Question

What is the name of the shape graphed by the function R = 1 -2 sin theta

Answer 1

Limacon of Pascal

Explanation:

The given polar functin is

r =1-2sint

We find that when r =0

2sint =1

sint =1/2

Or t = 60, 120

SO it passes through origin for 60, 120 degrees

There is a loop formed between 30 and 150 and again it completes a big round around the loop

When R=1, we find t =0, pi, 2pi

When R=2, we get sint =-1/2

Hence angles are 210 and 330 degrees

Since sin t lies between -1 and 1 only we have

`-1\leq (1-r)/(2) \leq1 \\-2\leq 1-r\leq 2\\`

When sint =-1, r has maximum 3

and when sint =1, r cannot have -1 but 0

So r can take values between 0 and 3