lados por

2

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

Simplificar

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

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

Aplique as propriedades trigonométricas inversas

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

Soluções gerais 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 as soluções

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

Dado que a equação é indefinida para:

2 πn, π + 2 πn

x = arccot (\frac{1}{2}) + πn

Mostrar soluções na forma decimal

x = 1.10714 \dots + πn

Gráfico

Exemplos populares

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

cos(x)= 20/29

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

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2cot(θ)+1=0