Q. 10. - Calculus

Question

Explain how to use parametric equations to transform a

polar function $r=f\left(\theta \right)$

Step-by-Step Solution

Answer

The parametric equations are $x=(1-\sin \theta )\cos \theta ,y=(1-\sin \theta )\sin \theta$

Step 1: Given information

The polar function is $r=f\left(\theta \right)$

Step 2: The objective is to transform the polar equation of the form r = f ( θ )   into the parametric equations.

The function $r=f\left(\theta \right)$is simply expressed using parametric equations of the form $x=x\left(\theta \right),y=y\left(\theta \right)$where $\theta$is the parameter.

To convert polar coordinates into rectangle coordinates $x=r\cos \theta ,y=r\sin \theta$

Step 3: write the parametric equations for r = 1 - sin θ  

If $r=f\left(\theta \right)$then,

$x=f\left(\theta \right)\cos \theta ,y=f\left(\theta \right)\sin \theta .$

The parametric equations for $r=1-\sin \theta$

The parametric equations have the following form:

$x=f\left(\theta \right)\cos \theta ,y=f\left(\theta \right)\sin \theta$

Then the parametric equations are:

$x=(1-\sin \theta )\cos \theta ,y=(1-\sin \theta )\sin \theta$