# Law of Sines

## Theorem

For any triangle $\triangle ABC$:

- $\dfrac a {\sin A} = \dfrac b {\sin B} = \dfrac c {\sin C}$

where $a$, $b$, and $c$ are the sides opposite $A$, $B$ and $C$ respectively.

## Proof 1

Construct the altitude from $B$.

It can be seen from the definition of sine that:

- $\sin A = \dfrac h c$ and $\sin C = \dfrac h a$

Thus:

- $h = c \sin A$ and $h = a \sin C$

This gives:

- $c \sin A = a \sin C$

So:

- $\dfrac a {\sin A} = \dfrac c {\sin C}$

Similarly, constructing the altitude from $A$ gives:

- $\dfrac b {\sin B} = \dfrac c {\sin C}$

$\blacksquare$

## Proof 2

Construct the circumcircle of $\triangle ABC$, let $O$ be the circumcenter and $R$ be the circumradius.

Construct $\triangle AOB$ and let $E$ be the foot of the altitude of $\triangle AOB$ from $O$.

By the Inscribed Angle Theorem:

- $\angle ACB = \dfrac{\angle AOB} 2$

From the definition of the circumcenter:

- $AO = BO$

From the definition of altitude and the fact that all right angles are congruent:

- $\angle AEO = \angle BEO$

Therefore from Pythagoras's Theorem:

- $AE = BE$

and then from Triangle Side-Side-Side Equality:

- $\angle AOE = \angle BOE$

Thus:

- $\angle AOE = \dfrac {\angle AOB} 2$

and so:

- $\angle ACB = \angle AOE$

Then by the definition of sine:

- $\sin C = \sin \left({\angle AOE}\right) = \dfrac {c / 2} R$

and so:

- $\dfrac c {\sin C} = 2 R$

Because the same argument holds for all three angles in the triangle:

- $\dfrac c {\sin C} = 2 R = \dfrac b {\sin B} = 2 R = \dfrac a {\sin A}$

Note that this proof also yields a useful extension of the law of sines:

- $\dfrac a {\sin A} = \dfrac b {\sin B} = \dfrac c {\sin C} = 2 R$

$\blacksquare$

## Also known as

This result is also known as the **sine law**, **sine rule** or **rule of sines**.

## Historical Note

The **Law of Sines** was documented by Nasir al-Din al-Tusi in his work *On the Sector Figure*, part of his five-volume *Kitāb al-Shakl al-Qattā* (*Book on the Complete Quadrilateral*).

## Sources

- 1968: Murray R. Spiegel:
*Mathematical Handbook of Formulas and Tables*... (previous) ... (next): $\S 5$: Trigonometric Functions: $5.92$ - 2008: Ian Stewart:
*Taming the Infinite*... (previous) ... (next): Chapter $5$: Eternal Triangles