Law of Sines
The Law of Sines (also called the Sine Rule) states that in any triangle, the ratio of the length of each side to the sine of the opposite angle is the same constant, equal to the diameter of the triangle's circumscribed circle. This law applies to all triangles, not just right triangles, making it a powerful tool for solving oblique triangles when two angles and a side (AAS or ASA) or two sides and a non-included angle (SSA) are known. It is fundamental in surveying, navigation, and geodesy.
Key Formula
a/sin(A) = b/sin(B) = c/sin(C) = 2R
LaTeX: \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R
| Symbol | Meaning | Unit |
|---|---|---|
| a, b, c | side lengths opposite to angles A, B, C respectively | length units |
| A, B, C | interior angles of the triangle | degrees or radians |
| R | circumradius (radius of circumscribed circle) | length units |
Worked Example
Problem
In triangle ABC, angle A = 40°, angle B = 75°, and side a = 12 cm. Find side b.
Solution
Step 1: Apply the Law of Sines: a/sin(A) = b/sin(B). Step 2: b = a × sin(B) / sin(A) = 12 × sin(75°) / sin(40°). Step 3: sin(75°) ≈ 0.9659, sin(40°) ≈ 0.6428. Step 4: b = 12 × 0.9659 / 0.6428 = 12 × 1.5027 = 18.03 cm.
Answer
Side b ≈ 18.0 cm (to 3 significant figures).
Law of Sines — When to Use It
| Given Information | Find | Case Name | Unique Solution? |
|---|---|---|---|
| 2 angles + 1 side (AAS) | 3rd angle and other 2 sides | AAS | Yes — always unique |
| 2 angles + included side (ASA) | Other 2 sides | ASA | Yes — always unique |
| 2 sides + non-included angle (SSA) | Other angles and side | SSA (ambiguous) | Sometimes 0, 1, or 2 solutions |
| 3 sides (SSS) | Not directly applicable | SSS | Use Law of Cosines instead |
| 2 sides + included angle (SAS) | Not directly applicable | SAS | Use Law of Cosines instead |
Interactive Tools
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Related Terms
Law of Cosines
The Law of Cosines generalises the Pythagorean theorem to any triangle, relating the square of one side to the sum of the squares of the other two sides minus twice their product times the cosine of the included angle. When the included angle is 90°, cos(90°) = 0 and the formula reduces to the Pythagorean theorem a² = b² + c². It is used to solve triangles when three sides (SSS) or two sides and the included angle (SAS) are known, and is fundamental in physics (vector addition), engineering, and 3D geometry.
Sine Function
The sine function is a fundamental trigonometric function defined for an angle θ in a right triangle as the ratio of the length of the side opposite the angle to the length of the hypotenuse, extended to all real numbers via the unit circle. It is a periodic function with period 2π, amplitude 1, and range [−1, 1], producing a smooth oscillating wave. Sine is essential in modelling wave phenomena including sound, light, alternating current, and simple harmonic motion.
Polar Coordinates
Polar coordinates are a two-dimensional coordinate system in which each point in the plane is specified by a radial distance r from a fixed origin (pole) and an angle θ measured from a fixed reference direction (polar axis), written as the ordered pair (r, θ). Unlike Cartesian coordinates that use perpendicular axes, polar coordinates are natural for describing curves with rotational symmetry such as circles, spirals, roses, and limaçons. They are widely used in physics (orbital mechanics, wave interference), engineering (antenna patterns), and complex number representation.
The rule was known to Islamic mathematicians al-Battani (c. 920 AD) and Abu Nasr Mansur (c. 1000 AD), who stated it for spherical triangles. The planar version was formalised by Nasir al-Din al-Tusi in the 13th century. The English name "Law of Sines" became standard in the 19th century textbook tradition.