Compute the tangent vectors to [tex]\vec r_1(t)[/tex] and [tex]\vec r_2(t)[/tex] at the origin - this intersection occurs when [tex]t=0[/tex].
[tex]\vec T_1(t)=\dfrac{\mathrm d}{\mathrm dt}\vec r_1(t)=(3,2t,4t^3)\implies\vec T_1(0)=(3,0,0)[/tex]
[tex]\vec T_2(t)=\dfrac{\mathrm d}{\mathrm dt}\vec r_2(t)=(\cos t,2\cos(2t),5)\implies\vec T_2(0)=(1,2,5)[/tex]
Then use the dot product to find the angle made by these two tangent vectors:
[tex](3,0,0)\cdot(1,2,5)=\|(3,0,0)\|\|(1,2,5)\|\cos\theta[/tex]
[tex]\implies\cos\theta=\dfrac3{3\sqrt{30}}=\dfrac1{\sqrt{30}}[/tex]
[tex]\implies\boxed{\theta\approx79^\circ}[/tex]
