Sine rule

Fig. 1. Sine rule: sinα:sinβ:sinγ=a:b:c

In trigonometry, the sine rule states that the ratio of the sines of the angles of a triangle is equal to the ratio of the lengths of the opposite sides, see Fig.1.

${\frac {\sin \alpha }{\sin {\beta }}}={\frac {a}{b}},\qquad {\frac {\sin \beta }{\sin {\gamma }}}={\frac {b}{c}}.$

Proof

The easiest proof is purely geometric.

Lemma

Fig. 2. The angles α and β share the chord a. The center of the circle is at C and its diameter is d.

In Fig. 2 the angle β satisfies,

$\sin \beta ={\frac {a}{d}}.$

Indeed, in Fig. 2 we see two angles, α and β, that share a segment of the circle (have the chord a in common). By a well-known theorem of plane geometry it follows that the angles are equal. The angle α, having the diameter of the circle d as one of its sides, has as opposite angle a right angle. Hence sin(α) = a/d, the length of chord a divided by the diameter d. It follows that the angle β, with a corner on the circumference of the same circle as α, but other than that arbitrary, has the same sine as α.

Fig. 3

Proof of sine rule

From the lemma follows that the angles in Fig. 3 are

$\sin \alpha ={\frac {a}{d}},\quad \sin \beta ={\frac {b}{d}},\quad \sin \alpha ={\frac {c}{d}},$

where d is the diameter of the circle. From this result the sine rule follows.