Рассмотрим сначала sin36° .
0\; ,\; \; 4t^2+2t-1=0\; ,\; \; D=20\; ," alt="sin36^\circ =sin(90^\circ -54^\circ )=cos54^\circ =cos(18^\circ +36^\circ )\; ,\\\\2sin18^\circ \cdot cos18^\circ =cos18^\circ \cdot cos36^\circ -sin18^\circ \cdot sin36^\circ \; ,\\\\2sin18^\circ \cdot cos18^\circ =cos18^\circ \cdot (\underbrace {1-2sin^218^\circ }_{cos36^\circ })-sin18^\circ \cdot \underbrace {2sin18^\circ \cdot cos18^\circ }_{sin36^\circ }\; ,\\\\2sin18^\circ =1-2sin^218^\circ -2sin^218^\circ \; ,\\\\4sin^218^\circ +2sin18^\circ -1=0\; ,\\\\t=sin18^\circ >0\; ,\; \; 4t^2+2t-1=0\; ,\; \; D=20\; ," align="absmiddle" class="latex-formula">
0\; \; \Rightarrow \; \; sin18^\circ =\frac{\sqrt5-1}{4}\; ." alt="t_{1,2}=\frac{-2\pm \sqrt{20}}{8}=\frac{-2\pm 2\sqrt5}{8}=\frac{-1\pm \sqrt5}{4}\; ,\\\\t_1=\frac{-1-\sqrt5}{4}<0\; \; ,\; \; t_2=\frac{-1+\sqrt5}{4}>0\; \; \Rightarrow \; \; sin18^\circ =\frac{\sqrt5-1}{4}\; ." align="absmiddle" class="latex-formula">
