Chapter 8.4, Problem 43E

To determine

To draw: the curve for the given parametric equations.

Answer to Problem 43E

The parametric graph is

  

Explanation of Solution

Given:

The following parameter equation is:

  $x=\sin t,y=2\cos 3t$

Calculation:

We need to draw the curve represented by the parametric equations $x=\sin t,y=2\cos 3t$ .

Above equation can be written as

  $x=\sin t$ and $\cos 3t=\frac{y}{2}$

Using the identical $\cos 3t=4{\cos }^{3}t-3\cos t$ ,

  $\cos 3t=\cos t\left(4{\cos }^{2}t-3\right)$ .

On using ${\cos }^{2}t=1-{\sin }^{2}t$ , we get

  $\begin{array}{l}\cos 3t=\left(\sqrt{1-{\sin }^{2}t}\right)\left(4-4{\sin }^{2}t-3\right)\\ =\left(\sqrt{1-{\sin }^{2}t}\right)\left(1-4{\sin }^{2}t\right)\end{array}$

On putting $\cos 3t$ and $\sin t$ in the above equation, we get

  $\frac{y}{2}=\left(1-4x^2\right)\sqrt{1-x^2}$

Thus,

  $y=2\left(1-4x^2\right)\sqrt{1-x^2}$

Since, $-1\le \sin t\le 1$ , therefore $-1\le x\le 1$

So the curve is represented between the lines $x=-1$ and $x=1$ .

  

Conclusion:

Therefore, the required graph is drawn.