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 A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, any three points, when non- colline ...

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

tangent In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. Mo ...

s states that :

\frac
= \frac
       .

The law of tangents, although not as commonly known as the

law of sines In trigonometry, the law of sines, sine law, sine formula, or sine rule is an equation relating the lengths of the sides of any triangle to the sines of its angles. According to the law, \frac \,=\, \frac \,=\, \frac \,=\, 2R, where , and ar ...

or 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 ...

, 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

: :

\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 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 ...

, 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 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 ...

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

. In the time before electronic calculators were available, this method was preferable to an application of the

, as this latter law necessitated an additional lookup in a

logarithm table In mathematics, the common logarithm is the logarithm with base 10. It is also known as the decadic logarithm and as the decimal logarithm, named after its base, or Briggsian logarithm, after Henry Briggs, an English mathematician who pioneered i ...

, 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.

Spherical version

On a sphere of unit radius, the sides of the triangle are arcs of great circles. 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 (1201–1274), who also presented the law of sines for plane triangles in his five-volume work ''Treatise on the Quadrilateral''.

Notes

{{reflist Trigonometry Articles containing proofs Theorems about triangles

Proof

Application

Spherical version

History

See also

Notes