The factor of tension and frequency increase in a conical pendulum?

1. PhoniexGuy

26
This, is just a concept question:

My teacher gave the Answer has something like this:

However I understand most of it, except the part where it says:

What is happening in this? Where does the 4*pi^2*L*sin(theta) come from, and how is v found? why is f = 1/t?