Period: One Wave Long

In $y = \sin\left(2x\right)$ the angle inside the sine runs twice as fast as x. When x has gone from 0 to π, 2x has already gone from 0 to 2π, a whole lap, so one full wave fits into a length of π instead of 2π. The wave is squeezed sideways; its height is unchanged.

The length of one wave is the period. For $y = \sin\left(b\cdot x\right)$ or $y = \cos\left(b\cdot x\right)$ with b positive, the inside reaches 2π when x reaches 2π/b:

$$period = \frac{2\pi}{b}$$

For sin 2x that is 2π/2 = π. The top of the wave comes a quarter of a period after the start, at π/4.