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4sin^2(x)tan(x)-4sin^2(x)-3tan(x)+3=0

Solução

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

Solução

x = \frac{π}{4} + πn, 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 = 4 5^{\circ} + 18 0^{\circ} n, 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 sin^{2} (x) tan (x) - 4 sin^{2} (x) - 3 tan (x) + 3 = 0

Fatorar

4 sin^{2} (x) tan (x) - 4 sin^{2} (x) - 3 tan (x) + 3 : (tan (x) - 1) (2 sin (x) + 3) (2 sin (x) - 3)

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

= (4 sin^{2} (x) tan (x) - 4 sin^{2} (x)) + (- 3 tan (x) + 3)

Fatorar

- 3

- 3 tan (x) + 3

- 3 (tan (x) - 1)

- 3 tan (x) + 3

Fatorar o termo comum

- 3

= - 3 (tan (x) - 1)

Fatorar

4 sin^{2} (x)

4 sin^{2} (x) tan (x) - 4 sin^{2} (x)

4 sin^{2} (x) (tan (x) - 1)

4 sin^{2} (x) tan (x) - 4 sin^{2} (x)

Fatorar o termo comum

4 sin^{2} (x)

= 4 sin^{2} (x) (tan (x) - 1)

= - 3 (tan (x) - 1) + 4 sin^{2} (x) (tan (x) - 1)

Fatorar o termo comum

tan (x) - 1

= (tan (x) - 1) (4 sin^{2} (x) - 3)

Fatorar

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

4 sin^{2} (x) - 3

Reescrever

4 sin^{2} (x) - 3

como

(2 sin (x))^{2} - (3)^{2}

4 sin^{2} (x) - 3

Reescrever

4

como

2^{2}

= 2^{2} sin^{2} (x) - 3

Aplicar as propriedades dos radicais:

a = (a)^{2}

3 = (3)^{2}

= 2^{2} sin^{2} (x) - (3)^{2}

Aplicar as propriedades dos expoentes:

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

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

= (2 sin (x))^{2} - (3)^{2}

= (2 sin (x))^{2} - (3)^{2}

Aplicar a regra da diferença de quadrados:

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

(2 sin (x))^{2} - (3)^{2} = (2 sin (x) + 3) (2 sin (x) - 3)

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

= (tan (x) - 1) (2 sin (x) + 3) (2 sin (x) - 3)

(tan (x) - 1) (2 sin (x) + 3) (2 sin (x) - 3) = 0

Resolver cada parte separadamente

tan (x) - 1 = 0 or 2 sin (x) + 3 = 0 or 2 sin (x) - 3 = 0

tan (x) - 1 = 0 : x = \frac{π}{4} + πn

tan (x) - 1 = 0

Mova

1

para o lado direito

tan (x) - 1 = 0

Adicionar

1

a ambos os lados

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

Simplificar

tan (x) = 1

tan (x) = 1

Soluções gerais para

tan (x) = 1

tan (x)

tabela de periodicidade com ciclo de

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} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

x = \frac{π}{4} + πn

x = \frac{π}{4} + πn

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

2 sin (x) + 3 = 0

Mova

3

para o lado direito

2 sin (x) + 3 = 0

Subtrair

3

de ambos os lados

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

2 sin (x) - 3 = 0

Mova

3

para o lado direito

2 sin (x) - 3 = 0

Adicionar

3

a ambos os lados

2 sin (x) - 3 + 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{π}{4} + πn, 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

Exemplos populares

cos(t)=-0,5

cos (t) = - 0, 5

sinh(x)=sinh(-x)

sinh (x) = sinh (- x)

pi/4 =arctan(t)

\frac{π}{4} = arctan (t)

sin(2x)sin(x)cos(x)=0

sin (2 x) sin (x) cos (x) = 0

sin(4x)-cos(4x)=sin(2x)-cos(2x)

sin (4 x) - cos (4 x) = sin (2 x) - cos (2 x)