First, notice that since cos θ is always between -1 and +1, the quantity r = 1 + cos θ is always between 0 and 2. In particular, r will never be negative. We should find points in every quadrant. Find the value of r at some nice angles. 
This gives us some points to start from. 
Now we need to figure out what's going on in between these points. Remember that r is never negative. From θ = 0 to  , the value of r will move from 2 to 1. 
From  to θ = π, the value of r will move from 1 to 0. 
From θ = π to  , the value of r will move from 0 to 1. 
From to θ = 2π, the value of r will move from 1 to 2. We end up with a heart shape that looks nothing like the graph r = cos θ. 
The moral of the story is that we typically use calculators when making anything but the simplest of graphs in polar coordinates.. |