Law of Sines
TrigonometryThe 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, if you divide a side by the sine of the angle across from it, you always get the same number for all three sides.
Example
In a triangle with angle $A = 30^\circ$, $a = 5$, and angle $B = 70^\circ$: $b/\sin(70^\circ) = 5/\sin(30^\circ) = 5/0.5 = 10$. So $b = 10 \times \sin(70^\circ) \approx 9.40$.
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.
Definition
For a triangle with sides $a$, $b$, $c$ opposite angles $A$, $B$, $C$: $a/\sin(A) = b/\sin(B) = c/\sin(C)$. This common ratio equals the diameter of the circumscribed circle ($2R$). Use the law of sines for ASA, AAS, and SSA (ambiguous case).
Example
AAS: $A = 40^\circ$, $B = 75^\circ$, $a = 8$. Then $C = 180^\circ - 40^\circ - 75^\circ = 65^\circ$. $b = 8\sin(75^\circ)/\sin(40^\circ) = 8 \times 0.9659/0.6428 \approx 12.02$. $c = 8\sin(65^\circ)/\sin(40^\circ) \approx 11.26$.
Key Insight
The SSA (two sides, non-included angle) case can give $0$, $1$, or $2$ solutions, called the "ambiguous case." When given SSA, always check: if $a < b\sin(A)$, no triangle exists; if $a \ge b$, exactly one; otherwise, two triangles may be possible.
Definition
The law of sines $a/\sin(A) = 2R$ follows from the inscribed angle theorem: the side $a$ subtends a central angle $2A$ in the circumscribed circle of radius $R$, and $a = 2R\sin(A)$ by the definition of sine in that circle. 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
The circumradius $R = a/(2\sin(A))$. For a triangle with $a = 10$, $A = 30^\circ$: $R = 10/(2 \times 0.5) = 10$. The area formula $K = abc/(4R)$ follows directly from the law of sines.
Key Insight
The law of sines connects triangle geometry to circle geometry via the circumradius. This connection is the basis of Ptolemy's theorem and is fundamental in triangle centers and cyclic quadrilateral theory in classical geometry.