Chapter 8.4, Problem 48E

To determine

To draw: the curve represented by the parametric equations.

Answer to Problem 48E

  

Explanation of Solution

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

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 $x=2\cos t+\cos 2t,y=2\sin t-\sin 2t$ , $-1\le x\le 3$ and $-2.5\le y\le 2.5$ , since both the sine and the cosine of any number will be between 1 and 1.

Thus, use the viewing the rectangle $\left[-1.5,3.5\right]$ and $\left[-3,3\right]$ .

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

Thus, the curves $x=2\cos t+\cos 2t$ and $y=2\sin t-2\sin t$ have the periods $2\pi$ .

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.