Chapter 8.4, Problem 44E

To determine

To draw: the curve represented by the parametric equations.

Answer to Problem 44E

  

Explanation of Solution

Given: $x=2\sin t,y=\cos 4t$

Calculation:

To draw the curve of the given parametric equations by using a graphing device, first set the viewing rectangle of the graphing device.

To set the viewing rectangle of the graphing device, find the range of x and y.

From the equations $x=2\sin t,y=\cos 4t$ , $-2\le x\le 2$ and $-1\le y\le 1$ , since both the sine

and the cosine of any number will be between 1 and 1.

Thus, use the viewing the rectangle $\left[-2.5,2.5\right]$ and $\left[-1.5,1.5\right]$ .

Now, the curves $y=a\sin kx$ and $y=a\cos bx$ have period $\frac{2\pi }{k}$

Thus, the curves $x=2\sin t$ and $y=\cos 4t$ have the periods $2\pi$ and $\frac{2\pi }{4}=\frac{\pi }{2}$ respectively.

Therefore, the value of $t$ over the interval $0\le t\le 2\pi$ gives the complete graph.

Now, enter the given parametric equation in the graphing device that displays the graph as shown below.

  

Conclusion:

Therefore, the required graph is drawn.