Answer
$\boldsymbol{T}(0)=\langle0,\frac{3}{5},\frac{4}{5}\rangle$.
Work Step by Step
$\boldsymbol{r}(t)=\langle \cos{t},3t,2\sin{2t}\rangle$
$\boldsymbol{T}(t)=\frac{\boldsymbol{r'}(t)}{|\boldsymbol{r'}(t)|}=\frac{1}{\sqrt{(-\sin{t})^2+(3)^2+(4\cos{2t})^2}}\langle-\sin{t},3,4\cos{2t}\rangle \Rightarrow \boldsymbol{T}(0)=\frac{1}{\sqrt{(-\sin{(0)})^2+(3)^2+(4\cos{2(0)})^2}}\langle-\sin{(0)},3,4\cos{2(0)}\rangle=\frac{1}{5}\langle0,3,4
\rangle=\langle0,\frac{3}{5},\frac{4}{5}\rangle$
