Question

Sendo √3 . tg x = 1 , com π/2 < x < 3π/2 , determine cos^2 x


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Answer 1

$\mathrm{Seja}\ x\in\left]\dfrac{\pi}{2},\dfrac{3\pi}{2}\right[\ \mathrm{tal\ que}\ \sqrt{3}\tan{x}=1.$

$\mathrm{\acute{E}\ preciso\ determinar}\ \cos^2{x}.$

$\mathrm{Primeiramente, determinaremos}\ \tan{x}.$

$\sqrt{3}\tan{x}=1\Longrightarrow \tan{x}=\dfrac{1}{\sqrt{3}}$

$\mathrm{Agora,\ \acute{e}\ poss\acute{\i}vel\ isolar}\ \sin{x}\ \mathrm{na\ express\tilde{a}o}\text{:}$

$\dfrac{\sin{x}}{\cos{x}}=\dfrac{1}{\sqrt{3}}\Longrightarrow \sin{x}=\dfrac{\cos{x}}{\sqrt{3}}$

$\mathrm{Podemos\ calcular}\ \cos^2{x}\ \mathrm{da\ seguinte\ maneira}\text{:}$

$\sin^2{x}+\cos^2{x}=1\Longrightarrow \bigg(\dfrac{\cos{x}}{\sqrt{3}}\bigg)^2+\cos^2{x}=1$

$\Longrightarrow \dfrac{\cos^2{x}}{3}+\dfrac{3\cos^2{x}}{3}=1\Longrightarrow \dfrac{4\cos^2{x}}{3}=1$

$\Longrightarrow \boxed{\cos^2{x}=\dfrac{3}{4}}$