Law of Tangents — Mathematics Knowledge System

Overview

The law of tangents relates the sum and difference of two sides of a triangle to the tangent of the half-sum and half-difference of their opposite angles. Historically it was valued (before calculators) because it let mathematicians solve the ambiguous SAS case of a triangle — two sides and the included angle — using only tangent tables, avoiding the awkward square roots of the law of cosines. Today it's mostly a curiosity and a good exercise in the sum-and-difference identities, since the law of cosines and law of sines handle triangle-solving more directly with modern computation.

Intuition

The law of tangents follows from the law of sines (a/sinA = b/sinB) by writing a and b as proportional to sinA and sinB, then applying sum-to-product identities to both a−b (∝ sinA−sinB) and a+b (∝ sinA+sinB). The sum-to-product identities convert each difference and sum of sines into a product involving sin and cos of the half-sum and half-difference of the angles; dividing the two results cancels the common factors and leaves a pure ratio of tangents.

Formal Definition

Definition

For a triangle with sides a, b opposite angles A, B respectively:

\frac{a-b}{a+b} = \frac{\tan\left(\frac{A-B}{2}\right)}{\tan\left(\frac{A+B}{2}\right)}

Law of tangents

Worked Examples

Compute the left side using the given sides.
$\frac{a-b}{a+b} = \frac{7-5}{7+5} = \frac{2}{12} = \frac{1}{6} \approx 0.1667$
Compute the right side using the angles A ≈ 76.10°, B ≈ 43.90°.
$\frac{\tan\left(\frac{76.10°-43.90°}{2}\right)}{\tan\left(\frac{76.10°+43.90°}{2}\right)} = \frac{\tan(16.10°)}{\tan(60°)} \approx \frac{0.2888}{1.7321} \approx 0.1667$

Answer: Both sides equal 1/6 ≈ 0.1667, confirming the law of tangents.

Practice Problems

Difficulty 4/10

A triangle has a = 10, b = 6, and A + B = 100°. Find tan((A−B)/2).

Difficulty 5/10

Verify the law of tangents for the 3-4-5 right triangle with the right angle at C, so A = arcsin(3/5) ≈ 36.87°, B ≈ 53.13°, a=3, b=4.

Difficulty 6/10

A surveyor measures two sides of a triangular plot as 120 m and 80 m with an included angle of 50°, so A+B = 130°. Use the law of tangents to set up the equation for (A−B)/2 (do not solve numerically).

Quiz

Summary

Law of tangents: (a−b)/(a+b) = tan((A−B)/2) / tan((A+B)/2), relating side differences to half-angle-difference tangents.
It follows from the law of sines via sum-to-product identities applied to a±b.
Historically used to solve the SAS triangle case with only tangent tables, avoiding the square root in the law of cosines.

References

WebsiteWikipedia — Law of tangents