Question

y = 4 cos (3/4) θ

a. amplitude = 4

b. number of cycles = ?

c. period = (8pi/3)

Answer 1

for cos kx, you will have k complete cycles from 0 to 2π , so .....

check:
http://www.wolframalpha.com/input/?i=plot+y+%3D+4+cos+(3x%2F4)

Answer 2

So there will be a cycle for every curve?

Answer 3

A cycle is one completed curve of the cosine curve, or any other cyclic curve.
Usually in questions like this, it might ask "how many cycles are there for the given curve in the interval from 0 to 360° or 2π radians.
in your case the graph I gave you shows that from 0 to 2π, there would be 3/4 of a complete cycle. Notice that agrees with the property I stated above.