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

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