Approximation for $\sin(\beta\sin(x))$

Question

Can someone explain why, assuming $\beta\ll 1$, we have

$$\cos(\beta \sin(2\pi f_mt))\approx 1$$

and

$$\sin(\beta \sin(2\pi f_mt))\approx \beta \sin(2\pi f_mt) $$

The equations are part of a FM narrow band model and I think I can understand the first one since $\cos(0)=1 $, but the second one eludes me.

Answer 1

The small-angle approximations for trigonometric functions are based on their Taylor series. Such as:

$$\sin x = x - \frac{x^3}{6} +\frac{x^5}{120} - \dots$$ $$\cos x = 1 - \frac{x^2}{2} +\frac{x^4}{24} - \dots$$

In your example, only the first, largest term of the expansion was used. Sometimes people use more, not only for greater accuracy but also greater complexity.