Details

Here is a derivation of the formula. Draw a perpendicular from the point to the side at . The triangle is now divided into two right triangles and . Let . Let the lengths of the two segments of be and .

Then, by trigonometry, , , . (*)

The area of the triangle is

(the base is and the height is )

()

(substituting from (*))

(factoring out )

(using the expansion of the sine of a sum in reverse)

. (adding the two angles at )