tangent half-angle formula
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trigonometry Trigonometry () is a branch of mathematics that studies relationships between side lengths and angles of triangles. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies. T ...
, 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 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 onto a line. Among these formulas are the following: : \begin \tan \tfrac12( \eta \pm \theta) &= \frac = \frac = -\frac, \\
0pt PT, Pt, or pt may refer to: Arts and entertainment * ''P.T.'' (video game), acronym for ''Playable Teaser'', a short video game released to promote the cancelled video game ''Silent Hills'' * Porcupine Tree, a British progressive rock group ...
\tan \tfrac12 \theta &= \frac = \frac = \frac, & & (\eta = 0) \\
0pt PT, Pt, or pt may refer to: Arts and entertainment * ''P.T.'' (video game), acronym for ''Playable Teaser'', a short video game released to promote the cancelled video game ''Silent Hills'' * Porcupine Tree, a British progressive rock group ...
\tan \tfrac12 \theta &= \frac = \frac = \csc\theta-\cot\theta, & & (\eta = 0) \\
0pt PT, Pt, or pt may refer to: Arts and entertainment * ''P.T.'' (video game), acronym for ''Playable Teaser'', a short video game released to promote the cancelled video game ''Silent Hills'' * Porcupine Tree, a British progressive rock group ...
\tan \tfrac12 \big(\theta \pm \tfrac12\pi \big) &= \frac = \sec\theta \pm \tan\theta = \frac, & & \big(\eta = \tfrac12\pi \big) \\
0pt PT, Pt, or pt may refer to: Arts and entertainment * ''P.T.'' (video game), acronym for ''Playable Teaser'', a short video game released to promote the cancelled video game ''Silent Hills'' * Porcupine Tree, a British progressive rock group ...
\tan \tfrac12 \big(\theta \pm \tfrac12\pi \big) &= \frac = \frac = \frac, & & \big(\eta = \tfrac12\pi \big) \\
0pt PT, Pt, or pt may refer to: Arts and entertainment * ''P.T.'' (video game), acronym for ''Playable Teaser'', a short video game released to promote the cancelled video game ''Silent Hills'' * Porcupine Tree, a British progressive rock group ...
\frac &= \pm\sqrt \\
0pt PT, Pt, or pt may refer to: Arts and entertainment * ''P.T.'' (video game), acronym for ''Playable Teaser'', a short video game released to promote the cancelled video game ''Silent Hills'' * Porcupine Tree, a British progressive rock group ...
\tan \tfrac12 \theta &= \pm \sqrt \\
0pt PT, Pt, or pt may refer to: Arts and entertainment * ''P.T.'' (video game), acronym for ''Playable Teaser'', a short video game released to promote the cancelled video game ''Silent Hills'' * Porcupine Tree, a British progressive rock group ...
\end From these one can derive identities expressing the sine, cosine, and tangent as functions of tangents of half-angles: : \begin \sin \alpha & = \frac \\ pt\cos \alpha & = \frac \\ pt\tan \alpha & = \frac \end


