The Law of Sines is one of the first most important relationships between trigonometric functions and geometry.

In Functions, we learned about trigonometry in right triangles. Now, let’s look at a triangle with unknown angles:

There are a few relationships that we can derive from this triangle. For example:

$\sin{a} = \frac{A}{Y} \rightarrow A=Y\sin{a}$

$\sin{b} = \frac{B}{X} \rightarrow B=X\sin{b}$

Now, we have:

$A=X \sin{B}= Y\sin{a}$

Omitting the first equality and dividing both sides by XY gives:

$\frac{\sin{a}}{X} = \frac{\sin{b}}{Y} $

If we repeat this process by drawing any other altitude, we get the following:

$\frac{\sin{a}}{X} = \frac{\sin{b}}{Y}= \frac{\sin{c}}{Z} $

This is called the Law of Sines. Can you conduct the remainder of the proof on your own?