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
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 alDin alTusi, who also discovered and proved the law of sines for plane triangles.
To prove the law of tangents we can start with the law of sines:
Let
so that
It follows that
Using the trigonometric identity
we get
As an alternative to using the identity for the sum or difference of two sines, one may cite the trigonometric identity
(see tangent halfangle formula).
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
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 alDin alTusi (120174), who also presented the law of sines for plane triangles in his five volume work Treatise on the Quadrilateral.^{[2]}^{[3]}
To prove the law of tangents we can start with the law of sines:
Let
so that
It follows that
Using the trigonometric identity
we get
As an alternative to using the identity for the sum or difference of two sines, one may cite the trigonometric identity
(see tangent halfangle formula).
