Задача 54690 ...
Условие
Решение
[m]arcsin\frac{77}{85}= α [/m] ⇒ [m]sin α =\frac{77}{85}[/m] ⇒[m] cos α =\sqrt(1-sin^2 α )=\sqrt(1-(\frac{77}{85})^2)=\frac{36}{85}[/m]
[m]arcsin\frac{8}{17}= β [/m]⇒ [m] sin β =\frac{8}{17}[/m] ⇒ [m]cos β =\sqrt(1-sin^2 α )=\sqrt(1-(\frac{8}{17})^2)=\frac{15}{17}[/m]
[m]arccos (-\frac{3}{5})= γ [/m] ⇒[m]cos γ =-\frac{3}{5}[/m] ⇒ [m]sin γ =sqrt(1-cos^2 α )=sqrt(1-(-\frac{3}{5})^2)=\frac{4}{5}[/m]
Найдем:
[m]sin(\frac{\pi}{2}+ arcsin\frac{77}{85})=sin(arcsin\frac{8}{17}+arccos (-\frac{3}{5})[/m]
[m]sin(\frac{\pi}{2}+ arcsin\frac{77}{85})=sin(\frac{\pi}{2}+ α )=cos α =\frac{36}{85}[/m]
[m]sin(arcsin\frac{8}{17}+arccos (-\frac{3}{5}))=sin( β + γ )=sin β \cdot cos γ +cos β\cdot sin γ =\frac{8}{17}\cdot (--\frac{3}{5})+\frac{15}{17}\cdot \frac{4}{5}=\frac{36}{85}[/m]
[m]\frac{36}{85}=\frac{36}{85}[/m] - верно, значит и данное равенство верно...