Law of tangents: Wikis

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Fig. 1 - A triangle.
 Trigonometry Reference Laws & Theorems Law of sines Law of cosines Law of tangents Pythagorean theorem Calculus

In trigonometry, the law of tangents[1] 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 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

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

Notes

1. ^ See Eli Maor, Trigonometric Delights, Princeton University Press, 2002.

In trigonometry, the law of tangents[1] is a statement about the relationship between the tangents of two angles of a triangle and the lengths of the opposite sides.

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

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

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 for spherical triangles was described in the 13th century by Persian mathematician, Nasir al-Din al-Tusi (1201-74), who also presented the law of sines for plane triangles in his five volume work Treatise on the Quadrilateral.[2][3]

Proof

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

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

Let

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

so that

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

It follows that

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

Using the trigonometric identity

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

we get

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

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

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

Notes

1. ^ See Eli Maor, Trigonometric Delights, Princeton University Press, 2002.
2. ^ Marie-Thérèse Debarnot (1996). "Trigonometry". In Rushdī Rāshid, Régis Morelon. Encyclopedia of the history of Arabic science, Volume 2. Routledge. p. 182. ISBN 0415124115.
3. ^ Q. Mushtaq, JL Berggren (2002). "Trigonometry". In C. E. Bosworth, M.S.Asimov. History of Civilizations of Central Asia, Volume 4, Part 2. Motilal Banarsidass Publ.. p. 190. ISBN 8120815963.