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Law of tangents

 
Sci-Tech Dictionary: law of tangents
(′lö əv ′tan·jəns)

(mathematics) Given a triangle with angles A, B, and C and sides a, b, c opposite these angles respectively: (a - b)/(a + b) = [tan ½(A - B)]/[tan ½(A + B)].


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Fig. 1 - A triangle.
Trigonometry

History
Usage
Functions
Inverse functions
Further reading

Reference

List of identities
Exact constants
Generating trigonometric tables

Euclidean theory

Law of sines
Law of cosines
Law of tangents
Pythagorean theorem

Calculus

Trigonometric substitution
Integrals of functions
Derivatives of functions
Integrals of inverses

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}

(see tangent half-angle formula).

See also

Notes

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

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