Law of Sines
The law of sines states that in any triangle, the ratio of each side to the sine of its opposite angle is constant.
Formula
\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}
Definition
The law of sines says that in any triangle, dividing a side by the sine of the angle opposite it always gives the same value for all three sides: $a/\sin(A) = b/\sin(B) = c/\sin(C)$. This common ratio equals the diameter of the circumscribed circle, $2R$, which follows from the inscribed angle theorem: side $a$ subtends a central angle $2A$ in the circumscribed circle of radius $R$, giving $a = 2R\sin(A)$. The law holds for all triangles and generalizes to spherical trigonometry as $\sin(a)/\sin(A) = \sin(b)/\sin(B) = \sin(c)/\sin(C)$ on a unit sphere.
Example
In a triangle with angle $A = 30^\circ$, $a = 5$, and angle $B = 70^\circ$: $b/\sin(70^\circ) = 5/\sin(30^\circ) = 10$, so $b \approx 9.40$. For AAS with $A = 40^\circ$, $B = 75^\circ$, $a = 8$: $C = 65^\circ$, $b = 8\sin(75^\circ)/\sin(40^\circ) \approx 12.02$. The circumradius $R = a/(2\sin(A))$, and the area formula $K = abc/(4R)$ follows directly from the law of sines.
Key Insight
The law of sines is a lifesaver for triangles that are not right triangles: as long as you know two angles and one side, or two sides and an angle opposite one, you can solve the triangle, though the SSA case can give $0$, $1$, or $2$ solutions (the "ambiguous case"). The law connects triangle geometry to circle geometry via the circumradius, the basis of Ptolemy's theorem and fundamental in triangle centers and cyclic quadrilateral theory in classical geometry.