Question

Encontrar o valor de x nas seguintes equações trigonométricas:
A) tgx=1
B) senx= raiz de três sobre dois
C) Sen ao quadrado x + 2 senx - 3=0

Answer 1

a) 
$\displaystyle \tan x=1\implies \arctan(\tan x)=\arctan(1)\implies x=45\°\therefore \frac{\pi}{4}$

b)
$\displaystyle \sin x =\frac{\sqrt3}{2}\implies \arcsin(\sin x)=\arcsin(\frac{\sqrt3}{2})\implies x=60\°\therefore \frac{\pi}{3}$

c)
$\displaystyle \sin^2x+2\sin x-3=0\implies \sin^2x+2\sin x=3\implies u=\sin x\\\\u^2+2u-3=0\implies U=\frac{-2\pm\sqrt{16}}{2}= \left \{ {{u'=\frac{-2+4}{2}=1} \atop {u''=\frac{-2-4}{2}=-3}} \right.\\\\\sin x=f(x)|\ f:(-\frac{\pi}{2},\frac{\pi}{2})\mapsto(-1,1)\\\\\nexists\sin(x)=-3\implies \sin(x)=1\\\\\arcsin(1)=\frac{\pi}{2}\\x=\frac{\pi}{2}\therefore 90\°$