Question

x=cos

3

t,y=sin

3

t,t∈[0,2π]

x=cos3⁡t,y=sin3⁡t,t∈[0,2π]

Answer 1

$$x=\cos(3t), y=\sin(3t), t\in[0,2\pi].$$

The length of the parametric curve is given by

$$L=\displaystyle\int_{0}^{2\pi}\sqrt{(x'(t))^2+(y'(t))^2}dt$$

$$=\displaystyle\int_{0}^{2\pi}\sqrt{(-3\sin(3t))^2+(3\cos(3t))^2}dt$$

$$=\displaystyle\int_{0}^{2\pi}3dt=[3t]\begin{matrix} 2\pi \\ 0 \end{matrix}=6\pi (units)$$

The length of curve is $6\pi.$