Question

Fie x ∈ (0,π/4) astfel incat tgx+ctgx=3. Sa se calculeze sinx-cosx.

Answer 1

$\mathrm{tg}\,x+ \mathrm{ctg}\, x = 3 \,\Leftrightarrow \,\dfrac{\sin x}{\cos x}+\dfrac{\cos x}{\sin x} = 3\,\Leftrightarrow\\ \\\Leftrightarrow \,\dfrac{\sin^2 x+\cos^2 x}{\sin x\cos x} =3\,\Leftrightarrow\,\dfrac{1}{\sin x \cos x} = 3 \, \Leftrightarrow\,\sin x\cos x = \dfrac{1}{3} \\ \\ \\ E = \sin x -\cos x\, \Leftrightarrow\,E^2 = \sin^2 x+\cos^2 x - 2\sin x\cos x \, \Leftrightarrow \\ \\ \Leftrightarrow\,E^2 = 1-\dfrac{2}{3} \, \Leftrightarrow\,E^2 = \dfrac{1}{3}$

$x \in\Big(0,\dfrac{\pi}{4}\Big) \Rightarrow \sin x <\cos x\,\Leftrightarrow\\ \\ \Leftrightarrow\, \sin x - \cos x < 0 \Rightarrow E = -\sqrt{\dfrac{1}{3}}\Rightarrow \boxed{\sin x-\cos x = -\dfrac{\sqrt 3}{3}}$