Решебник по геометрии 11 класс. Атанасян ФГОС 359

\[\boxed{\mathbf{359.}ОК\ ГДЗ\ –\ домашка\ на\ 5}\]

\[Дано:\]

\[AS = l;\]

\[AO = r.\]

\[Найти:\]

\[\angle ASB.\]

\[Решение.\]

\[\frac{\pi l^{2}\alpha}{360{^\circ}} = \pi rl\]

\[r = \frac{\pi l^{2}\alpha}{360{^\circ}\pi l} = \frac{\text{αl}}{360{^\circ}}.\]

\[В\ треугольнике\ ASO:\]

\[\sin{\angle ASO} = \frac{\text{AO}}{\text{SA}} = \frac{r}{l}.\]

\[\textbf{а)}\ \alpha = 180{^\circ}:\]

\[r = \frac{180l}{360} = \frac{l}{2}.\]

\[\sin{\angle ASO} = \frac{1}{2};\]

\[\angle ASO = 30{^\circ};\]

\[\angle ASB = 2\angle ASO = 60{^\circ}.\]

\[\textbf{б)}\ \alpha = 90{^\circ}:\]

\[r = \frac{90l}{360} = \frac{l}{4};\]

\[\sin{\angle ASO} = \frac{1}{4};\]

\[\angle ASO = \arcsin\frac{1}{4};\]

\[\angle ASB = 2\arcsin\frac{1}{4}.\]

\[\textbf{в)}\ \alpha = 60{^\circ}:\]

\[r = \frac{60l}{360} = \frac{l}{6};\]

\[\sin{\angle ASO} = \frac{1}{6};\]

\[\angle ASO = \arcsin\frac{1}{6};\]

\[\angle ASB = 2\arcsin\frac{1}{6}.\]