Question

$r\left(t\right)=⟨\sin 3t,\cos 4t⟩$

Short answer

The normal component of acceleration,

$$\begin{gathered} a_N=\frac{\Vert v\times a\Vert }{\Vert v\mid } \\ \text{thus}{a}_N=\frac{|48\cos 3t\cos 4t+36\sin 3t\sin 4t|}{\sqrt{9{\cos }^{2}3t+16{\sin }^{2}4t}} \end{gathered}$$

Step-by-step solution

Introduction 

Consider $r\left(t\right)=⟨\sin 3t,\cos 4t⟩$

The objective is to find the tangential and normal components of acceleration for $r\left(t\right)$

Given information

Acceleration's tangential and normal components

$r\left(t\right)$is a twice - differentiable vector function with $r^{\text{'}}\left(t\right)=v\left(t\right)$and $r^{\text{'}\text{'}}\left(t\right)=a\left(t\right)$.

The tangential component of acceleration, $a_T=\frac{v·a}{\Vert v\Vert }$. The normal component of acceleration,

$$\begin{gathered} a_N=\frac{\Vert v\times a\Vert }{\Vert v\Vert } \\ r\left(t\right)=⟨\sin 3t,\cos 4t⟩ \\ v\left(t\right)=r^{\text{'}}\left(t\right)=⟨3\cos 3t,-4\sin 4t⟩ \\ a\left(t\right)=r^{\text{'}\text{'}}\left(t\right)=⟨-9\sin 3t,-16\cos 4t⟩ \end{gathered}$$

Explanation 

$$\begin{gathered} \Vert v\Vert =\Vert ⟨3\cos 3t,-4\sin 4t⟩\Vert \\ =\sqrt{(3\cos 3t{)}^2+(-4\sin 4t{)}^2} \\ =\sqrt{9{\cos }^{2}3t+16{\sin }^{2}4t} \\ v·a=v\left(t\right)·a\left(t\right) \\ =⟨3\cos 3t,-4\sin 4t⟩·⟨-9\sin 3t,-16\cos 4t⟩ \\ =\left(3\cos 3t\right)(-9\sin 3t)+(-4\sin 4t)(-16\cos 4t) \\ =-27\sin 3t\cos 3t+64\sin 4t\cos 4t \end{gathered}$$

Explanation

$v\times a=⟨3\cos 3t,-4\sin 4t⟩\times ⟨-9\sin 3t,-16\cos 4t⟩$

$=\left|\begin{array}{c}ijk\\ 3cso3t-4\sin 4t0\\ -9\sin 3t-16\cos 4t0\end{array}\right|$

$=i\left(0\right)-j\left(0\right)+k(-48\cos 3t\cos 4t-36\sin 3t\sin 4t)$

$=⟨0,0,-48\cos 3t\cos 4t-36\sin 3t\sin 4t)$

$\Vert v\times a\Vert =\sqrt{(-(48\cos 3t\cos 4t+36\sin 3t\sin 4t){)}^2}$

$=|48\cos 3t\cos 4t+36\sin 3t\sin 4t|$

The tangential component of acceleration

$a_{\tau }=\frac{v·a}{\Vert v\Vert }$

thus $a_r=\frac{-27\sin 3t\cos 3t+64\sin 4t\cos 4t}{\sqrt{9{\cos }^{2}3t+16{\sin }^{2}4t}}$

The normal component of acceleration,

$a_N=\frac{\Vert v\times a\Vert }{\Vert v\Vert }$

thus $a_N=\frac{|48\cos 3t\cos 4t+36\sin 3t\sin 4t|}{\sqrt{9{\cos }^{2}3t+16{\sin }^{2}4t}}$