# Law of Sines

## Theorem

Let $\triangle ABC$ be a triangle whose sides $a, b, c$ are such that $a$ is opposite $A$, $b$ is opposite $B$ and $c$ is opposite $C$.

Then:

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

where $R$ is the circumradius of $\triangle ABC$.

## Proof 1

Construct the altitude from $B$.

From the definition of sine:

- $\sin A = \dfrac h c$ 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:

- $\sin C = \map \sin {\angle AOE} = \dfrac {c / 2} R$

and so:

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

The same argument holds for all three angles in the triangle, and so:

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

$\blacksquare$

## Proof 3

### Acute Case

Let $\triangle ABC$ be acute.

Construct the circumcircle of $\triangle ABC$.

Let its radius be $R$.

Construct its diameter $BX$ through $B$.

By Thales' Theorem, $\angle BAX$ is a right angle.

From Angles in Same Segment of Circle are Equal:

- $\angle AXB = \angle ACB$

Then:

\(\ds \sin \angle AXB\) | \(=\) | \(\ds \dfrac {AB} {BX}\) | Definition of Sine of Angle | |||||||||||

\(\ds \leadsto \ \ \) | \(\ds \sin \angle ACB\) | \(=\) | \(\ds \dfrac c {2 R}\) | |||||||||||

\(\ds \leadsto \ \ \) | \(\ds 2 R\) | \(=\) | \(\ds \dfrac c {\sin C}\) |

The same construction can be applied to each of the remaining vertices of $\triangle ABC$.

Hence the result.

$\Box$

Let $\triangle ABC$ be obtuse.

As for the acute case, construct the circumcircle of $\triangle ABC$.

Let its radius be $R$.

Construct its diameter $BX$ through $B$.

By Thales' Theorem, $\angle BAX$ is a right angle.

We note that $\Box ABXC$ is a cyclic quadrilateral.

From Opposite Angles of Cyclic Quadrilateral sum to Two Right Angles:

- $\angle BAC = 108 \degrees - A$

Hence using a similar argument to the acute case:

\(\ds a\) | \(=\) | \(\ds 2 R \map \sin {180 \degrees - A}\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds 2 R \sin A\) |

and the result follows.

$\blacksquare$

## Also presented as

Some sources do not include the relation with the circumradius, but instead merely present:

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

## Also known as

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

## Also see

## 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

- 1953: L. Harwood Clarke:
*A Note Book in Pure Mathematics*... (previous) ... (next): $\text V$. Trigonometry: Formulae $(8)$ - 1968: Murray R. Spiegel:
*Mathematical Handbook of Formulas and Tables*... (previous) ... (next): $\S 5$: Trigonometric Functions: $5.92$ - 1989: Ephraim J. Borowski and Jonathan M. Borwein:
*Dictionary of Mathematics*... (previous) ... (next):**sine law**or**sine rule** - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**sine rule (law of sines)**:**1.** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**sine rule (law of sines)**:**1.** - 2008: Ian Stewart:
*Taming the Infinite*... (previous) ... (next): Chapter $5$: Eternal Triangles: Logarithms - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next):**sine rule** - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next):**triangle**(ii)