Chapter 10.1, Problem 5E

To determine

To find: The equation that relates $x$ and $y$ directly.

Answer to Problem 5E

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

Explanation of Solution

Given information:

The equations $x=\sin t$ and

  $y=\cos (2t)$ for $t$ in the interval $[0,2\pi ]$ .

Calculation:

The given equations

  $\begin{array}{c}x=\sin t,\\ y=\cos (2t),\end{array}$

And

  $t\in [0,2\pi ]$

To use the previous result, the parametric curve in the following graph.

  

Thus may infer from the graph that the parametric curve generates a line segment, which indicates that it defines $y$ as a function of $x$ .

To use the trigonometric identity,

  ${\sin }^{2}t=\frac{1-\cos (2t)}{2}$

Thus can conclude from the previous findings that the equation connecting $x$ and $y$ is equivalent to:

  $\begin{array}{c}{x}^2=\frac{1-y}{2}\\ \Rightarrow 2x^2=1-y\\ \Rightarrow y=1-2x^2\end{array}$

Therefore, the required parametric curve defines $y$ as a function of $x$ , while the equation that relates $x$ and $y$ is equal to $y=1-2x^2$ .