Home > Tangent half-angle formula

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).

[A diagram would be nice here.]

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.

1 Equations

The tangent half-angle formulae are as follows. Let

Then we have

as well as

By inverting the exponential formula for t and finding φ in terms of t, one arrives at the following useful relationship for the arctangent in terms of the natural logarithm

2 Hyperbolic identities

One can play an entirely analogous game with the hyperbolic functionIn mathematics, the hyperbolic functions are analogs of the ordinary trigonometric, or circular, functions. They are: sinh, cosh and tanh csch, sech and coth : :: hyperbolic sine pronounced "shine" or "sinch") : :: hyperbolic cosine pronounced "cosh") : :s. A point on (the right branch of) a hyperbolaFor hyperbole, the figure of speech, see hyperbole. In mathematics, a hyperbola is a type of conic section. Geometrically, it is defined as the intersection between a cone and a plane which cuts through both halves of the cone. Analytically, it is defined is given by (cosh θ, sinh θ). Projecting this onto y-axis from the center (−1,0) gives the following:

with the identities

and

Finding θ in terms of t leads to following relationship between the hyperbolic arctangent and the natural logarithm:

3 The Gudermannian function

Comparing the hyperbolic identities to the circular ones, one notices that they involve the same functions of t, just permuted. If we identify the parameter t in both cases we arrive at a relationship between the circular functions and the hyperbolic ones. That is, if

then

The function gd(θ) is called the Gudermannian functionThe Gudermannian function relates the circular and hyperbolic trigonometric functions without resorting to complex numbers. It is defined by : :: :: Note that : The following identities also hold: : : : : : : The inverse Gudermannian function is given by. The Gudermannian function gives a direct relationship between the circular functions and the hyperbolic ones that does not involve complex numbers. The above descriptions of the tangent half-angle formulae (projection the unit circle and standard hyperbola onto the y-axis) give a nice geometric interpretation of this function.