The t-Formulas

The \(t\)-formulas are part of the Further integration topic in NSW Year 12 Mathematics Extension 2 (NESA outcome ME2-12-04). With \(t = \tan\dfrac{A}{2}\), this subtopic derives \(\sin A = \dfrac{2t}{1+t^2}\), \(\cos A = \dfrac{1-t^2}{1+t^2}\) and \(\tan A = \dfrac{2t}{1-t^2}\), turning trigonometric expressions into rational functions of \(t\).

Students and teachers will cover using the \(t\)-substitution to evaluate integrals such as \(\displaystyle\int \dfrac{dx}{5+3\cos x}\) and to solve trigonometric equations on a given interval — a powerful method in the HSC Extension 2 course.