Small-angle approximations

The idea

When $\theta$ is small and in radians: $\sin\theta \approx \theta$, $\tan\theta \approx \theta$ and $\cos\theta \approx 1 - \frac{\theta^2}{2}$.

1. Replace each trig function with its approximation.
2. For $\sin k\theta$, use $k\theta$; for $\cos k\theta$, use $1 - \frac{(k\theta)^2}{2}$.
3. Simplify, cancelling powers of $\theta$.

Watch out The angle must be in radians, and $\cos\theta \approx 1 - \frac{\theta^2}{2}$, not $1$, when the $\theta^2$ term matters.