solution:
[tex]a) r(t)=5sint\widehat{i}+2cost\widehat{j}\\
\Rightarrow r'(t)=\frac{d}{dt}(5sint\widehat{i}+2cost\widehat{j})\\
\Rightarrow r'(t)=\frac{d}{dt}(5sint)\widehat{i}+\frac{d}{dt}(2cost)\widehat{j}\\
\Rightarrow r'(t)=5cost\widehat{i}-2sint\widehat{j}\\[/tex][tex]b) at (t)=\frac{\pi }{4},we have\\
r'(\frac{\pi }{4})=\frac{5}{\sqrt{2}}\widehat{i}-\frac{2}{\sqrt{2}}\widehat{j}\\
so, the tangent vector r'(\frac{\pi }{4}) start at(\frac{5}{\sqrt{2}},\frac{-2}{\sqrt{2}})\\
and moves \frac{5}{\sqrt{2}} right and \frac{2}{\sqrt{2}}down[/tex]
