# Sine rule

## Statement

The ratio between a triangle's side length and the sine of its angle is the same for all sides:

$$\frac{a}{\sin(\alpha)} = \frac{b}{\sin(\beta)} = \frac{c}{\sin(\gamma)}$$

## Proof

Construct altitudes $h$ and $k$ like the image below:

### Altitude $h$

Write the following equation from the definition of sine:

$$\sin(\alpha) = \frac{h}{b}$$

Flip the fraction and multiply both sides by $a$.

$$\frac{1}{\sin(\alpha)} = \frac{b}{h}$$

$$\frac{a}{\sin(\alpha)} = \frac{ab}{h}$$

Now write another equation using the definition of sine:

$$\sin(\beta) = \frac{h}{a}$$

Flip the fraction and multiply both sides by $b$.

$$\frac{1}{\sin(\beta)} = \frac{a}{h}$$

$$\frac{b}{\sin(\beta)} = \frac{ab}{h}$$

Note that both $\frac{a}{\sin(\alpha)}$ and $\frac{b}{\sin(\beta)}$ are equal to $\frac{ab}{h}$, giving:

$$\frac{a}{\sin(\alpha)} = \frac{b}{\sin(\beta)}$$

### Altitude $k$

Write the following equation from the definition of sine:

$$\sin(\alpha) = \frac{k}{c}$$

Flip the fraction and multiply both sides by $a$.

$$\frac{1}{\sin(\alpha)} = \frac{c}{k}$$

$$\frac{a}{\sin(\alpha)} = \frac{ac}{k}$$

Now write another equation using the definition of sine:

$$\sin(\gamma) = \frac{k}{a}$$

Flip the fraction and multiply both sides by $c$.

$$\frac{1}{\sin(\gamma)} = \frac{a}{k}$$

$$\frac{c}{\sin(\gamma)} = \frac{ac}{h}$$

Note that both $\frac{a}{\sin(\alpha)}$ and $\frac{c}{\sin(\gamma)}$ are equal to $\frac{ac}{h}$, giving:

$$\frac{a}{\sin(\alpha)} = \frac{c}{\sin(\gamma)}$$

## Conclusion

Since $\frac{a}{\sin(\alpha)} = \frac{b}{\sin(\beta)}$ and $\frac{a}{\sin(\alpha)} = \frac{c}{\sin(\gamma)}$:

$$\frac{a}{\sin(\alpha)} = \frac{b}{\sin(\beta)} = \frac{c}{\sin(\gamma)}$$