Q. 4 - Calculus

Question

Sketching parametric equations: Sketch the curves defined by the given sets of parametric equations. Indicate the direction of motion on each curve.

$x=\sin t$, $y={\cos }^2 t$, $t\in \left[0,4\mathrm{\pi }\right]$.

Step-by-Step Solution

Answer

The graph is parabola.

Step 1. Given Information.

The equation are $x=\sin t$ and $y={\cos }^2 t$ such that $t\in \left[0,4\mathrm{\pi }\right]$

Step 2. Explanation.

Start by eliminating the parameter. Square the equations for x and add it with y.

$x^2={\sin }^2 t$

Apply the Pythagorean identity. 

$$\begin{gathered} {\sin }^{2}t+{\cos }^{2}t=1 \\ x^2+y=1 \end{gathered}$$

Rewrite the equation.

$y=-x^2+1$

This is the equation of parabola.

Step 3. Calculation.

Make a table as follows,

t	0	$\frac{\mathrm{\pi }}{2}$	$-\frac{\mathrm{\pi }}{2}$	$4\mathrm{\pi }$
$x=\sin t$	$0$	1	-1	0
$y={\cos }^{2}t$	1	0	0	1

Step 4. Draw the graph.

The required curve is as follows,