Chapter 11.1, Problem 4E

To determine

To calculate: The equation which relates x and y directly.

Answer to Problem 4E

The required equation is $x^2+{\left(\frac{y}{2}\right)}^2=1$ .

Explanation of Solution

Given information:

The parametric equations: $x=\sin t$ , $y=2\cos t$ for t in the interval $\left[0,\pi \right]$

Calculation:

The parametric equations are $x=\sin t$ , $y=2\cos t$ .

It is known that ${\cos }^{2}t+{\sin }^{2}t=1$ .

Rewrite the above expression $y=2\cos t$ as:

  $\begin{array}{c}y=2\cos t\\ \cos t=\frac{y}{2}\end{array}$

Substitute $\frac{y}{2}$ for y and $\sin t$ for x in the above expression ${\cos }^{2}t+{\sin }^{2}t=1$ .

  $\begin{array}{c}{\cos }^{2}t+{\sin }^{2}t=1\\ {\left(\frac{y}{2}\right)}^2+x^2=1\\ x^2+{\left(\frac{y}{2}\right)}^2=1\end{array}$

Hence, the required equation is $x^2+{\left(\frac{y}{2}\right)}^2=1$ .