Question

In Exercises 16–23 sketch the parametric curve by plotting points.

23.$x={\cos }^{5}t,y={\sin }^{5}t,t\in [0,2\pi ]$

Short answer

We've plotted the required graph

Step-by-step solution

Given information

Given :$x={\cos }^{5}t,y={\sin }^{5}t,t\in [0,2\pi ]$

:Substituting different values in the parametric equations 

Find the values of $x,y$by substituting different values in the parametric equations :

(a) When $t=0$, the point $(x,y)$is :

$$\begin{gathered} (x,y)=({\sin }^{5}t,{\cos }^{5}t) \\ =({\sin }^{5}0,{\cos }^{5}0)...(\sin cet=0) \\ =\boxed{(0,1)} \end{gathered}$$

(b) When $t=\mathrm{\pi }$$(x,y)$is :

$$\begin{gathered} (x,y)=({\sin }^{5}t,{\cos }^{5}t) \\ =({\sin }^5\mathrm{\pi },{\cos }^5\mathrm{\pi })...[byinsertingt=\mathrm{\pi }], \\ =\boxed{(0,-1)} \end{gathered}$$

:Substituting different values in the parametric equations

(c) When $t=\frac{3\mathrm{\pi }}{2},(x,y)$is :

$$\begin{gathered} (x,y)=({\sin }^{5}t,{\cos }^{5}t) \\ =({\sin }^5\frac{3\mathrm{\pi }}{2},{\cos }^5\frac{3\mathrm{\pi }}{2})...(\sin cet=\frac{3\mathrm{\pi }}{2}) \\ =\boxed{(-1,0)} \end{gathered}$$

Plotting graph

The required graph is :