Q. 23 - Calculus

Question

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

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

Step-by-Step Solution

Answer

We've plotted the required graph

Step 1 : Given information

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

Step 2 :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 :

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

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

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

Step 3 :Substituting different values in the parametric equations

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

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

Step 4 : Plotting graph

The required graph is :