Question

Sendo tgx + cotgx = 5/2, calcule sen2x

Answer 1

Resposta:

$\boxed{\sin(2x) = \frac{4}{5}}$

Explicação passo-a-passo:

$\tan(x) + \cot(x) = \frac{5}{2}$

$\frac{\sin(x)}{\cos(x)} + \frac{\cos(x)}{\sin(x)} = \frac{5}{2}$

$\frac{\sin(x)\sin(x)}{\cos(x)\sin(x)} + \frac{\cos(x)\cos(x)}{\cos(x)\sin(x)} = \frac{5}{2}$

$\frac{\sin^2(x)}{\cos(x)\sin(x)} + \frac{\cos^2(x)}{\cos(x)\sin(x)} = \frac{5}{2}$

$\frac{\sin^2(x)+\cos^2(x)}{\cos(x)\sin(x)} = \frac{5}{2}$

$\frac{1}{\cos(x)\sin(x)} = \frac{5}{2}$

$\frac{1}{\frac{1}{2}\sin(2x)} = \frac{5}{2}$

$2 = \frac{5}{2}\sin(2x)$

$4 = 5\sin(2x)$

$\boxed{\sin(2x) = \frac{4}{5}}$

Identidades úteis:

$1) \cos^2x+\sin^2x = 1$

$2) \tan x = \frac{\sin x}{\cos x}$

$3) \cot x = \frac{\cos x}{\sin x}$

$4) \cos x \sin x = \frac{1}{2}\sin 2x$