Law of tangents

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

: $frac{a-b}{a+b} = frac{ an [frac{1}{2}(alpha-eta)] }{ an [frac{1}{2}(alpha+eta)] }.$

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.

Proof

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

: $frac{a}{sin{alpha = frac{b}{sin{eta$

We can say there's a "q" that equals to,

: $q = frac{a}{sin{alpha = frac{b}{sin{eta$

With this identity, we can solve for both "b" and "a" as such,

: $a = q sin{alpha} ext{ and }b = q sin{eta}$

Substituting in the original equation for "a" and "b" we get,

: $frac{a-b}{a+b} = frac{q sin alpha -qsineta}{qsinalpha+qsineta} = frac{ sin alpha -sineta}{sinalpha+sineta}$

Cancelling the "q"'s, and using the trigonometric identity

: $sin(alpha) + sin(eta) = 2 sinleft( frac{alpha + eta}{2}
ight) cosleft( frac{alpha - eta}{2}
ight) ;$

for $scriptstyle{x,=,alpha}$ and $scriptstyle{y,=,pmeta}$ we get

: $frac{a-b}{a+b} = frac{ 2 sinleft( frac{alpha -eta}{2}
ight) cosleft( frac{alpha+eta}{2}
ight) }{ 2 sinleft( frac{alpha +eta}{2}
ight) cosleft( frac{alpha-eta}{2}
ight)} = frac{ an [frac{1}{2}(alpha-eta)] }{ an [frac{1}{2}(alpha+eta)] }. qquadlacksquare$

Alternatively, one may cite the trigonometric identity

: $anleft( frac{alpha pm eta}{2}
ight) = frac{sinalpha pm sineta}{cosalpha + coseta}$

(see tangent half-angle formula).

ee also

*Law of sines
*Law of cosines

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