Tangent half-angle substitution
In integral calculus, the tangent half-angle substitution is a change of variables used for evaluating integrals, which converts a rational function of trigonometric functions of
This is the one-dimensional stereographic projection of the unit circle parametrized by angle measure onto the real line.
The tangent of half an angle is important in spherical trigonometry and was sometimes known in the 17th century as the half tangent or semi-tangent.
[2] Leonhard Euler used it to evaluate the integral
in his 1768 integral calculus textbook,[3] and Adrien-Marie Legendre described the general method in 1817.
[4] The substitution is described in most integral calculus textbooks since the late 19th century, usually without any special name.
[7] Michael Spivak called it the "world's sneakiest substitution".
sines and cosines can be expressed as rational functions of
Similar expressions can be written for tan x, cot x, sec x, and csc x.
and introducing denominators equal to one by the Pythagorean identity
We can confirm the above result using a standard method of evaluating the cosecant integral by multiplying the numerator and denominator by
csc x − cot x = tan
The secant integral may be evaluated in a similar manner.
The singularity (in this case, a vertical asymptote) of
Alternatively, first evaluate the indefinite integral, then apply the boundary values.
{\displaystyle {\begin{aligned}\int {\frac {dx}{a\cos x+b\sin x+c}}&=\int {\frac {2\,dt}{a(1-t^{2})+2bt+c(t^{2}+1)}}\\[6pt]&=\int {\frac {2\,dt}{(c-a)t^{2}+2bt+a+c}}\\[6pt]&={\frac {2}{\sqrt {c^{2}-(a^{2}+b^{2})}}}\arctan \left({\frac {(c-a)\tan {\tfrac {x}{2}}+b}{\sqrt {c^{2}-(a^{2}+b^{2})}}}\right)+C\end{aligned}}}
As x varies, the point (cos x, sin x) winds repeatedly around the unit circle centered at (0, 0).
As t goes from −∞ to −1, the point determined by t goes through the part of the circle in the third quadrant, from (−1, 0) to (0, −1).
As t goes from −1 to 0, the point follows the part of the circle in the fourth quadrant from (0, −1) to (1, 0).
As t goes from 0 to 1, the point follows the part of the circle in the first quadrant from (1, 0) to (0, 1).
Finally, as t goes from 1 to +∞, the point follows the part of the circle in the second quadrant from (0, 1) to (−1, 0).
Draw the unit circle, and let P be the point (−1, 0).
Furthermore, each of the lines (except the vertical line) intersects the unit circle in exactly two points, one of which is P. This determines a function from points on the unit circle to slopes.
As with other properties shared between the trigonometric functions and the hyperbolic functions, it is possible to use hyperbolic identities to construct a similar form of the substitution,
Similar expressions can be written for tanh x, coth x, sech x, and csch x. Geometrically, this change of variables is a one-dimensional stereographic projection of the hyperbolic line onto the real interval, analogous to the Poincaré disk model of the hyperbolic plane.
There are other approaches to integrating trigonometric functions.
For example, it can be helpful to rewrite trigonometric functions in terms of eix and e−ix using Euler's formula.
Later authors, citing Stewart, have sometimes referred to this as the Weierstrass substitution, for instance: Weierstrass, Karl (1915) [1875].
Bestimmung des Integrals ...".
Mathematische Werke von Karl Weierstrass (in German).
