More generally one can say that, for piecewise monotonic functions, it is sufficient to consider the endpoints formula_45, formula_68of an interval, together with the so-called "critical points" within the interval, being those points where the monotonicity of the function changes direction. For the sine and cosine functions, the critical points are at formula_80 or formula_81 for formula_82, respectively. Thus, only up to five points within an interval need to be considered, as the resulting interval is formula_83 if the interval includes at least two extrema. For sine and cosine, only the endpoints need full evaluation, as the critical points lead to easily pre-calculated values—namely -1, 0, and 1.