To derive the equation for the law of sines, observe the shared perpendicular line by two angles on any triangle. The third angle can be included by repeating the process.

Take two of the angles and apply the definition of the sine function:

This gives the two equations which share the common length of D, representing the opposite side of the right triangle.

Solve both equations for D and set them equal to each other.

Simplifying, the result is the first part of the law of sines.

This same process can be repeated to relate to the third angle.

Combined, this gives us the equation for the law of sines.