Question

encontre o valor de x no triângulo, sabendo que o sen 120° = sen 60°
A=120°
B=45°

Answer 1

Vamos aplicar a Lei dos Senos:

$\boxed{\dfrac{Lado~Oposto~a~A}{sen(\hat{A})}~=~\dfrac{Lado~Oposto~a~B}{sen(\hat{B})}}$

Substituindo os valores dados, temos:

$\dfrac{x}{sen(120^\circ)}~=~\dfrac{100}{sen(45^\circ)}\\\\\\\dfrac{x}{sen(60^\circ)}~=~\dfrac{100}{sen(45^\circ)}\\\\\\Multiplicando~cruzado:\\\\\\x\cdot sen(45^\circ)~=~100\cdot sen(60^\circ)\\\\\\x~=~\dfrac{100\cdot sen(60^\circ)}{sen(45^\circ)}$

Substituindo os valores dos senos dos arcos notáveis seguindo a tabela abaixo, temos:

$\begin{array}{c|c|c|c|c|c|}\boxed{_{Funcao}\backslash^{Angulo}}&^{~~0^\circ}_{0~rad}&^{~~~30^\circ}_{\pi/6~rad}&^{~~~45^\circ}_{\pi/4~rad}&^{~~~60^\circ}_{\pi/3~rad}&^{~~~90^\circ}_{\pi/2~rad}\\Seno&0&\dfrac{1}{2}&\dfrac{\sqrt{2}}{2}&\dfrac{\sqrt{3}}{2}&1\\Cosseno&1&\dfrac{\sqrt{3}}{2}&\dfrac{\sqrt{2}}{2}&\dfrac{1}{2}&0\\Tangente&0&\dfrac{\sqrt{3}}{3}&1&\sqrt{3}&^{~nao}_{existe}\end{array}$

$x~=~\dfrac{100\cdot \frac{\sqrt{3}}{2}}{\frac{\sqrt{2}}{2}}\\\\\\x~=~\dfrac{100\cdot \sqrt{3}}{\sqrt{2}}\\\\\\x~=~\dfrac{100\cdot \sqrt{3}}{\sqrt{2}}~\cdot~\dfrac{\sqrt{2}}{\sqrt{2}}\\\\\\x~=~\dfrac{100\cdot \sqrt{3}\cdot\sqrt{2}}{\sqrt{2}^{\,2}}\\\\\\x~=~\dfrac{100\cdot \sqrt{6}}{2}\\\\\\\boxed{x~=~50\sqrt{6}~m}$

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