Chapter 10.1, Problem 6E

To determine

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

Answer to Problem 6E

The equation is $x=\sin (3y)$ .

Explanation of Solution

Given information:

The equations $x=\sin 6t$ and

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

Calculation:

The given equations

  $\begin{array}{c}x=\sin (6t),\\ y=2t\end{array}$

And

  $t\in [0,\pi /2]$

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 $x$ .

From the equation $y=2t$

So,

  $t=y/2$

Put,

  $t=y/2$

In the equation

  $x=\sin (6t)$

So,

  $\begin{array}{c}x=\sin (6t)\\ =\sin \left(6\cdot \frac{y}{2}\right)\\ =\sin (3y)\end{array}$

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

  $x=\sin (3y)$

Therefore, the required parametric curve defines $y$ as a function of $x$ , while the equation that relates $x$ and $y$ is equal to $x=\sin (3y)$ .