Verified answer

r = 1 + 2cosθ

http://www3.wolframalpha.com/input/?i=polar+plot+r...

Lower half: r = 0, r = ±1

r = 0

1 + 2cosθ = 0

2 cosθ = -1

cosθ = -1/2

θ = 2π/3, 4π/3

Since polar graphs are drawn counter-clockwise, lower half of inner loop gets drawn first, so we need first occurrence of θ

θ = 2π/3

Now checking out graph we notice that when we reach rightmost point of inner loop, we've gone past 2π/3, which means that θ must be to the left of vertical axis. However rightmost point of inner loop is to the right of vertical axis, on horizontal axis

θ = π (with r = -1)

Interval: [2π/3, π]

Check:http://www3.wolframalpha.com/input/?i=polar+plot+r...

Mαthmφm

When theta is 2 pi/3 the radius will be 0, since cosine 2 pi/3 is -1/2. Then it makes the lower half of the inner loop until you get to pi. At pi, the radius is -1, which means that it goes straight right instead of straight left like you would for pi radians. Then the upper half of the inner loop is from pi to 4pi/3.

r = 1 + 2cosθ

r² = r + 2rcosθ.........<--multiply through by r

r = r² - 2rcosθ..........<--rearrange

r² = (r² - 2rcosθ)².....<--square both sides

cosθ = x / r...............<--r = hypotenuse, x = side, θ = angle

2rcosθ = 2r(x/r) = 2x

and remember, equation of a circle:

x²+y² = r²

r² = (r² - 2rcosθ)²

x²+y² = (x²+y² - 2x)² <--final answer

So, It's simple.