Question

izvestno chto tg=2/3 *$\sqrt{3+2 \sqrt{2} }$ * $\sqrt{3-2 \sqrt{2} }$ nayti
$\frac{3sinx+4cosx}{4sinx + 3 cosx}$

Answer 1

$tgx= \frac{2}{3}\cdot \sqrt{3+2\sqrt2}\cdot \sqrt{3-2\sqrt2} = \frac{2}{3}\cdot \sqrt{(3+2\sqrt2)(3-2\sqrt2)} =\\\\= \frac{2}{3} \cdot \sqrt{9-4\cdot 2} = \frac{2}{3}\cdot \sqrt1=\frac{2}{3} \\\\\\ \frac{3sinx+4cosx}{4sinx+3cosx} =\Big [ \frac{:cosx}{:cosx}\Big ]= \frac{3tgx+4}{4tgx+3}= \frac{3\cdot \frac{2}{3}+4}{4\cdot \frac{2}{3}+3} = \frac{2+4}{\frac{8}{3}+3} = \frac{6\cdot 3}{8+9}= \frac{18}{17}=1\frac{1}{17}$