Proofs


Algebraic proofs

Using
double-angle formulae 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 ...
and the Pythagorean identity 1 + \tan^2 \alpha = 1 \big/ \cos^2 \alpha gives : \sin \alpha = 2\sin \tfrac12 \alpha \cos \tfrac12 \alpha = \frac = \frac, \quad \text : \cos \alpha = \cos^2 \tfrac12 \alpha - \sin^2 \tfrac12 \alpha = \frac = \frac, \quad \text Taking the quotient of the formulae for sine and cosine yields : \tan \alpha = \frac. Combining the Pythagorean identity with the double-angle formula for the cosine, \cos 2\alpha = \cos^2 \alpha - \sin^2 \alpha = 1 - 2\sin^2 \alpha = 2\cos^2 \alpha - 1 , rearranging, and taking the square roots yields : , \sin \alpha, = \sqrt and , \cos \alpha, = \sqrt which, upon division gives : , \tan \alpha, = \frac = \frac =\frac = \frac. Alternatively, : , \tan \alpha, = \frac = \frac = \frac = \frac. It turns out that the absolute value signs in these last two formulas may be dropped, regardless of which quadrant 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: : \cos (a+b) = \cos a \cos b - \sin a \sin b : \cos (a-b) = \cos a \cos b + \sin a \sin b : \sin (a+b) = \sin a \cos b + \cos a \sin b : \sin (a-b) = \sin a \cos b - \cos a \sin b Pairwise addition of the above four formulae yields: : \begin &\sin (a+b) + \sin (a-b) \\ mu&\quad= \sin a \cos b + \cos a \sin b + \sin a \cos b - \cos a \sin b \\ mu&\quad = 2 \sin a \cos b \\ 5mu &\cos (a+b) + \cos (a-b) \\ mu&\quad= \cos a \cos b - \sin a \sin b + \cos a \cos b + \sin a \sin b \\ mu&\quad= 2 \cos a \cos b \end Setting a= \tfrac12 (p+q) and b= \tfrac12 (p-q) and substituting yields: : \begin & \sin p + \sin q \\ mu&\quad= \sin \left(\tfrac12 (p+q) + \tfrac12 (p-q)\right) + \sin\left(\tfrac12(p+q) - \tfrac12 (p-q)\right) \\ mu&\quad= 2 \sin \tfrac12(p+q) \, \cos \tfrac12(p-q) \\ 5mu& \cos p + \cos q \\ mu&\quad= \cos\left(\tfrac12(p+q) + \tfrac12 (p-q)\right) + \cos\left(\tfrac12(p+q) - \tfrac12(p-q)\right) \\ mu&\quad= 2 \cos\tfrac12(p+q) \, \cos\tfrac12(p-q) \end Dividing the sum of sines by the sum of cosines one arrives at: : \frac = \frac = \tan \tfrac12(p+q)


Geometric proofs

Applying the formulae derived above to the rhombus figure on the right, it is readily shown that : \tan \tfrac12 (a+b) = \frac = \frac. In the unit circle, application of the above shows that t = \tan \tfrac12 \varphi. By similarity of triangles, : \frac = \frac. It follows that : t = \frac = \frac = \frac.


The tangent half-angle substitution in integral calculus

In various applications of
trigonometry Trigonometry () is a branch of mathematics that studies relationships between side lengths and angles of triangles. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies. T ...
, it is useful to rewrite the
trigonometric function 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 ...
s (such as
sine In mathematics, sine and cosine are trigonometric functions of an angle. The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side that is oppo ...
and cosine) in terms of
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 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 Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithm ...
for converting rational functions in sine and cosine to functions of in order to find their
antiderivative In calculus, an antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral of a function is a differentiable function whose derivative is equal to the original function . This can be stated symbolically ...
s. Geometrically, the construction goes like this: for any point on 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 ...
, draw the line passing through it and the point . This point crosses the -axis at some point . One can show using simple geometry that . The equation for the drawn line is . The equation for the intersection of the line and circle is then a
quadratic equation In algebra, a quadratic equation () is any equation that can be rearranged in standard form as ax^2 + bx + c = 0\,, where represents an unknown (mathematics), unknown value, and , , and represent known numbers, where . (If and then the equati ...
involving . The two solutions to this equation are and . This allows us to write the latter as rational functions of (solutions are given below). The parameter represents the
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 point onto the -axis with the center of projection at . Thus, the tangent half-angle formulae give conversions between the stereographic coordinate on the unit circle and the standard angular coordinate . Then we have : \begin & \sin\varphi = \frac, & & \cos\varphi = \frac, \\ pt& \tan\varphi = \frac & & \cot\varphi = \frac, \\ pt& \sec\varphi = \frac, & & \csc\varphi = \frac, \end and : e^ = \frac, \qquad e^ = \frac. By eliminating phi between the directly above and the initial definition of t, one arrives at the following useful relationship for the
arctangent In mathematics, the inverse trigonometric functions (occasionally also called arcus functions, antitrigonometric functions or cyclometric functions) are the inverse functions of the trigonometric functions (with suitably restricted domains). Spec ...
in terms of the
natural logarithm The natural logarithm of a number is its logarithm to the base of the mathematical constant , which is an irrational and transcendental number approximately equal to . The natural logarithm of is generally written as , , or sometimes, if ...
:2 \arctan t = -i \ln\frac. In
calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithm ...
, the Weierstrass substitution is used to find antiderivatives of
rational functions 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 rati ...
of and . After setting :t=\tan\tfrac12\varphi. This implies that :\varphi=2\arctan(t)+2\pi n , for some integer , and therefore :d\varphi = .


