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Popular Trigonometria >

2sec(x)tan(x)-sec^2(x)=0

Solução

2 sec (x) tan (x) - sec^{2} (x) = 0

Solução

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

+1

Graus

x = 3 0^{\circ} + 36 0^{\circ} n, x = 15 0^{\circ} + 36 0^{\circ} n

Passos da solução

2 sec (x) tan (x) - sec^{2} (x) = 0

Fatorar

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

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

Aplicar as propriedades dos expoentes:

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

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

= 2 sec (x) tan (x) - sec (x) sec (x)

Fatorar o termo comum

sec (x)

= sec (x) (2 tan (x) - sec (x))

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

Resolver cada parte separadamente

sec (x) = 0 or 2 tan (x) - sec (x) = 0

sec (x) = 0 :

Sem solução

sec (x) = 0

sec (x) \leq - 1 or sec (x) \geq 1

Sem solu \overset{c}{\c} \tilde{a} o

2 tan (x) - sec (x) = 0 : x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

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

Expresar com seno, cosseno

- sec (x) + 2 tan (x)

Utilizar a Identidade Básica da Trigonometria:

sec (x) = \frac{1}{cos ( x )}

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

Utilizar a Identidade Básica da Trigonometria:

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

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

Simplificar

- \frac{1}{cos ( x )} + 2 \cdot \frac{sin ( x )}{cos ( x )} : \frac{- 1 + 2 sin ( x )}{cos ( x )}

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

Multiplicar

2 \cdot \frac{sin ( x )}{cos ( x )} : \frac{2 sin ( x )}{cos ( x )}

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

Multiplicar frações:

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

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

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

Aplicar a regra

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

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

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

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

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

- 1 + 2 sin (x) = 0

Mova

1

para o lado direito

- 1 + 2 sin (x) = 0

Adicionar

1

a ambos os lados

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

Simplificar

2 sin (x) = 1

2 sin (x) = 1

Dividir ambos os lados por

2

2 sin (x) = 1

Dividir ambos os lados por

2

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

Simplificar

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

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

Soluções gerais para

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

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 = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

Combinar toda as soluções

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

Gráfico

Exemplos populares

csc^2(x)=cot(x)+1

csc^2(x)=2cot(x)

21csc^2(2x)-5=23

cos(2x-pi/4)=(sqrt(2))/2