Question

найдите значение выражения соs(
$\alpha - \beta$
если:
* фото *
​

Answer 1

$sin\alpha \cdot sin\beta =\frac{1}{4}\; \; ,\; \; \alpha +\beta =\frac{5\pi }{4}$

$sin\alpha \cdot sin\beta =\frac{1}{2}\cdot \Big (cos(\alpha -\beta )-cos(\alpha +\beta )\Big )\\\\\frac{1}{4}=\frac{1}{2}\cdot \Big (cos(\alpha -\beta )-cos\frac{5\pi }{4}\Big )\\\\\frac{1}{4}=\frac{1}{2}\cdot cos(\alpha -\beta )-\frac{1}{2}\cdot cos(\pi +\frac{\pi}{4})\\\\\frac{1}{4}=\frac{1}{2}\cdot cos(\alpha -\beta )-\frac{1}{2}\cdot (-cos\frac{\pi}{4})\\\\\frac{1}{4}=\frac{1}{2}\cdot cos(\alpha -\beta )+\frac{1}{2}\cdot \frac{\sqrt2}{2}\\\\\frac{1}{2}\cdot cos(\alpha -\beta )=\frac{1}{4}-\frac{\sqrt2}{4}$

$cos(\alpha -\beta )=2\cdot \frac{1-\sqrt2}{4}\\\\cos(\alpha -\beta )=\frac{1-\sqrt2}{2}$