Area of Triangles

27.

K  =  area of triangle  =  \(\frac12\) ( base )( height )

Let the base  =  a

and the height  =  c sin B

Let's get  c  in terms of what we know using the Law of Sines

\(\frac{c}{\sin C}\,=\,\frac{a}{\sin A}\\~\\ c\,=\,\frac{a\sin C}{\sin A}\)

So the height  =  c sin B  =  (\(\frac{a\sin C}{\sin A}\)) sin B  =  \(\frac{a\sin B\sin C}{\sin A}\)

K  =  area of triangle  =  \(\frac12\) ( base )( height )  =  \(\frac12\) ( a )( \(\frac{a\sin B\sin C}{\sin A}\) )   =   \(\frac{a^2\sin B\sin C}{2\sin A}\)

\(K\,=\,\frac{a^2\sin B\sin C}{2\sin A}\)