Let ${\mathrm{a}}_1=\cos (\sin \mathrm{\pi x}),{\mathrm{a}}_2=\sin (\cos \mathrm{\pi x})$ and ${\mathrm{a}}_3=\cos \left(\mathrm{\pi }\right(\mathrm{x}+1\left)\right),\text{ where }-\frac{1}{2}<\mathrm{x}<0$ .What are the relative sizes of  ${\mathrm{a}}_1,{\mathrm{a}}_2\text{ and }{\mathrm{a}}_3$ ?

Let y = - x, then we have $0<\mathrm{y}<\frac{1}{2}$

${\mathrm{a}}_1=\cos (\sin \mathrm{\pi y})>0,{\mathrm{a}}_2=\sin (\cos \mathrm{\pi y})>0$ and ${\mathrm{a}}_3=\cos (1-\mathrm{y})\mathrm{\pi }=-\cos \mathrm{\pi y}<0$

now, $\sin \mathrm{\pi y}+\cos \mathrm{\pi y}\le \sqrt{2}<\frac{\mathrm{\pi }}{2}$

$\begin{array}{l}\Rightarrow \sin (\cos \mathrm{\pi y})<\cos (\sin \mathrm{\pi y})\\ \Rightarrow {\mathrm{a}}_2<{\mathrm{a}}_1\\ \therefore {\mathrm{a}}_3<{\mathrm{a}}_2<{\mathrm{a}}_1\end{array}$