Law of sines

Fig.1 ∠BAC ≡ α, ∠ABC ≡ β, ∠ ACB ≡ γ

In trigonometry, the law of sines (also known as sine rule) relates in a triangle the sines of the three angles and the lengths of their opposite sides,

${\frac {a}{\sin \alpha }}={\frac {b}{\sin \beta }}={\frac {c}{\sin {\gamma }}}=d\quad {\hbox{or}}\quad {\frac {\sin \alpha }{a}}={\frac {\sin \beta }{b}}={\frac {\sin {\gamma }}{c}}={\frac {1}{d}},$

where d is the diameter of the circle circumscribing the triangle and the angles and the lengths of the sides are defined in Fig. 1 for an obtuse-angled triangle and in Fig. 2 for an acute-angled triangle. From the law of sines follows that the ratio of the sines of the angles of a triangle is equal to the ratio of the lengths of the opposite sides.

Fig. 2. Sine rule: sinα:sinβ:sinγ=a:b:c

The rule is useful to determine unknown angles and sides of a triangle when any of the following three elements is given:

• One side, the opposite angle, and one adjacent angle.
• One side and two adjacent angles
• Two sides and an angle not included by the sides.

The rule

$\alpha +\beta +\gamma =180^{\circ }$

may be useful in such a determination.

The use of the law of sines is complementary to the use of the law of cosines.

Proof

The easiest proof is purely geometric, not algebraic.

Lemma

Fig. 3. The angles α and α' share the chord a. The center of the circle is at C and its diameter is d.

In Fig. 3 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. To prove this we consider the angle α' that has the diameter of the circle as one of its sides, see Fig. 3. The two angles, α and α' share a segment of the circle (have the chord a in common). 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. A well-known theorem of plane geometry states that α = α' and it follows that the angle α has the same sine as α'.

Fig. 4

Proof of sine rule

From the lemma follows that the angles in Fig. 4 are

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

where d is the diameter of the circle. This proves the sine rule.

External link

Life lecture on Sine rule