Q. 36 - Calculus

Question

The curve is a circle centered at the origin. It is traced once, clockwise, starting at the point $(0,1)$ with $t\in [0,2\pi ]$.

Step-by-Step Solution

Answer

The required parametric equations are $x=\sin t,y=\cos t$. 

Step 1: Given information

A curve starting at the point $(0,1)$ with $t\in [0,2\pi ]$. 

Step 2: Calculation

Consider a curve starting at the point $(0,1)$with $t\in [0,2\pi ]$.

The objective is to find the parametric equations which represent the given condition.

Given that the curve is centered at the origin ,traced once in clockwise direction.

The curve is a unit circle starting from the point $(0,1)$ so the radius of the curve is 1 .

The parametric equations which move in a clockwise direction starting at $(0,1)$ is given by

$(x,y)=(r\sin t,r\cos t)$

Here$r=1$ then,

$$\begin{gathered} (x,y)=(1·\sin t,1·\cos t) \\ (x,y)=(\sin t,\cos t) \end{gathered}$$

Thus,

$x=\sin t$ and  forms a circle with center $(0,0)$ and radius is Iwhich starts at $(0,1)$ moving in clockvise direction.

Therefore, the required parametric equations are $x=\sin t,y=\cos t$.