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Popular Trigonometria >

(1+tan^2(x))cos^2(x)=sec(x)

Solução

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

Solução

x = 2 πn

+1

Graus

x = 0^{\circ} + 36 0^{\circ} n

Passos da solução

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

Subtrair

sec (x)

de ambos os lados

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

Reeecreva usando identidades trigonométricas

- sec (x) + (1 + tan^{2} (x)) cos^{2} (x)

Utilizar a Identidade Básica da Trigonometria:

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

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

Simplificar

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

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

Aplicar as propriedades dos expoentes:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

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

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

Expandir

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

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

Colocar os parênteses utilizando:

a (b + c) = ab + a c

a = cos^{2} (x), b = 1, c = \frac{sin ^{2} ( x )}{cos ^{2} ( x )}

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

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

Simplificar

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

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

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

1 \cdot cos^{2} (x)

Multiplicar:

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

= cos^{2} (x)

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

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

Multiplicar frações:

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

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

Eliminar o fator comum:

cos^{2} (x)

= sin^{2} (x)

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

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

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

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

Utilizar a identidade trigonométrica pitagórica:

cos^{2} (x) + sin^{2} (x) = 1

= - sec (x) + 1

- sec (x) + 1 = 0

Mova

1

para o lado direito

- sec (x) + 1 = 0

Subtrair

1

de ambos os lados

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

Simplificar

- sec (x) = - 1

- sec (x) = - 1

Dividir ambos os lados por

- 1

- sec (x) = - 1

Dividir ambos os lados por

- 1

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

Simplificar

sec (x) = 1

sec (x) = 1

Soluções gerais para

sec (x) = 1

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 = 0 + 2 πn

x = 0 + 2 πn

Resolver

x = 0 + 2 πn : x = 2 πn

x = 0 + 2 πn

0 + 2 πn = 2 πn

x = 2 πn

x = 2 πn

Gráfico

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