Law of Tangents
Law of tangents formula (a−b)/(a+b) = tan[(A−B)/2] / tan[(A+B)/2] solves triangles when two sides and the included angle are known.
Alternative to cosine rule.
The Formula
The law of tangents provides another way to solve triangles when you know two sides and the included angle. It can be more numerically stable than the law of cosines for certain calculations.
Variables
| Symbol | Meaning |
|---|---|
| a, b | Two sides of the triangle |
| A, B | Angles opposite to sides a and b respectively |
Example 1
In a triangle: a = 8, b = 5, angle C = 60°. Find angles A and B.
A + B = 180° - 60° = 120°, so (A+B)/2 = 60°
(8-5)/(8+5) = tan((A-B)/2) / tan(60°)
3/13 = tan((A-B)/2) / 1.732
tan((A-B)/2) = 0.2308 × 1.732 = 0.3997
(A-B)/2 = 21.8°, so A-B = 43.6°
A = (120 + 43.6)/2 = 81.8°, B = (120 - 43.6)/2 = 38.2°
Example 2
a = 12, b = 12 (isosceles triangle), C = 40°
(12-12)/(12+12) = 0/24 = 0
tan((A-B)/2) = 0, so A-B = 0
A = B = (180 - 40)/2 = 70° (confirms the triangle is isosceles)
When to Use It
Use the law of tangents when:
- Solving triangles with two known sides and the included angle (SAS)
- Needing a direct formula without intermediate cosine calculations
- Working with surveying and navigation problems
- Checking results obtained from the law of sines or cosines
Key Notes
- Formula: (a − b) / (a + b) = tan½(A − B) / tan½(A + B): Relates two sides of a triangle to the tangent of half the difference and half the sum of their opposite angles. An alternative to the Law of Cosines for SAS triangles.
- Historical significance: Before calculators, multiplying and dividing logarithms was faster than extracting square roots. The Law of Tangents, using only tangent functions (easier in log tables), was preferred for SAS problems in navigation and surveying.
- When to use it: Given two sides and the included angle (SAS), the Law of Tangents can find the remaining angles. However, the Law of Cosines achieves the same result and is now the standard approach.
- Half-angle substitution avoids ambiguity: Using ½(A − B) and ½(A + B) ensures the arguments stay within a manageable range and avoids the ambiguous case that can arise in the Law of Sines (SSA configuration).
- Applications: Primarily of historical and educational interest today. Still appears in some celestial navigation texts, advanced trigonometry courses, and problems where logarithmic calculation techniques are explicitly required.