Law of tangents

Fig. 1 - A triangle.

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

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 two sides and an angle, or two angles and a side are known.

The law of tangents used for spherical triangles was discovered and proven by the 13th century Persian mathematician, Nasir al-Din al-Tusi, who also discovered and proved the law of sines for plane triangles.

Proof

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

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

Let

$d = \frac{a}{\sin\alpha} = \frac{b}{\sin\beta},$

so that

$a = d \sin\alpha \text{ and }b = d \sin\beta. \,$

It follows that

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

Using the trigonometric identity

$\sin(\alpha) \pm \sin(\beta) = 2 \sin\left( \frac{\alpha \pm \beta}{2} \right) \cos\left( \frac{\alpha \mp \beta}{2} \right), \;$

we get

$\frac{a-b}{a+b} = \frac{ 2 \sin\left( \frac{\alpha -\beta}{2} \right) \cos\left( \frac{\alpha+\beta}{2}\right) }{ 2 \sin\left( \frac{\alpha +\beta}{2} \right) \cos\left( \frac{\alpha-\beta}{2}\right)} = \frac{\tan[\frac{1}{2}(\alpha-\beta)]}{\tan[\frac{1}{2}(\alpha+\beta)]}. \qquad\blacksquare$

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

$\tan\left( \frac{\alpha \pm \beta}{2} \right) = \frac{\sin\alpha \pm \sin\beta}{\cos\alpha + \cos\beta}$

(see tangent half-angle formula).

See also

• Law of sines
• Law of cosines
• Mollweide's formula

Notes

ar:قانون الظل ca:Teorema de la tangent cs:Tangentová věta eo:Leĝo de tangentoj nl:Tangensregel km:ទ្រឹស្តីបទតង់សង់ pl:Twierdzenie tangensów sl:Tangensni izrek ta:தாஞ்சன்களின் விதி uk:Теорема тангенсів