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Popular Trigonometria >

4tan^2(x)-3sec^2(x)=0

Solução

4 tan^{2} (x) - 3 sec^{2} (x) = 0

Solução

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

Graus

x = 24 0^{\circ} + 36 0^{\circ} n, x = 30 0^{\circ} + 36 0^{\circ} n, x = 6 0^{\circ} + 36 0^{\circ} n, x = 12 0^{\circ} + 36 0^{\circ} n

Passos da solução

4 tan^{2} (x) - 3 sec^{2} (x) = 0

Fatorar

4 tan^{2} (x) - 3 sec^{2} (x) : (2 tan (x) + 3 sec (x)) (2 tan (x) - 3 sec (x))

4 tan^{2} (x) - 3 sec^{2} (x)

Reescrever

4 tan^{2} (x) - 3 sec^{2} (x)

como

(2 tan (x))^{2} - (3 sec (x))^{2}

4 tan^{2} (x) - 3 sec^{2} (x)

Reescrever

4

como

2^{2}

= 2^{2} tan^{2} (x) - 3 sec^{2} (x)

Aplicar as propriedades dos radicais:

a = (a)^{2}

3 = (3)^{2}

= 2^{2} tan^{2} (x) - (3)^{2} sec^{2} (x)

Aplicar as propriedades dos expoentes:

a^{m} b^{m} = (ab)^{m}

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

= (2 tan (x))^{2} - (3)^{2} sec^{2} (x)

Aplicar as propriedades dos expoentes:

a^{m} b^{m} = (ab)^{m}

(3)^{2} sec^{2} (x) = (3 sec (x))^{2}

= (2 tan (x))^{2} - (3 sec (x))^{2}

= (2 tan (x))^{2} - (3 sec (x))^{2}

Aplicar a regra da diferença de quadrados:

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

(2 tan (x))^{2} - (3 sec (x))^{2} = (2 tan (x) + 3 sec (x)) (2 tan (x) - 3 sec (x))

= (2 tan (x) + 3 sec (x)) (2 tan (x) - 3 sec (x))

(2 tan (x) + 3 sec (x)) (2 tan (x) - 3 sec (x)) = 0

Resolver cada parte separadamente

2 tan (x) + 3 sec (x) = 0 or 2 tan (x) - 3 sec (x) = 0

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

2 tan (x) + 3 sec (x) = 0

Expresar com seno, cosseno

2 tan (x) + sec (x) 3

Utilizar a Identidade Básica da Trigonometria:

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

= 2 \cdot \frac{sin ( x )}{cos ( x )} + sec (x) 3

Utilizar a Identidade Básica da Trigonometria:

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

= 2 \cdot \frac{sin ( x )}{cos ( x )} + \frac{1}{cos ( x )} 3

Simplificar

2 \cdot \frac{sin ( x )}{cos ( x )} + \frac{1}{cos ( x )} 3 : \frac{2 sin ( x ) + 3}{cos ( x )}

2 \cdot \frac{sin ( x )}{cos ( x )} + \frac{1}{cos ( x )} 3

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 )} 3 = \frac{3}{cos ( x )}

\frac{1}{cos ( x )} 3

Multiplicar frações:

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

= \frac{1 \cdot 3}{cos ( x )}

Multiplicar:

1 \cdot 3 = 3

= \frac{3}{cos ( x )}

= \frac{2 sin ( x )}{cos ( x )} + \frac{3}{cos ( x )}

Aplicar a regra

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

= \frac{2 sin ( x ) + 3}{cos ( x )}

= \frac{2 sin ( x ) + 3}{cos ( x )}

\frac{3 + 2 sin ( x )}{cos ( x )} = 0

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

3 + 2 sin (x) = 0

Mova

3

para o lado direito

3 + 2 sin (x) = 0

Subtrair

3

de ambos os lados

3 + 2 sin (x) - 3 = 0 - 3

Simplificar

2 sin (x) = - 3

2 sin (x) = - 3

Dividir ambos os lados por

2

2 sin (x) = - 3

Dividir ambos os lados por

2

\frac{2 sin ( x )}{2} = \frac{- 3}{2}

Simplificar

sin (x) = - \frac{3}{2}

sin (x) = - \frac{3}{2}

Soluções gerais para

sin (x) = - \frac{3}{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{4 π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

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

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

2 tan (x) - 3 sec (x) = 0

Expresar com seno, cosseno

2 tan (x) - sec (x) 3

Utilizar a Identidade Básica da Trigonometria:

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

= 2 \cdot \frac{sin ( x )}{cos ( x )} - sec (x) 3

Utilizar a Identidade Básica da Trigonometria:

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

= 2 \cdot \frac{sin ( x )}{cos ( x )} - \frac{1}{cos ( x )} 3

Simplificar

2 \cdot \frac{sin ( x )}{cos ( x )} - \frac{1}{cos ( x )} 3 : \frac{2 sin ( x ) - 3}{cos ( x )}

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

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 )} 3 = \frac{3}{cos ( x )}

\frac{1}{cos ( x )} 3

Multiplicar frações:

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

= \frac{1 \cdot 3}{cos ( x )}

Multiplicar:

1 \cdot 3 = 3

= \frac{3}{cos ( x )}

= \frac{2 sin ( x )}{cos ( x )} - \frac{3}{cos ( x )}

Aplicar a regra

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

= \frac{2 sin ( x ) - 3}{cos ( x )}

= \frac{2 sin ( x ) - 3}{cos ( x )}

\frac{- 3 + 2 sin ( x )}{cos ( x )} = 0

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

- 3 + 2 sin (x) = 0

Mova

3

para o lado direito

- 3 + 2 sin (x) = 0

Adicionar

3

a ambos os lados

- 3 + 2 sin (x) + 3 = 0 + 3

Simplificar

2 sin (x) = 3

2 sin (x) = 3

Dividir ambos os lados por

2

2 sin (x) = 3

Dividir ambos os lados por

2

\frac{2 sin ( x )}{2} = \frac{3}{2}

Simplificar

sin (x) = \frac{3}{2}

sin (x) = \frac{3}{2}

Soluções gerais para

sin (x) = \frac{3}{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}{3} + 2 πn, x = \frac{2 π}{3} + 2 πn

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

Combinar toda as soluções

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

Gráfico

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