A question came up today regarding why we need to approximate
sin(theta) as theta for small theta when deriving the angular
frequency for the pendulum. Here is why: The m*g*sin(theta) force
vector I drew in the diagram pointed down and to the left, when
strictly speaking, for SHM, it should point only left (that is, only
in the -x direction, since x is the displacement). When theta is
small, that vector points almost left, so the approximation is
OK. When theta gets big, that vector has too much of a component
downward and the approximation is no good. So that's why we need to
consider only small theta (small displacements) for the pendulum when
using SHM for the analysis.