Найдите значение выражения \( \frac{4}{\sqrt{2+\sqrt{3}}sin15} \)
Решение
\( \frac{4}{\sqrt{2+\sqrt{3}}sin(45-30)} \)
\( \frac{4}{\sqrt{2+\sqrt{3}}(sin45*cos30-sin30*cos45)} \)
\( \frac{4}{\sqrt{2+\sqrt{3}}(\frac{\sqrt{6}}{4}-\frac{\sqrt{2}}{4})}=\frac{16}{(\sqrt{2+\sqrt{3}})(\sqrt{6}-\sqrt{2})} \)
\( \frac{16*(\sqrt{2+\sqrt{3}})}{(2+\sqrt{3})(\sqrt{6}-\sqrt{2})}=\frac{16*(\sqrt{2+\sqrt{3}})}{(\sqrt{6}+\sqrt{2})} \)
\( \frac{16*(\sqrt{2+\sqrt{3}})*(\sqrt{6}-\sqrt{2})}{6-2}=4*(\sqrt{2+\sqrt{3}})*(\sqrt{6}-\sqrt{2}) \)
\( 4*(\sqrt{2+\sqrt{3}})*(\sqrt{6}-\sqrt{2})=4*(\sqrt{12+6\sqrt{3}}-\sqrt{4+2\sqrt{3}})=4*\sqrt{(3+\sqrt{3})^2})-4\sqrt{(1+\sqrt{3})^2}=8 \)
Ответ: 8