Two trigonometric quadrature formulae, I one of non-interpolatory type and one of interpolatory type for computing the hypersingular integral integral(-1)(1) w(tau)g(tau)/(tau - t)(2) dtau are developed on the basis of trigonometric quadrature formulae for Cauchy principal value integrals. The formulae use the cosine change of variables and trigonometric polynomial interpolation at the practical abscissae. Fast three-term recurrence relations for evaluating the quadrature weights are derived. Numerical tests are carried out using the current formula. As applications, two simple crack problems are considered. One is a semi-infinite plane containing an internal crack perpendicular to its boundary and the other is a centre cracked panel subjected to both normal and shear tractions. It is found that the present method generally gives superior results. Copyright (C) 2002 John Wiley Sons, Ltd.