Law of Sines

Topics within Trigonometry

LAW OF SINES

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?