Hyperbolic identities

One can play an entirely analogous game with the
hyperbolic function In mathematics, hyperbolic functions are analogues of the ordinary trigonometric functions, but defined using the hyperbola rather than the circle. Just as the points form a circle with a unit radius, the points form the right half of the u ...
s. A point on (the right branch of) a
hyperbola In mathematics, a hyperbola (; pl. hyperbolas or hyperbolae ; adj. hyperbolic ) is a type of smooth curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, cal ...
is given by . Projecting this onto -axis from the center gives the following: :t = \tanh\tfrac12\psi = \frac = \frac with the identities : \begin & \sinh\psi = \frac, & & \cosh\psi = \frac, \\ pt& \tanh\psi = \frac, & & \coth\psi = \frac, \\ pt& \operatorname\,\psi = \frac, & & \operatorname\,\psi = \frac, \end and : e^\psi = \frac, \qquad e^ = \frac. Finding in terms of leads to following relationship between the
inverse hyperbolic tangent In mathematics, the inverse hyperbolic functions are the inverse functions of the hyperbolic functions. For a given value of a hyperbolic function, the corresponding inverse hyperbolic function provides the corresponding hyperbolic angle. The ...
\operatorname and the natural logarithm: :2 \operatorname t = \ln\frac.


The Gudermannian function

Comparing the hyperbolic identities to the circular ones, one notices that they involve the same functions of , just permuted. If we identify the parameter in both cases we arrive at a relationship between the circular functions and the hyperbolic ones. That is, if :t = \tan\tfrac12 \varphi = \tanh\tfrac12 \psi then :\varphi = 2\arctan \bigl(\tanh \tfrac12 \psi\,\bigr) \equiv \operatorname \psi. where is the
Gudermannian function In mathematics, the Gudermannian function relates a hyperbolic angle measure \psi to a circular angle measure \phi called the ''gudermannian'' of \psi and denoted \operatorname\psi. The Gudermannian function reveals a close relationship betwee ...
. 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 -axis) give a geometric interpretation of this function.


Rational values and Pythagorean triples

If is a
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ration ...
then each of , , , , , and will be a rational number (or be infinite). Likewise if is a rational number then each of , , , , , and will be a rational number (or be infinite). This is helpful with Pythagorean triples; each interior angle has a rational sine because of the SAS area formula for a triangle and has a rational cosine because of the
Law of Cosines In trigonometry, the law of cosines (also known as the cosine formula, cosine rule, or al-Kashi's theorem) relates the lengths of the sides of a triangle to the cosine of one of its angles. Using notation as in Fig. 1, the law of cosines states ...
. These imply that the half-angle tangent is necessarily rational. Vice versa, when a half-angle tangent is a rational number in the interval then the full-angle sine and cosine will both be rational, and there is a right triangle that has the full angle and that has side lengths that are a Pythagorean triple. Generally, if is a subfield of the complex numbers then implies that . A similar statement can be made about .


See also

*
List of trigonometric identities 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 ...
*
Half-side formula In spherical trigonometry, the half side formula relates the angles and lengths of the sides of spherical triangles, which are triangles drawn on the surface of a sphere and so have curved sides and do not obey the formulas for plane triangles. Fo ...


External links


''Tangent Of Halved Angle''
at
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