Seja 0 = x = π /2 uma medida de angulo em radianos tal que cos x + sen x = √5/2 cos - sen x = √3/2.... Pergunta de ideia deEM4N03L

December 2019 1 314 Report

Seja 0 = x = π /2 uma medida de angulo em radianos tal que

cos x + sen x = √5/2

cos - sen x = √3/2

O valor de tg 2x é:

GABARITO: √15/15

Lista de comentários

DanJR

Verified answer

Olá!

$\\ \displaystyle \mathsf{\tan (2x) = \frac{\sin (2x)}{\cos (2x)}} \\\\\\ \mathsf{\tan (2x) = \frac{2 \cdot \sin x \cdot \cos x}{\cos^2 x - \sin^2 x}} \\\\\\ \mathsf{\tan (2x) = \frac{2 \cdot \sin x \cdot \cos x}{(\cos x + \sin x)(\cos x - \sin x)}}$

Resolvendo o sistema abaixo (adição):

$\mathsf{\begin{cases} \mathsf{\cos x + \sin x = \frac{\sqrt{5}}{2}} \\\\ \mathsf{\cos x - \sin x = \frac{\sqrt{3}}{2}}\end{cases}}$

$\\ \displaystyle \mathsf{2 \cdot \cos x = \frac{\sqrt{5}}{2} + \frac{\sqrt{3}}{2}} \\\\\\ \boxed{\mathsf{\cos x = \frac{\sqrt{5} + \sqrt{3}}{4}}}$

E,

$\\ \displaystyle \mathsf{\cos x + \sin x = \frac{\sqrt{5}}{2}} \\\\\\ \mathsf{\sin x = \frac{\sqrt{5}}{2} - \frac{\sqrt{5} + \sqrt{3}}{4}} \\\\\\ \boxed{\mathsf{\sin x = \frac{\sqrt{5} - \sqrt{3}}{4}}}$

Com efeito,

$\\ \displaystyle \mathsf{\tan (2x) = \left ( 2 \cdot \frac{\sqrt{5} + \sqrt{3}}{4} \cdot \frac{\sqrt{5} - \sqrt{3}}{4} \right ) : \left ( \frac{\sqrt{5}}{2} \cdot \frac{\sqrt{3}}{2} \right )} \\\\\\ \mathsf{\tan (2x) = \frac{2 \cdot (5 - 3)}{16} \cdot \frac{4}{\sqrt{15}}} \\\\\\ \mathsf{\tan (2x) = \frac{16}{16\sqrt{15}}} \\\\\\ \mathsf{\tan (2x) = \frac{1}{\sqrt{15}} \cdot \frac{\sqrt{15}}{\sqrt{15}}} \\\\\\ \boxed{\boxed{\mathsf{\tan (2x) = \frac{\sqrt{15}}{15}}}}$

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