Stretching the Wave

A b between 0 and 1 slows the angle down. In $y = \sin\left(\frac{x}{2}\right)$ the inside is only half of x, so it reaches 2π when x reaches 4π: one wave is twice as long as before.

The rule is the same: dividing by 2 is multiplying by b = 1/2, and $\frac{2\pi}{\frac{1}{2}} = 4\pi$. The top comes a quarter of the way along, at π, instead of at π/2.

So for b positive, a b bigger than 1 squeezes the wave and a b between 0 and 1 stretches it. Either way the heights stay between −1 and 1.