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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. ...
, the law of tangents is a statement about the relationship between the tangents of two angles of a triangle and the lengths of the opposing sides. In Figure 1, , , and are the lengths of the three sides of the triangle, and , , and are the angles ''opposite'' those three respective sides. The law of tangents states that : \frac = \frac . The law of tangents, although not as commonly known as the law of sines or the law of cosines, is equivalent to the law of sines, and can be used in any case where two sides and the included angle, or two angles and a side, are known.


Proof

To prove the law of tangents one can start with the law of sines: : \frac = \frac. Let : d = \frac = \frac so that : a = d \sin\alpha \quad\text \quad b = d \sin\beta. It follows that : \frac = \frac = \frac . Using the trigonometric identity, the factor formula for sines specifically : \sin\alpha \pm \sin\beta = 2 \sin\tfrac12(\alpha \pm \beta) \, \cos\tfrac12( \alpha \mp \beta), we get :\frac = \frac = \frac \Bigg/ \frac = \frac . As an alternative to using the identity for the sum or difference of two sines, one may cite the trigonometric identity : \tan \tfrac12 (\alpha \pm \beta) = \frac (see tangent half-angle formula).


Application

The law of tangents can be used to compute the missing side and angles of a triangle in which two sides and and the enclosed angle are given. From : \tan\tfrac12(\alpha-\beta) = \frac \tan\tfrac12(\alpha+\beta) = \frac \cot\tfrac12\gamma one can compute ; together with this yields and ; the remaining side can then be computed using the law of sines. In the time before electronic calculators were available, this method was preferable to an application of the law of cosines , as this latter law necessitated an additional lookup in a logarithm table, in order to compute the square root. In modern times the law of tangents may have better numerical properties than the law of cosines: If is small, and and are almost equal, then an application of the law of cosines leads to a subtraction of almost equal values, incurring
catastrophic cancellation In numerical analysis, catastrophic cancellation is the phenomenon that subtracting good approximations to two nearby numbers may yield a very bad approximation to the difference of the original numbers. For example, if there are two studs, one L_ ...
.


Spherical version

On a sphere of unit radius, the sides of the triangle are arcs of
great circle In mathematics, a great circle or orthodrome is the circular intersection of a sphere and a plane passing through the sphere's center point. Any arc of a great circle is a geodesic of the sphere, so that great circles in spherical geometry ...
s. Accordingly, their lengths can be expressed in radians or any other units of angular measure. Let , , be the angles at the three vertices of the triangle and let , , be the respective lengths of the opposite sides. The spherical law of tangents saysDaniel Zwillinger, ''CRC Standard Mathematical Tables and Formulae'', 32nd Edition, CRC Press, 2011, page 219. : \frac = \frac .


History

The law of tangents for planar triangles was described in the 11th century by Ibn Muʿādh al-Jayyānī. The law of tangents for spherical triangles was described in the 13th century by Persian mathematician
Nasir al-Din al-Tusi Muhammad ibn Muhammad ibn al-Hasan al-Tūsī ( fa, محمد ابن محمد ابن حسن طوسی 18 February 1201 – 26 June 1274), better known as Nasir al-Din al-Tusi ( fa, نصیر الدین طوسی, links=no; or simply Tusi in the West ...
(1201–1274), who also presented the law of sines for plane triangles in his five-volume work ''Treatise on the Quadrilateral''.


See also

* Law of sines * Law of cosines * Law of cotangents * Mollweide's formula * Half-side formula * Tangent half-angle formula


Notes

{{reflist Trigonometry Articles containing proofs Theorems about triangles