Question

Срочно 20 баллов
найменшее значение​

Answer 1

$1)\; \; cosa-sina=\sqrt2\cdot (\frac{\sqrt2}{2}\cdot cosa-\frac{\sqrt2}{2}\cdot sina)=\\\\=\sqrt2\cdot (sin\frac{\pi}{4}\cdot cosa-cos\frac{\pi}{4}\cdot sina)=\sqrt2\cdot sin(\frac{\pi}{4}-a)\; ;\\\\\\-1\leq sin(\frac{\pi}{4}-a)\leq 1\; \; \Rightarrow \; \; \; -\sqrt2\leq \sqrt2\cdot sin(\frac{\pi }{4}-a)\leq \sqrt2\\\\naimenshee\; \; zyachenie:\; \; -\sqrt2\; .$

$2)\; \; 8\, cosa-15\, sina=\Big [\; 8^2+15^2=289=17^2\; \; \Rightarrow \; \; (\frac{8}{17})^2+(\frac{15}{17})^2=1\; \Rightarrow \\\\tak\; kak\; \;sin^2\beta +cos^2\beta =1\; ,\; \; to\; \; \frac{8}{17}=sin\beta \; ,\; \; \frac{15}{17}=cos\beta \; \Big ]=\\\\=17\cdot (\frac{8}{17}\cdot cosa-\frac{15}{17}\cdot sina)=\\\\=17\cdot (sin\beta \cdot cosa-cos\beta \cdot sina)=17\cdot sin(\beta -a)\\\\\\-1\leq sin(\beta -\alpha )\leq 1\; \; \Rightarrow \; \; \; -17\leq 17\cdot sin(\beta -\alpha )\leq 17\\\\naimenshee\; znachenie:\; \; -17\; .$