Question

Mostre que:

$\mathsf\displaystyle{\sqrt{37}}=\mathsf{\dfrac{1}{2}~\cdot\sqrt{282-2\sqrt{4505}~\cdot~cos\left(arctan\left(\dfrac{-4}{67}\right)\right)}}}}$

Responder de forma detalhada, explicando o raciocínio utilizado.

@Hamilton Brito

Answer 1

Explicação passo-a-passo:

$\sqrt{37} =\frac{1}{2} *\sqrt{282-2\sqrt{4505}*cos(arctan(\frac{-4}{67}))}$

Elevamos os dois lados da igualdade ao quadrado:

$(\sqrt{37})^{2} =[\frac{1}{2} *\sqrt{282-2\sqrt{4505}*cos(arctan(\frac{-4}{67}))}]^{2} \\\\\\\sqrt{37^{2} }=\frac{1^{2} }{2^{2} } *\sqrt{[282-2\sqrt{4505}*cos(arctan(\frac{-4}{67}))]^{2}}\\\\\\37 = \frac{1}{4} *[282-2\sqrt{4505}*cos(arctan(\frac{-4}{67}))]\\\\\\37=\frac{282-2\sqrt{4505}*cos(arctan(\frac{-4}{67}))}{4} \\\\\\37=\frac{282-2\sqrt{4505}*cos(arctan(-0,0597))}{4}\\\\\\37=\frac{282-2\sqrt{4505}*cos(-3,4165)}{4}\\\\\\37=\frac{282-2\sqrt{4505}*0,9982}{4}$

$37=\frac{282-2*67,1192*0,9982}{4}\\\\\\37=\frac{282-134,2385*0,9982}{4}\\\\\\\\37=\frac{282-133,9969}{4}$

Vamos arredondar 133,9969 para 134, pois afinal estamos trabalhando com apenas quatro casas, se trabalhássemos com todas, teríamos 134.

$37=\frac{282-134}{4}\\\\\\37=\frac{148}{4}\\\\\\37=37$