Chapter 11.1, Problem 5E

To determine

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

Answer to Problem 5E

The required equation is $y=1-2x^2$ .

Explanation of Solution

Given information:

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

Calculation:

The parametric equations are $x=\sin t$ , $y=\cos \left(2t\right)$ .

It is known that $\cos \left(2t\right)=1-2{\sin }^{2}t$ .

Rewrite the above expression as:

  $\begin{array}{l}y=1-2{\sin }^{2}t\\ y=1-2{\left(\sin t\right)}^2\end{array}$

Substitute $\sin t$ for x in the above expression.

  $y=1-2x^2$

Hence, the required equation is $y=1-2x^2$ .