Q. 35 - Calculus

Question

Finish Example 1 (b) by showing that the graph of the parametric equations $x=\cos t,y=\sin t,t\in [\pi ,2\pi ]$ is the bottom half of the unit circle centered at the origin with a counterclockwise direction of motion.

Step-by-Step Solution

Answer

The required graph is 

Step 1: Given Information

The parametric equations , $x=\cos t,y=\sin t,t\in [\pi ,2pi]$.

Step 2: Calculation

Consider the parametric equations $,x=\cos t,y=\sin t,t\in [\pi ,2\pi ]$.

The objective is to sketch the parametric curve by eliminating the parameter.

Take the equation: $x=\cos t$.

Squaring the equation on both sides.

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

Now take the equation $y=\sin t$.

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

Add the equations $x^2={\cos }^{2}t$ and $y^2={\sin }^{2}t$.

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

$x^2+y^2=1\left[\text{ since }{\cos }^{2}t+{\sin }^{2}t=1\right]$

The equation after eliminating the parameter t is $x^2+y^2=1$.

Now when $t=\pi$,

When $t=2\pi$

$$\begin{gathered} (x,y)=(\cos t,\sin t) \\ (x,y)=(\cos \pi ,\sin \pi ) \\ (x,y)=(-1,0) \end{gathered}$$

Now when $t=\frac{3\pi }{2}$,

$$\begin{gathered} (x,y)=(\cos t,\sin t) \\ (x,y)=\left(\cos \frac{3\pi }{2},\sin \frac{3\pi }{2}\right) \\ (x,y)=(0,-1) \end{gathered}$$

Step 3: Further calculation

When $t=2\pi$

$$\begin{gathered} (x,y)=(\cos t,\sin t) \\ (x,y)=(\cos 2\pi ,\sin 2\pi ) \\ (x,y)=(1,0) \end{gathered}$$

The graphical representation by using the points $(-1,0)(0,-1)(1,0)$ is as follows,

The equation after elimination of the parameter is $x^2+y^2=1$.