Chapter 10.7, Problem 25AYU

To determine

The graph of curve of the parametric equation $x={\sin }^{2}t,y={\cos }^{2}t;0\le t\le 2\pi$, show its orientation. Find the rectangular equation of the curve.

Answer to Problem 25AYU

Solution:

The graph of the parametric equations $x={\sin }^{2}t,y={\cos }^{2}t;0\le t\le 2\pi$ is

The rectangular equation of the parametric equation $x={\sin }^{2}t,y={\cos }^{2}t;0\le t\le 2\pi$ is $x+y=1$.

Explanation of Solution

Given information:

The parametric equation is $x={\sin }^{2}t,y={\cos }^{2}t;0\le t\le 2\pi$.

Explanation:

To draw the given parametric equation, plug some values of $t$ in the given equation and find few points on the curve.

The interval for $t$ is $0\le t\le 2\pi$.

$\begin{array}{cccc}t& x\left(t\right)& y\left(t\right)& \left(x,y\right)\\ 0& 0& 1& \left(0,1\right)\\ \frac{\pi }{2}& 1& 0& \left(1,0\right)\\ \pi & 0& 1& \left(0,1\right)\\ \frac{3\pi }{2}& 1& 0& \left(1,0\right)\\ 2\pi & 0& 1& \left(0,1\right)\end{array}$

The orientation is the curve traced out in a certain direction by the corresponding succession of points $\left(x,y\right)$.

The arrow shows the orientation along the curve as $t$ varies from $0\le t\le 2\pi$.

As $t$ takes values from $0\le t\le 2\pi$, the corresponding orientation is from $\left(0,1\right)$ to $\left(1,0\right)$

Use points in the table to sketch the curve.

The graph of the parametric equation $x={\sin }^{2}t,y={\cos }^{2}t;0\le t\le 2\pi$ is

The orientation is back and forth movement along the line.

Let $x={\sin }^{2}t,y={\cos }^{2}t$

By using Pythagorean identity ${\sin }^{2}t+{\cos }^{2}t=1$,

$\Rightarrow x+y=1$

The equation $x+y=1$ represents a line.