A Rational Trigonometric Approximation for the Inverse Sinc Function

I present a family of rational approximations for the inverse sinc function α/sin α expressed entirely in terms of cos α. The zeroth-order approximation, (1 + 2cos α)/(3cos α), requires only one trigonometric evaluation and three further floating-point operations (one multiplication, one addition, and one division), yet achieves less than 0.025% relative error for |α| < π/12. Successive correction terms in v = 1 − cos α improve accuracy by approximately 1.9 orders of magnitude per term at small angles, with exact rational coefficients d1 = 3/5, d2 = 8/35, and d3 = 2/21. The method uses zero memory, is numerically stable for |α| < π/2, and is trivial to implement on any platform. Its principal advantage arises when cos α is already available from a preceding computation (CORDIC output, rotation matrix entry, dot product), eliminating the need for an additional trigonometric evaluation. I demonstrate applications to DAC sinc compensation and spherical distance calculations.