Question

Resolva em R a equação tg(x-pi/4) = 1

Answer 1

Tg(x-$\pi$/4) = 1

sen(x-$\pi$/4) / cos(x-$\pi$/4) = 1 

sen(x-$\pi$/4) = cos(x-$\pi$/4)

Elevando ao quadrado : 

$sen^{2}$(x-$\pi$/4) = $cos^{2}$(x-$\pi$/4)

$sen^{2}$(x-$\pi$/4) - $cos^{2}$(x-$\pi$/4) = 0 (Eq.1)

Pelo teorema fund. da trigonometria sabemos que :

$sen^{2}$(x-$\pi$/4) + $cos^{2}$(x-$\pi$/4) = 1 (Eq.2)

Somando Eq.1 e Eq.2 , temos :

$sen^{2}$(x-$\pi$/4) + $cos^{2}$(x-$\pi$/4) + $sen^{2}$(x-$\pi$/4) - $cos^{2}$(x-$\pi$/4) = 1

2.$sen^{2}$(x-$\pi$/4) = 1 

$sen^{2}$(x-$\pi$/4) = 1/2

sen(x-$\pi$/4) =  +√2/2 ⇒ cos(x-$\pi$/4) = +√2/2

logo (x-$\pi$/4) = $\pi$/4
         x = $\pi$/4 + $\pi$/4 = $\pi$/2



OU  sen(x-$\pi$/4) = -√2/2 ⇒ cos(x-$\pi$/4) = -√2/2

logo (x-$\pi$/4) = 5$\pi$/4
         x = 5$\pi$/4 + $\pi$/4 = 6$\pi$/4 = 3$\pi$/2