Law of Tangents
Relate the sides and angles of any triangle using tangent.
An alternative to the law of cosines for solving triangles.
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