Question

sin п/12+ cos п/12
вычислите

Answer 1

$sin\frac{\pi }{12}+cos\frac{\pi }{12}=\sqrt2\cdot (\frac{1}{\sqrt2}\cdot sin\frac{\pi }{12}+\frac{1}{\sqrt2}\cdot cos\frac{\pi }{12})=\Big [\frac{1}{\sqrt2}=\frac{\sqrt2}{2}=cos\frac{\pi}{4}=sin\frac{\pi}{4}\Big ]=\\\\=\sqrt2\cdot (cos\frac{\pi }{4}\cdot sin\frac{\pi }{12}+sin\frac{\pi }{4}\cdot cos\frac{\pi }{12})=\sqrt2\cdot sin(\frac{\pi}{4}+\frac{\pi}{12})=\\\\=\sqrt2\cdot sin\frac{4\pi }{12}=\sqrt2\cdot sin\frac{\pi}{3}=\sqrt2\cdot \frac{\sqrt3}{2}=\frac{\sqrt6}{2}\; \; \Big (=\sqrt{\frac{3}{2}}\, \Big )$

$\star \; \; sin\alpha \cdot cos\beta +sin\beta \cdot cos\alpha =sin(\alpha +\beta )\; \; \star$