Small Angle Approximations for sin, cos and tan

Functions can be approximated by polynomials by taking a Taylor Expansion (what is a Taylor Expansion?). These expansions are, of course, dependent on where we are taking the approximation. Recall that for the small angle approximations above, they are only valid for values of $\theta$ around zero. For other ranges, the Taylor approximation will look different. Note that when the Taylor series is taken for values around zero, we call it a Maclaurin series.

It can be shown that the Maclaurin series for $\sin(\theta)$ and $\cos(\theta)$ ($\tan(\theta)$ isn’t as simple) are given by:

$\sin(\theta)\approx\theta-\frac{\theta^3}{3!}+\frac{\theta^5}{5!}-\frac{\theta^7}{7!}+…$

$\cos(\theta)\approx 1-\frac{\theta^2}{2}+\frac{\theta^4}{4!}-\frac{\theta^6}{6!}+…$

Notice that $\sin(\theta)$ has only odd powers of $\theta$ whereas $\cos(\theta)$ has only even powers. As we can see, both expansions have alternating signs and increasing powers of $\theta$. Hence, the more terms included in the expansion, the larger the range of $\theta$ values the expansion will be a good approximation for. This explains why the small angle approximation for $\cos(\theta)$ works for a larger range of $\theta$ values – it is quadratic whereas the approximations for $\sin(\theta)$ and $\tan(\theta)$ are linear.