TheoremProved

Law of sines

Statement

In any triangle $ABC$ with side lengths $a$ , $b$ , $c$ opposite the interior angles $A$ , $B$ , $C$ and circumradius $R$ , $\dfrac{a}{\sin A} = \dfrac{b}{\sin B} = \dfrac{c}{\sin C} = 2R$ .

Why is it true?

Inscribe the triangle $ABC$ in its circumcircle of diameter $2R$ . Each side, such as $a = BC$ , is a chord of that circle subtending the inscribed angle $A$ . Rotating the vertex $A$ along the circle to the endpoint of a diameter turns the triangle into a right triangle whose hypotenuse is the diameter $2R$ , immediately revealing that the chord length $a$ is $2R\sin A$ — and since all three sides sit inside the same circle of diameter $2R$ , the ratio $\dfrac{a}{\sin A}$ must equal $2R$ for every side.

Proof sketch

Draw the circumcircle of $\triangle ABC$ with radius $R$ , and let $BD$ be a diameter, so $BD = 2R$ and $\angle BCD = 90^\circ$ . When $A$ is acute, the inscribed angles $\angle A$ and $\angle BDC$ subtend the same arc $BC$ and are equal, so in the right triangle $\triangle BDC$ we have $\sin A = \sin(\angle BDC) = \dfrac{a}{2R}$ , or $\dfrac{a}{\sin A} = 2R$ (if $A$ is obtuse, $\angle BDC = 180^\circ - A$ has the same sine). Repeating for $b$ and $c$ yields $\dfrac{a}{\sin A} = \dfrac{b}{\sin B} = \dfrac{c}{\sin C} = 2R$ .

Topics that use this theorem

Trigonometric relations in triangles Trigonometry

Related theorems

Law of cosines Inscribed angle theorem Ptolemy's theorem

Step-by-step proofs

No step-by-step proof yet for this theorem.

References

Glen Van Brummelen (2009). The Mathematics of the Heavens and the Earth: The Early History of Trigonometry
H. S. M. Coxeter, S. L. Greitzer (1967). Geometry Revisited