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1. Law of Tangents By Group 4 ES11KA1
2. In trigonometry, the law of tangents 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 a - b tan ½(α – β) = a + b tan ½(α + β)
3. 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. α b c γ β a
4. Derivation To prove the law of tangents we can start with the law of sines: a b = sin α sin β Let a b = = d sin α sin β It follows that a - b dsinα – dsinβ = a + b dsinα + dsinβ sinα – sinβ = sinα + sinβ
5. Derivation Using the trigonometric identity = sin α ± sin β We get 2sin ½ (α ± β)cos(α β) ± a - b 2sin ½ (α - β)cos(α + β) = a + b 2sin ½ (α + β)cos(α - β) tan ½(α – β) = tan ½(α + β)
6. Derivation As an alternative to using the identity for the sum or difference of two sines, one may cite the trigonometric identity. sin α ± sin β = tan ½ (α ± β) cosα +cosβ
7. Reference http://en.wikipedia.org/