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Popular Trigonometria >

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

Solução

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

Solução

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

Graus

x = 13 5^{\circ} + 18 0^{\circ} n, x = 4 5^{\circ} + 18 0^{\circ} n

Passos da solução

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

Subtrair

1 - sin^{2} (x)

de ambos os lados

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

Reeecreva usando identidades trigonométricas

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

Utilizar a identidade trigonométrica pitagórica:

tan^{2} (x) + 1 = sec^{2} (x)

tan^{2} (x) = sec^{2} (x) - 1

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

Simplificar

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

- 1 + sin^{2} (x) + sin^{2} (x) (sec^{2} (x) - 1)

Expandir

sin^{2} (x) (sec^{2} (x) - 1) : sin^{2} (x) sec^{2} (x) - sin^{2} (x)

sin^{2} (x) (sec^{2} (x) - 1)

Colocar os parênteses utilizando:

a (b - c) = ab - a c

a = sin^{2} (x), b = sec^{2} (x), c = 1

= sin^{2} (x) sec^{2} (x) - sin^{2} (x) \cdot 1

= sin^{2} (x) sec^{2} (x) - 1 \cdot sin^{2} (x)

Multiplicar:

1 \cdot sin^{2} (x) = sin^{2} (x)

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

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

Simplificar

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

- 1 + sin^{2} (x) + sin^{2} (x) sec^{2} (x) - sin^{2} (x)

Agrupar termos semelhantes

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

Somar elementos similares:

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

= sin^{2} (x) sec^{2} (x) - 1

= sin^{2} (x) sec^{2} (x) - 1

= sin^{2} (x) sec^{2} (x) - 1

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

Fatorar

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

- 1 + sec^{2} (x) sin^{2} (x)

Aplicar as propriedades dos expoentes:

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

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

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

Aplicar a regra da diferença de quadrados:

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

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

= (sec (x) sin (x) + 1) (sec (x) sin (x) - 1)

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

Resolver cada parte separadamente

sec (x) sin (x) + 1 = 0 or sec (x) sin (x) - 1 = 0

sec (x) sin (x) + 1 = 0 : x = \frac{3 π}{4} + πn

sec (x) sin (x) + 1 = 0

Reeecreva usando identidades trigonométricas

sec (x) sin (x) + 1

sec (x) sin (x) = tan (x)

sec (x) sin (x)

Expresar com seno, cosseno

sec (x) sin (x)

Utilizar a Identidade Básica da Trigonometria:

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

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

Simplificar

\frac{1}{cos ( x )} sin (x) : \frac{sin ( x )}{cos ( x )}

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

Multiplicar frações:

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

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

Multiplicar:

1 \cdot sin (x) = sin (x)

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

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

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

Utilizar a Identidade Básica da Trigonometria:

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

= tan (x)

= 1 + tan (x)

1 + tan (x) = 0

Mova

1

para o lado direito

1 + tan (x) = 0

Subtrair

1

de ambos os lados

1 + tan (x) - 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{3 π}{4} + πn

x = \frac{3 π}{4} + πn

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

sec (x) sin (x) - 1 = 0

Reeecreva usando identidades trigonométricas

sec (x) sin (x) - 1

sec (x) sin (x) = tan (x)

sec (x) sin (x)

Expresar com seno, cosseno

sec (x) sin (x)

Utilizar a Identidade Básica da Trigonometria:

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

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

Simplificar

\frac{1}{cos ( x )} sin (x) : \frac{sin ( x )}{cos ( x )}

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

Multiplicar frações:

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

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

Multiplicar:

1 \cdot sin (x) = sin (x)

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

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

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

Utilizar a Identidade Básica da Trigonometria:

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

= tan (x)

= - 1 + tan (x)

- 1 + tan (x) = 0

Mova

1

para o lado direito

- 1 + tan (x) = 0

Adicionar

1

a ambos os lados

- 1 + tan (x) + 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

Combinar toda as soluções

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

Gráfico

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