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| {{unreferenced|date=November 2011}}
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| In [[trigonometry]], the '''tangent half-angle formulas''' relate the tangent of one half of an angle to trigonometric functions of the entire angle. There are a number of different forms:
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| :<math>
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| \begin{align}
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| \tan\left(\frac{\eta}{2} \pm \frac{\theta}{2}\right) & = \frac{\sin\eta \pm \sin\theta}{\cos\eta + \cos\theta} = -\frac{\cos\eta - \cos\theta}{\sin\eta \mp \sin\theta}, \\[10pt]
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| \tan\left(\pm\frac{\theta}{2}\right) & = \frac{\pm\sin\theta}{1 + \cos\theta} = \frac{\pm\tan\theta}{\sec\theta + 1} = \frac{\pm 1}{\csc\theta + \cot\theta}, ~~~~(\eta = 0) \\[10pt]
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| \tan\left(\pm\frac{\theta}{2}\right) & = \frac{1-\cos\theta}{\pm\sin\theta} = \frac{\sec\theta-1}{\pm\tan\theta} = \pm(\csc\theta-\cot\theta), ~~~~(\eta=0) \\[10pt]
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| \tan\left(\frac{\pi}{4} \pm \frac{\theta}{2} \right) & = \frac{1 \pm \sin\theta}{\cos\theta} = \sec\theta \pm \tan\theta = \frac{\csc\theta \pm 1}{\cot\theta}, ~~~~(\eta=\frac{\pi}{2}) \\[10pt]
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| \tan\left(\frac{\pi}{4} \pm \frac{\theta}{2} \right) & = \frac{\cos\theta}{1 \mp \sin\theta} = \frac{1}{\sec\theta \mp \tan\theta} = \frac{\cot\theta}{\csc\theta \mp 1}, ~~~~(\eta=\frac{\pi}{2}) \\[10pt]
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| \frac{1 - \tan(\theta/2)}{1 + \tan(\theta/2)} & = \sqrt{\frac{1 - \sin\theta}{1 + \sin\theta}}.
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| \end{align}
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| </math> | |
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| == Geometric proofs ==
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| {{expand section|date=January 2014}}
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| [[Image:Weierstrass substitution.svg|right|400px|thumb|A [[geometric]] proof of the tangent half-angle formula]]
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| [[File:Tan.half.svg|right|400px|thumb|The sides of this rhombus have length 1. The angle between the horizontal line and the shown diagonal is (''a'' + ''b'')/2. This is a geometric way to prove a tangent half-angle formula.]]
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| == The tangent half-angle substitution in integral calculus ==
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| {{Main|Tangent half-angle substitution}}
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| In various applications of [[trigonometry]], it is useful to rewrite the [[trigonometric function]]s (such as [[sine]] and [[cosine]]) in terms of [[rational function]]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]] for converting rational functions in sine and cosine to functions of ''t'' in order to find their [[antiderivative]]s.
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| Technically, the existence of the tangent half-angle formulae stems from the fact that the [[circle]] is an [[algebraic curve]] of [[Genus (mathematics)|genus]] 0. One then expects that the 'circular functions' should be reducible to rational functions.
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| 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).
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| 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.
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| Then we have
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| {| cellpadding=5 style="margin-left: 2em;"
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| |<math>\cos\varphi = \frac{1 - t^2}{1 + t^2},</math>
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| |<math>\sin\varphi = \frac{2t}{1 + t^2},</math>
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| |-
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| |<math>\tan\varphi = \frac{2t}{1 - t^2},</math>
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| |<math>\cot\varphi = \frac{1 - t^2}{2t},</math>
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| |-
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| |<math>\sec\varphi = \frac{1 + t^2}{1 - t^2},</math>
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| |<math>\csc\varphi = \frac{1 + t^2}{2t},</math>
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| |}
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| and | |
| {| cellpadding=5 style="margin-left: 2em;"
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| |<math>e^{i \varphi} = \frac{1 + i t}{1 - i t},</math>
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| |<math>e^{-i \varphi} = \frac{1 - i t}{1 + i t}.</math>
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| |}
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| By eliminating phi between the directly above and the initial definition of ''t'', one arrives at the following useful relationship for the [[arctangent]] in terms of the [[natural logarithm]]
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| :<math>\arctan t = \frac{1}{2i}\ln\frac{1+it}{1-it}.</math>
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| In [[calculus]], the Weierstrass substitution is used to find antiderivatives of [[rational functions]] of sin(''φ'') and cos(''φ''). After setting
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| :<math>t=\tan\tfrac{1}{2}\varphi.</math>
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| This implies that
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| :<math>\varphi=2\arctan t, \, </math>
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| and therefore
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| :<math>d\varphi = {{2\,dt} \over {1 + t^2}}.</math>
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| ==Hyperbolic identities==
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| One can play an entirely analogous game with the [[hyperbolic function]]s. A point on (the right branch of) a [[hyperbola]] is given by (cosh ''θ'', sinh ''θ''). Projecting this onto ''y''-axis from the center (−1, 0) gives the following:
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| :<math>t = \tanh\tfrac{1}{2}\theta = \frac{\sinh\theta}{\cosh\theta+1} = \frac{\cosh\theta-1}{\sinh\theta}</math>
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| with the identities
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| {| cellpadding=5 style="margin-left: 2em;"
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| |<math>\cosh\theta = \frac{1 + t^2}{1 - t^2},</math>
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| |<math>\sinh\theta = \frac{2t}{1 - t^2},</math>
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| |-
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| |<math>\tanh\theta = \frac{2t}{1 + t^2},</math>
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| |<math>\coth\theta = \frac{1 + t^2}{2t},</math>
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| |-
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| |<math>\mathrm{sech}\,\theta = \frac{1 - t^2}{1 + t^2},</math>
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| |<math>\mathrm{csch}\,\theta = \frac{1 - t^2}{2t},</math>
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| |}
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| and | |
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| {| cellpadding=5 style="margin-left: 2em;"
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| |<math>e^{\theta} = \frac{1 + t}{1 - t},</math>
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| |<math>e^{-\theta} = \frac{1 - t}{1 + t}.</math>
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| |}
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| The use of this substitution for finding antiderivatives was introduced by [[Karl Weierstrass]].
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| Finding ''θ'' in terms of ''t'' leads to following relationship between the hyperbolic arctangent and the natural logarithm:
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| :<math>\operatorname{artanh} t = \frac{1}{2}\ln\frac{1+t}{1-t}.</math>
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| ==The Gudermannian function==
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| {{Main|Gudermannian function}}
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| 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
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| :<math>t = \tan\tfrac{1}{2}\varphi = \tanh\tfrac{1}{2}\theta</math>
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| then
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| :<math>\varphi = 2\tan^{-1}\tanh\tfrac{1}{2}\theta \equiv \mathrm{gd}\,\theta.</math>
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| The function gd(''θ'') is called the [[Gudermannian function]]. 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. | |
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| ==See also==
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| *[[List of trigonometric identities]]
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| *[[Half-side formula]]
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| ==External links==
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| <!-- * {{springer|title=Tangent formula|id=p/t092150}} Invalid link. -->
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| * [http://planetmath.org/encyclopedia/TangentOfHalvedAngle.html ''Tangent Of Halved Angle''] at [[Planetmath]]
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|
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| {{DEFAULTSORT:Tangent Half-Angle Formula}}
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| [[Category:Trigonometry]]
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| [[Category:Conic sections]]
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| [[Category:Mathematical identities]]
| |
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