Tangent Half-angle Formula - The Weierstrass Substitution

The Weierstrass Substitution

In various applications of trigonometry, it is useful to rewrite the trigonometric functions (such as sine and cosine) in terms of rational functions of a new variable t. These identities are known collectively as the tangent half-angle formulae because of the definition of t. These identities can be useful in calculus for converting rational functions in sine and cosine to functions of t in order to find their antiderivatives.

Technically, the existence of the tangent half-angle formulae stems from the fact that the circle is an algebraic curve of genus 0. One then expects that the 'circular functions' should be reducible to rational functions.

Geometrically, the construction goes like this: for any point (cos φ, sin φ) on the unit circle, draw the line passing through it and the point (−1,0). This point crosses the y-axis at some point y = t. One can show using simple geometry that t = tan(φ/2). The equation for the drawn line is y = (1 + x)t. The equation for the intersection of the line and circle is then a quadratic equation involving t. The two solutions to this equation are (−1, 0) and (cos φ, sin φ). This allows us to write the latter as rational functions of t (solutions are given below).

Note also that the parameter t represents the stereographic projection of the point (cos φ, sin φ) onto the y-axis with the center of projection at (−1,0). Thus, the tangent half-angle formulae give conversions between the stereographic coordinate t on the unit circle and the standard angular coordinate φ.

Then we have

and

By eliminating phi between the directly above and the initial definition of t, one arrives at the following useful relationship for the arctangent in terms of the natural logarithm

In calculus, the Weierstrass substitution is used to find antiderivatives of rational functions of sin(φ) and cos(φ). After setting

This implies that

and therefore