Start with ABC and circumscribe it about a circle with center at O and with radius equal to R units.
Draw in the diameter CJ and the chord BJ. <CBJ is a right angle, since it is inscribed in a semicircle.
Therefore in both figures,
In the first diagram <J=<A because they are both inscribed in the same arc of the circle.
In the second diagram <J=180-<A, because opposite angles of an inscribed quadrilateral are supplementary.
Remember that --> in both figures. Therefore, -->
The same procedure applied to the other angles of ABC yields
and
Combining these results we get the extended Law of Sines,
For a triangle ABC with circumradius R,