Tangent half-angle formula

In trigonometry, tangent half-angle formulas relate the tangent of half of an angle to trigonometric functions of the entire angle.

Using double-angle formulae and the Pythagorean identity

Taking the quotient of the formulae for sine and cosine yields

Combining the Pythagorean identity with the double-angle formula for the cosine,

rearranging, and taking the square roots yields

It turns out that the absolute value signs in these last two formulas may be dropped, regardless of which quadrant α is in.

With or without the absolute value bars these formulas do not apply when both the numerator and denominator on the right-hand side are zero.

Also, using the angle addition and subtraction formulae for both the sine and cosine one obtains:

Pairwise addition of the above four formulae yields:

Applying the formulae derived above to the rhombus figure on the right, it is readily shown that

In the unit circle, application of the above shows that

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

These identities are known collectively as the tangent half-angle formulae because of the definition of

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.

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 allows us to write the latter as rational functions of t (solutions are given below).

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.

Equating these gives the arctangent in terms of the natural logarithm

In calculus, the tangent half-angle substitution is used to find antiderivatives of rational functions of sin φ and cos φ. Differentiating

One can play an entirely analogous game with the hyperbolic functions.

Projecting this onto y-axis from the center (−1, 0) gives the following:

Finding ψ in terms of t leads to following relationship between the inverse hyperbolic tangent

The hyperbolic tangent half-angle substitution in calculus uses

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.

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 geometric interpretation of this function.

Starting with a Pythagorean triangle with side lengths a, b, and c that are positive integers and satisfy a2 + b2 = c2, it follows immediately that each interior angle of the triangle has rational values for sine and cosine, because these are just ratios of side lengths.

Thus each of these angles has a rational value for its half-angle tangent, using tan φ/2 = sin φ / (1 + cos φ).