Law of Tangents Calculator
Calculate law tangents instantly with our math tool. Shows detailed work, formulas used, and multiple solution methods.
Formula
(a - b) / (a + b) = tan((A - B) / 2) / tan((A + B) / 2)
Where a and b are two known sides of the triangle, A and B are the angles opposite those sides, and (A + B) / 2 is found from the angle sum property since A + B = 180 - C. This formula lets you find the unknown angles directly from two sides and the included angle.
Worked Examples
Example 1: SAS Triangle with Sides 7 and 5, Included Angle 60 degrees
Problem: Given a triangle with sides a = 7, b = 5, and included angle C = 60 degrees, find angles A and B using the Law of Tangents.
Solution: Half-sum = (A + B) / 2 = (180 - 60) / 2 = 60 degrees\ntan((A - B)/2) = ((7 - 5)/(7 + 5)) * tan(60)\n= (2/12) * 1.7321 = 0.2887\n(A - B)/2 = arctan(0.2887) = 16.1021 degrees\nA = 60 + 16.1021 = 76.1021 degrees\nB = 60 - 16.1021 = 43.8979 degrees
Result: Angle A = 76.1021 degrees, Angle B = 43.8979 degrees, Side c = 6.2450
Example 2: Isosceles Triangle with Equal Sides
Problem: Given a triangle with sides a = 10, b = 10, and included angle C = 40 degrees, verify that the Law of Tangents gives equal angles.
Solution: Half-sum = (180 - 40) / 2 = 70 degrees\ntan((A - B)/2) = ((10 - 10)/(10 + 10)) * tan(70)\n= 0 * 2.7475 = 0\n(A - B)/2 = arctan(0) = 0 degrees\nA = 70 + 0 = 70 degrees\nB = 70 - 0 = 70 degrees
Result: Angle A = 70.0000 degrees, Angle B = 70.0000 degrees (isosceles confirmed)
Frequently Asked Questions
What is the Law of Tangents and how does it differ from the Law of Sines?
The Law of Tangents is a trigonometric identity that relates the sides and angles of a triangle using tangent functions. It states that (a - b) / (a + b) equals tan((A - B) / 2) divided by tan((A + B) / 2). Unlike the Law of Sines, which requires knowing at least one side-angle pair, the Law of Tangents is particularly useful when you know two sides and the included angle (SAS case). It provides a direct way to find the difference between the two unknown angles without needing to use the Law of Cosines first. Historically, it was preferred over the Law of Cosines because it avoids the need for computing square roots and is better suited for logarithmic computation.
When should I use the Law of Tangents instead of the Law of Cosines?
The Law of Tangents is most useful in the SAS (Side-Angle-Side) case, where you know two sides and the included angle. While the Law of Cosines can also solve this case, the Law of Tangents directly gives you the two unknown angles without needing to find the third side first. This can be computationally more efficient and introduces fewer rounding errors in hand calculations. In the era before electronic calculators, the Law of Tangents was strongly preferred because it could be solved entirely with logarithmic tables, whereas the Law of Cosines required extracting a square root. Today, with digital computation, both methods yield equally accurate results, but the Law of Tangents remains valuable for theoretical understanding.
What is the formula derivation behind the Law of Tangents?
The Law of Tangents can be derived from the Law of Sines. Starting with a/sin(A) = b/sin(B), we get a/b = sin(A)/sin(B). Applying the sum-to-product identities, (a - b)/(a + b) = (sin(A) - sin(B))/(sin(A) + sin(B)). Using the identities sin(A) - sin(B) = 2 cos((A+B)/2) sin((A-B)/2) and sin(A) + sin(B) = 2 sin((A+B)/2) cos((A-B)/2), this simplifies to tan((A-B)/2) / tan((A+B)/2). This elegant derivation connects three fundamental trigonometric laws and demonstrates the deep interconnection between sine, cosine, and tangent relationships in triangles.
What are the limitations and edge cases of the Law of Tangents?
The Law of Tangents has several important limitations to consider. First, it requires that the included angle C is strictly between 0 and 180 degrees, exclusive. Second, the sum (a + b) must be positive, which is always true for valid triangles. Third, when a equals b, the formula simplifies trivially as discussed above. Fourth, the tangent function has asymptotes at 90 degrees, which can cause numerical issues when (A + B) / 2 is near 90 degrees (meaning C is near 0). Fifth, very obtuse included angles can lead to one of the unknown angles being very small, potentially causing precision loss. For most practical triangle configurations, the formula works reliably and produces accurate results.
How does the Law of Tangents relate to navigation and surveying?
Historically, the Law of Tangents was one of the most important tools in navigation and land surveying before modern GPS technology. Surveyors frequently encountered the SAS configuration when measuring distances and bearings between landmarks. They could measure two distances from a known point and the angle between them, then use the Law of Tangents to calculate the remaining angles and determine the complete triangle. Because the computation relied solely on multiplication, division, and tangent lookups, it could be performed efficiently using logarithmic tables and mechanical calculators. Even today, the principles behind the Law of Tangents inform geodetic calculations and triangulation methods used in geographic information systems.
Can the Law of Tangents be extended to non-Euclidean geometry?
Yes, the Law of Tangents has analogues in both spherical and hyperbolic geometry, though the formulas are modified to account for curvature. In spherical trigonometry, Napier analogies serve a similar purpose, relating sides and angles of spherical triangles using tangent functions. In hyperbolic geometry, the hyperbolic law of tangents uses hyperbolic tangent functions for the sides while keeping circular tangent functions for the angles. These extended forms are essential in astronomy, where celestial triangles are spherical, and in the theory of relativity, where hyperbolic geometry describes certain spacetime relationships. Understanding the Euclidean version provides a solid foundation for grasping these more advanced generalizations.