Q. 3 - Calculus

Question

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

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

Step-by-Step Solution

Answer

The graph is circle in clockwise.

Step 1. Given Information.

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

Step 2. Explanation.

Start by eliminating the parameter. Square the equations for x and y,

So,

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

$y^2={\cos }^2 t$

Apply the Pythagorean identity.

${\sin }^2 t+{\cos }^2 t=1$

So, $x^2+y^2=1$

So, this is a circle of radius 1 and centered at the origin. 

Step 3. Calculation

To know the direction make a table as follows,

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

Draw the graph as follows,