\(sin\left(a+\frac{2019\pi}{2}\right)=sin\left(a-\frac{\pi}{2}+1010\pi\right)=sin\left(a-\frac{\pi}{2}\right)=-cosa\)
\(sin\left(a-\frac{9\pi}{2}\right)=sin\left(a-\frac{\pi}{2}-4\pi\right)=sin\left(a-\frac{\pi}{2}\right)=-cosa\)
\(cos\left(a+999\pi\right)=cos\left(a+\pi+998\pi\right)=cos\left(a+\pi\right)=-cosa\)
\(sin\left(a+999\pi\right)=sin\left(a+\pi+998\pi\right)=sin\left(a+\pi\right)=-sina\)