The dot product of two vectors is given by

where is the angle between the vectors.

|p|= while |q|= [tex]\sqrt{cos^2(2t)+ sin^2(2t)+ 1/4}= \sqrt{2}{2}.

Sincepq= cos(3t)- 1, we must have

so the problem is simply "what is the maximum value of cos(3t)-1 for t between 0 and [tex]2\pi[/itex]" and then take the arccosine of that. Since cosine ranges between -1 and 1, cos(3t)-1 ranges between -2 and 0. -1, the smallest possible value of cosine, is included in that and so the largest possible angle betweenpandqis simply the largest possible angle between any two vectors, 180 degrees.