Law of tangents
In trigonometry, 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

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:

 

Let

 

so that

 

It follows that

 

Using the trigonometric identity, the factor formula for sines specifically

 

we get

As an alternative to using the identity for the sum or difference of two sines, one may cite the trigonometric identity

 

(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 

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, which implies a loss of significant digits.

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 says

History
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.