Law of tangents

From Wikipedia, the free encyclopedia

Fig. 1 - A triangle.
Fig. 1 - A triangle.

In trigonometry, the law of tangents is a statement about arbitrary triangles in the plane.

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

\frac{a-b}{a+b} = \frac{\tan[\frac{1}{2}(\alpha-\beta)]}{\tan[\frac{1}{2}(\alpha+\beta)]}.

The law of tangents, although not as commonly known as the law of sines or the law of cosines, is just as useful, and can be used in any case where you know either two sides and an angle, or two angles and a side.

To prove the law of tangents we start with the law of sines:

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

This implies that

a \,sin \beta = b \,sin \alpha

Using trigonometric identities we can say

{\sin{\alpha}} = {\sin{\alpha + \beta \over 2}\cos{\alpha - \beta \over 2}} + {\cos{\alpha + \beta \over 2}\sin{\alpha - \beta \over 2}}
{\sin{\beta}} = {\sin{\alpha + \beta \over 2}\cos{\alpha - \beta \over 2}} - {\cos{\alpha + \beta \over 2}\sin{\alpha - \beta \over 2}}

Thus we have

{a}\left({\sin{\alpha + \beta \over 2}\cos{\alpha - \beta \over 2}} - {\cos{\alpha + \beta \over 2}\sin{\alpha - \beta \over 2}}\right) =
= {b}\left({\sin{\alpha + \beta \over 2}\cos{\alpha - \beta \over 2}} + {\cos{\alpha + \beta \over 2}\sin{\alpha - \beta \over 2}}\right).

Dividing both sides by {\cos{\alpha - \beta \over 2}\cos{\alpha + \beta \over 2}} we attain

{a}\left({\tan{\alpha + \beta \over 2} - \tan{\alpha - \beta \over 2}}\right) = {b}\left(\tan{\alpha + \beta \over 2} + \tan{\alpha - \beta \over 2}\right).

Cross-dividing shows that

{a \over b} = {\tan{\alpha + \beta \over 2} + \tan{\alpha - \beta \over 2} \over \tan{\alpha + \beta \over 2} - \tan{\alpha - \beta \over 2}}

Hence the law of tangents:

{{a - b} \over {a + b}} = {{\tan{\alpha - \beta \over 2}} \over {\tan{\alpha + \beta \over 2}}}.

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