Sine rule

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

In trigonometry, the sine rule (also known as Law of Sines) 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. Equivalently,

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

where d is the diameter of the circle circumscribing the triangle.

Proof

The easiest proof is purely geometric.

Lemma

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

In Fig. 2 the arbitrary angle α' satisfies,

$\sin \alpha '={\frac {a}{d}},$

where d is the diameter of the circle and a is the chord opposite α'. This follows because the two angles, α and α', in Fig. 2 share a segment of the circle (have the chord a in common). By a well-known theorem of plane geometry it follows that the two 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 α' 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.

External link

Life lecture on Sine Law