What exactly is a "Double Angle"?

I had a question about Trig. Double angle formulas….why is it called “double angle”? For instance, say we have Cos 2x. Does this mean we are multiplying the value of the angle by 2 and then taking the Cosine of it? So 150 degrees would 300 degrees? And then we take the Cosine of it? I understand how to calculate these using the formulas, I just don't understand what a double angle is (visually I guess).

Also, why does Cosine have 3 ways of calculating a double angle and Sine only has 1?