Chapter 0.4, Problem 2E

To determine

To match: the parametric equation $x={\sin }^{3}t$ , $y={\cos }^{3}t$ with their graph. Also state the approximate dimension of the viewing window and given a parameter interval that traces the curve exactly once.

Answer to Problem 2E

The graph of the given parametric equation is the graph (a).

The viewing window is: $\left[-2,2\right]$ by $\left[-2,2\right]$

The parameter interval is: $0\le t\le 2\pi$

Explanation of Solution

Given information:

The given graphs are:

  

Formula Used:

  $\sin \left(0\right)=0$

  $\cos \left(0\right)=1$

Period of $\sin \left(x\right)=2\pi$

Period of $\cos \left(x\right)=2\pi$

Consider the given parametric equation,

  $x={\sin }^{3}t$ , $y={\cos }^{3}t$

Substitute $t=0$, then the corresponding values of x and y are:

  $\begin{array}{c}x={\sin }^3\left(0\right)\\ ={\left[0\right]}^3\\ =0\end{array}$

  $\begin{array}{c}y={\cos }^3\left(0\right)\\ ={\left(1\right)}^3\\ =1\end{array}$

This implies that the graph of this parametric equation must contain the point $\left(0,1\right)$ . This eliminates the choices (b), (c) and (d).

So, the correct answer is graph (a).

  

Now, on observing the graph (a), it can be noticed that the point $\left(0,1\right)$ lie on the the first mark on the positive y-axis. So, it can be implied that each mark denotes 1 unit. So, the viewing window is $\left[-2,2\right]$ by $\left[-2,2\right]$ .

Now, it can also be observed that this graph is a closed curve in the window. This is possible when the parameter of t is such that gives one full cycle for both x and y .

Now, the period of both $\sin t$ and $\cos t$ is $2\pi$ .

Thus, for the parameter t to be able to complete one full cycle it must lie between 0 and $2\pi$ .

That is, $0\le t\le 2\pi$