<p><it>Abstract</it>—The calculation of a trigonometric function of a large argument <it>x</it> is effectively carried out by finding the integer <it>N</it> and 0 ≤α < 1 such that $<tmath>x = (N + \alpha){\pi\over 4}</tmath>$. This reduction modulo $<tmath>{\pi\over 4}</tmath>$ makes it possible to calculate a trigonometric function of a reduced argument, either $<tmath>\alpha{\pi\over 4}</tmath>$ or $<tmath>(1 - \alpha){\pi\over 4}</tmath>$, which lies in the interval $<tmath>(0,{\pi\over 4})</tmath>$. Payne and Hanek [<ref rid="BIBC13481" type="bib">1</ref>] described an efficient algorithm for computing α to a predetermined level of accuracy. They noted that if <it>x</it> differs only slightly from an integral multiple of $<tmath>{\pi\over 2}</tmath>$, the reduction must be carried out quite accurately to avoid a large loss of significance in the reduced argument. We present a simple method using continued fractions for determining, for all numbers <it>x</it> represented in an IEEE floating-point format, the specific <it>x</it> for which the greatest number of insignificant leading bits occur. Applications are made to IEEE single-precision and double-precision formats and two extended-precision formats.</p>