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sec^2(x)csc(x)-2csc(x)=2-sec^2(x)

Solução

sec^{2} (x) csc (x) - 2 csc (x) = 2 - sec^{2} (x)

Solução

x = \frac{3 π}{4} + 2 πn, x = \frac{5 π}{4} + 2 πn, x = \frac{π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

Graus

x = 13 5^{\circ} + 36 0^{\circ} n, x = 22 5^{\circ} + 36 0^{\circ} n, x = 4 5^{\circ} + 36 0^{\circ} n, x = 31 5^{\circ} + 36 0^{\circ} n

Passos da solução

sec^{2} (x) csc (x) - 2 csc (x) = 2 - sec^{2} (x)

Subtrair

2 - sec^{2} (x)

de ambos os lados

sec^{2} (x) csc (x) - 2 csc (x) - 2 + sec^{2} (x) = 0

Fatorar

sec^{2} (x) csc (x) - 2 csc (x) - 2 + sec^{2} (x) : (sec (x) + 2) (sec (x) - 2) (csc (x) + 1)

sec^{2} (x) csc (x) - 2 csc (x) - 2 + sec^{2} (x)

= (sec^{2} (x) csc (x) - 2 csc (x)) + (sec^{2} (x) - 2)

Fatorar

csc (x)

sec^{2} (x) csc (x) - 2 csc (x)

csc (x) (sec^{2} (x) - 2)

sec^{2} (x) csc (x) - 2 csc (x)

Fatorar o termo comum

csc (x)

= csc (x) (sec^{2} (x) - 2)

= (sec^{2} (x) - 2) + csc (x) (sec^{2} (x) - 2)

Fatorar o termo comum

sec^{2} (x) - 2

= (sec^{2} (x) - 2) (csc (x) + 1)

Fatorar

sec^{2} (x) - 2 : (sec (x) + 2) (sec (x) - 2)

sec^{2} (x) - 2

Aplicar as propriedades dos radicais:

a = (a)^{2}

2 = (2)^{2}

= sec^{2} (x) - (2)^{2}

Aplicar a regra da diferença de quadrados:

x^{2} - y^{2} = (x + y) (x - y)

sec^{2} (x) - (2)^{2} = (sec (x) + 2) (sec (x) - 2)

= (sec (x) + 2) (sec (x) - 2)

= (sec (x) + 2) (sec (x) - 2) (csc (x) + 1)

(sec (x) + 2) (sec (x) - 2) (csc (x) + 1) = 0

Resolver cada parte separadamente

sec (x) + 2 = 0 or sec (x) - 2 = 0 or csc (x) + 1 = 0

sec (x) + 2 = 0 : x = \frac{3 π}{4} + 2 πn, x = \frac{5 π}{4} + 2 πn

sec (x) + 2 = 0

Mova

2

para o lado direito

sec (x) + 2 = 0

Subtrair

2

de ambos os lados

sec (x) + 2 - 2 = 0 - 2

Simplificar

sec (x) = - 2

sec (x) = - 2

Soluções gerais para

sec (x) = - 2

sec (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} sec (x) 1 \frac{2 3}{3} 22 Undefined - 2 - 2 - \frac{2 3}{3} 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} sec (x) - 1 - \frac{2 3}{3} - 2 - 2 Undefined 22 \frac{2 3}{3}

x = \frac{3 π}{4} + 2 πn, x = \frac{5 π}{4} + 2 πn

x = \frac{3 π}{4} + 2 πn, x = \frac{5 π}{4} + 2 πn

sec (x) - 2 = 0 : x = \frac{π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

sec (x) - 2 = 0

Mova

2

para o lado direito

sec (x) - 2 = 0

Adicionar

2

a ambos os lados

sec (x) - 2 + 2 = 0 + 2

Simplificar

sec (x) = 2

sec (x) = 2

Soluções gerais para

sec (x) = 2

sec (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} sec (x) 1 \frac{2 3}{3} 22 Undefined - 2 - 2 - \frac{2 3}{3} 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} sec (x) - 1 - \frac{2 3}{3} - 2 - 2 Undefined 22 \frac{2 3}{3}

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

csc (x) + 1 = 0 : x = \frac{3 π}{2} + 2 πn

csc (x) + 1 = 0

Mova

1

para o lado direito

csc (x) + 1 = 0

Subtrair

1

de ambos os lados

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

Simplificar

csc (x) = - 1

csc (x) = - 1

Soluções gerais para

csc (x) = - 1

csc (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} csc (x) Undefiend 22 \frac{2 3}{3} 1 \frac{2 3}{3} 22 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} csc (x) Undefiend - 2 - 2 - \frac{2 3}{3} - 1 - \frac{2 3}{3} - 2 - 2

x = \frac{3 π}{2} + 2 πn

x = \frac{3 π}{2} + 2 πn

Combinar toda as soluções

x = \frac{3 π}{4} + 2 πn, x = \frac{5 π}{4} + 2 πn, x = \frac{π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn, x = \frac{3 π}{2} + 2 πn

Dado que a equação é indefinida para:

\frac{3 π}{2} + 2 πn

x = \frac{3 π}{4} + 2 πn, x = \frac{5 π}{4} + 2 πn, x = \frac{π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

Exemplos populares

2sin(x)=-sqrt(2),0<= x<= 2pi

2 sin (x) = - 2, 0 \leq x \leq 2 π

csc(a)=3

csc (a) = 3

csc(a)=5

csc (a) = 5

1-1/2 cos(2x-pi/2)=0

1 - \frac{1}{2} cos (2 x - \frac{π}{2}) = 0

sin(pix)=-sin(pi-pix)

sin (π x) = - sin (π - π x)