Using the Sine and Cosine

Jim Wilson


In any triangle, the altitude to a side is equal to the product of the sine of the angle subtending the altitude and a side from the angle to the vertex of the triangle. In this picutre, the altitude to side c is    b sin A    or  a sin B. (Setting these equal and rewriting as ratios leads to the demonstration of the Law of Sines)



In the triangle pictured     so 


Now we look for a substitution for sin A in terms of a, b, and c. It is readily (if messy) available from the Law of Cosines  --  i.e. the projection of sides   a   and   b  onto  c.

and substitution in the identity

Factor (easier than multiplying it out) to get

Rewrite with common denominators.


Factor and rearrange

Now where the semiperimeter s is defined by

the four expressions under the radical are 2s, 2(s - a), 2(s - b), and 2(s - c).   So


we have