Question

Let a and b be real numbers. Describe the graph of r = a cos (theta + b) as much as possible. (The intent of this question is to get as specific a description as possible that holds for all possible values of a and b.)

How would I proceed on this problem? I know r = cos theta is a plot of a circle
but how would I insert a and b in there? Thanks

Answer 1

Have some fun here, changing the values of a and b
I used a=4 and b=3

http://www.wolframalpha.com/input/?i=plot+r+%3D4cos%28Ø%2B3%29+%2C-6.28+%3C+Ø+%3C+6.28

Notice I also placed a domain on Ø from -2π to +2π
things really get interesting if you put a coefficient in front of the Ø , such as

r = 4cos(7Ø + 3)

Answer 2

cos(θ+b) just rotates the figure around the origin by b.
In effect, it rotates the x- and y-axes by b.

acos(θ+b) scales the rotated figure by a factor of a.