In
integral calculus
In mathematics, an integral assigns numbers to Function (mathematics), functions in a way that describes Displacement (geometry), displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding ...
, the tangent half-angle substitution is a
change of variables
Change or Changing may refer to:
Alteration
* Impermanence, a difference in a state of affairs at different points in time
* Menopause, also referred to as "the change", the permanent cessation of the menstrual period
* Metamorphosis, or change, ...
used for evaluating
integrals
In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with di ...
, which converts a
rational function
In mathematics, a rational function is any function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be rat ...
of
trigonometric functions
In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all ...
of
into an ordinary rational function of
by setting
. This is the one-dimensional
stereographic projection
In mathematics, a stereographic projection is a perspective projection of the sphere, through a specific point on the sphere (the ''pole'' or ''center of projection''), onto a plane (geometry), plane (the ''projection plane'') perpendicular to ...
of the
unit circle
In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Eucl ...
parametrized by
angle measure
In Euclidean geometry, an angle is the figure formed by two rays, called the '' sides'' of the angle, sharing a common endpoint, called the ''vertex'' of the angle.
Angles formed by two rays lie in the plane that contains the rays. Angles are ...
onto the
real line
In elementary mathematics, a number line is a picture of a graduated straight line (geometry), line that serves as visual representation of the real numbers. Every point of a number line is assumed to correspond to a real number, and every real ...
. The general transformation formula is:
The tangent of half an angle is important in
spherical trigonometry
Spherical trigonometry is the branch of spherical geometry that deals with the metrical relationships between the sides and angles of spherical triangles, traditionally expressed using trigonometric functions. On the sphere, geodesics are gr ...
and was sometimes known in the 17th century as the half tangent or semi-tangent.
Leonhard Euler
Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ma ...
used it to evaluate the integral
in his
1768 integral calculus textbook, and
Adrien-Marie Legendre
Adrien-Marie Legendre (; ; 18 September 1752 – 9 January 1833) was a French mathematician who made numerous contributions to mathematics. Well-known and important concepts such as the Legendre polynomials and Legendre transformation are named ...
described the general method in 1817.
The substitution is described in most integral calculus textbooks since the late 19th century, usually without any special name. It is known in Russia as the universal trigonometric substitution, and also known by variant names such as ''half-tangent substitution'' or ''half-angle substitution''. It is sometimes misattributed as the Weierstrass substitution.
Michael Spivak
Michael David Spivak (25 May 19401 October 2020)Biographical sketch in Notices of the AMS', Vol. 32, 1985, p. 576. was an American mathematician specializing in differential geometry, an expositor of mathematics, and the founder of Publish-or-Per ...
called it the "world's sneakiest substitution".
The substitution
Introducing a new variable
sines and cosines can be expressed as
rational function
In mathematics, a rational function is any function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be rat ...
s of
and
can be expressed as the product of
and a rational function of
as follows:
Derivation
Using the
double-angle formula
In trigonometry, trigonometric identities are equalities that involve trigonometric functions and are true for every value of the occurring variables for which both sides of the equality are defined. Geometrically, these are identities involvin ...
s, introducing denominators equal to one thanks to the
Pythagorean theorem
In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite t ...
, and then dividing numerators and denominators by
one gets
Finally, since
,
differentiation rules imply
:
and thus
:
Examples
Antiderivative of cosecant
We can confirm the above result using a standard method of evaluating the cosecant integral by multiplying the numerator and denominator by
and performing the substitution
.
These two answers are the same because
The
secant integral may be evaluated in a similar manner.
A definite integral
In the first line, one cannot simply substitute
for both
limits of integration
In calculus and mathematical analysis the limits of integration (or bounds of integration) of the integral
: \int_a^b f(x) \, dx
of a Riemann integrable function f defined on a closed and bounded interval are the real numbers a and b , in w ...
. The
singularity (in this case, a
vertical asymptote
In analytic geometry, an asymptote () of a curve is a line such that the distance between the curve and the line approaches zero as one or both of the ''x'' or ''y'' coordinates tends to infinity. In projective geometry and related contexts, ...
) of
at
must be taken into account. Alternatively, first evaluate the indefinite integral, then apply the boundary values.
By symmetry,
which is the same as the previous answer.
Third example: both sine and cosine
if
Geometry
![Weierstrass](https://upload.wikimedia.org/wikipedia/commons/d/d2/Weierstrass.substitution.svg)
As ''x'' varies, the point (cos ''x'', sin ''x'') winds repeatedly around the
unit circle
In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Eucl ...
centered at (0, 0). The point
goes only once around the circle as ''t'' goes from −∞ to +∞, and never reaches the point (−1, 0), which is approached as a limit as ''t'' approaches ±∞. 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).
Here is another geometric point of view. Draw the unit circle, and let ''P'' be the point . A line through ''P'' (except the vertical line) is determined by its slope. 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. The trigonometric functions determine a function from angles to points on the unit circle, and by combining these two functions we have a function from angles to slopes.
Gallery
File:Weierstrass substitution.svg, (1/2) The tangent half-angle substitution relates an angle to the slope of a line.
File:WeierstrassSubstitution.svg, (2/2) The tangent half-angle substitution illustrated as stereographic projection
In mathematics, a stereographic projection is a perspective projection of the sphere, through a specific point on the sphere (the ''pole'' or ''center of projection''), onto a plane (geometry), plane (the ''projection plane'') perpendicular to ...
of the circle.
Hyperbolic functions
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,
:
Geometrically, this change of variables is a one-dimensional analog of the
Poincaré disk
Poincaré is a French surname. Notable people with the surname include:
* Henri Poincaré (1854–1912), French physicist, mathematician and philosopher of science
* Henriette Poincaré (1858-1943), wife of Prime Minister Raymond Poincaré
* L ...
projection.
See also
*
Rational curve
In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane c ...
*
Stereographic projection
In mathematics, a stereographic projection is a perspective projection of the sphere, through a specific point on the sphere (the ''pole'' or ''center of projection''), onto a plane (geometry), plane (the ''projection plane'') perpendicular to ...
*
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. The tangent of half an angle is the stereographic projection of the circle onto a line. Among these formulas are th ...
*
Trigonometric substitution
In mathematics, trigonometric substitution is the replacement of trigonometric functions for other expressions. In calculus, trigonometric substitution is a technique for evaluating integrals. Moreover, one may use the trigonometric identities ...
*
Euler substitution
Euler substitution is a method for evaluating integrals of the form
\int R(x, \sqrt) \, dx,
where R is a rational function of x and \sqrt. In such cases, the integrand can be changed to a rational function by using the substitutions of Euler.
...
Further reading
*
*
* Second edition 1916
pp. 52–62
*
Notes and references
External links
Weierstrass substitution formulasat
PlanetMath
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{{Integrals
Integral calculus