Law of Tangent

In trigonometry , the law of tangent is a statement about the relationship between the tangents of two angles of a triangle and the lengths of opposing sides.

In Figure 1, a , b , and c are the lengths of the three sides of a triangle, and α , β , and are angles opposite those three corresponding sides. The law of tangents states that

{\displaystyle {\frac {ab}{a+b}}={\frac {\tan {\tfrac {1}{2}}(\alpha -\beta )}{\tan {\tfrac {1}{ 2}}(\alpha +\beta )}}.}

The law of tangent, although not commonly known as the law of sine or the law of cosines , is equivalent to the law of sine , and can be used in any case where two sides and one angle are involved, or two angles and one sides are known.


One can start with the law of tangents to prove the law of chords :

{\frac {a}{\sin \alpha }}={\frac {b}{\sin \beta }}.


{\displaystyle d={\frac {a}{\sin \alpha }}={\frac {b}{\sin \beta }}}

So that

{\displaystyle a=d\sin \alpha \quad {\text{and}}\quad b=d\sin \beta .}

it follows that

{\displaystyle {\frac {a-b}{a+b}}={\frac {d\sin \alpha -d\sin \beta }{d\sin \alpha +d\sin \beta }}={\frac {\sin \alpha -\sin \beta }{\sin \alpha +\sin \beta }}.}

Using trigonometric identities , especially factor formulas for sine

{\displaystyle \sin \alpha \pm \sin \beta =2\sin {\tfrac {1}{2}}(\alpha \pm \beta )\,\cos {\tfrac {1}{2}}(\alpha \mp \beta ),}

we find

{\displaystyle {\frac {a-b}{a+b}}={\frac {2\sin {\tfrac {1}{2}}(\alpha -\beta )\,\cos {\tfrac {1}{2}}(\alpha +\beta )}{2\sin {\tfrac {1}{2}}(\alpha +\beta )\,\cos {\tfrac {1}{2}}(\alpha -\beta )}}={\frac {\sin {\tfrac {1}{2}}(\alpha -\beta )}{\cos {\tfrac {1}{2}}(\alpha -\beta )}}{\Bigg /}{\frac {\sin {\tfrac {1}{2}}(\alpha +\beta )}{\cos {\tfrac {1}{2}}(\alpha +\beta )}}={\frac {\tan {\tfrac {1}{2}}(\alpha -\beta )}{\tan {\tfrac {1}{2}}(\alpha +\beta )}}.}

As an alternative to using the identity for the sum or difference of two chords, one can cite the trigonometric identity.

{\displaystyle \tan {\tfrac {1}{2}}(\alpha \pm \beta )={\frac {\sin \alpha \pm \sin \beta }{\cos \alpha +\cos \beta } }}

( see tangent half-angle formula ).


The law of tangent can be used to calculate the missing side and angles of a triangle in which both sides a and b and the adjacent angle are given. From

{\displaystyle \tan {\tfrac {1}{2}}(\alpha -\beta )={\frac {a-b}{a+b}}\tan {\tfrac {1}{2}}(\alpha +\beta )={\frac {a-b}{a+b}}\cot {\tfrac {1}{2}}\gamma }

One can calculate α – β ; Together with α + β = 180° – this yields alpha and β ; The remaining side c can then be calculate 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 c = 2 + b 2 – 2 Now because , this latter law required an additional look into the logarithm table as In order to calculate the square root. In modern times the law of tangent may be better numerical properties than the law of cosines: if is small, and a subtraction of values ​​a and b approximately equal to the mean, then an application of the law of cosine leads to an important Points loss .

Circular Version

On a sphere of unit radius, the sides of the triangle are arcs of larger circles . Accordingly, their length can be expressed in radians or in any other unit of angular measurement. Let A , B , C be the angles at the three vertices of the triangle and let a , b , c be the respective lengths of the opposite sides. The circular law of tangents states [2]

{\displaystyle {\frac {\tan {\tfrac {1}{2}}(A-B)}{\tan {\tfrac {1}{2}}(A+B)}}={\frac {\tan {\tfrac {1}{2}}(a-b)}{\tan {\tfrac {1}{2}}(a+b)}}.}


The law of tangent for spherical triangles was described in the 13th century by the Persian mathematician Nasir al-Din al-Tusi (1201–1274), who in his five-volume work Treatise on the Quadrilaterals of Sine for Plane Triangles Rules were also introduced.