Consider any triangle (see figure). Define sin theta = h/c, show that the area of triangle is A_Delta = 1/2 (b*c) sintheta where b and c are any two sides of the traingle that make the angle theta?

Oct 9, 2016

See below.

Explanation:

It is well know that

${A}_{\Delta} = \frac{1}{2} b h = \frac{1}{2} b h \frac{c}{c} = \frac{1}{2} b c \left(\frac{h}{c}\right) = \frac{1}{2} b c \sin \theta$

the same regarding angle $\hat{A C B}$

${A}_{\Delta} = \frac{1}{2} b h = \frac{1}{2} b h \frac{a}{a} = \frac{1}{2} b a \left(\frac{h}{a}\right) = \frac{1}{2} b a \sin \beta$

Also can be established

$\frac{1}{2} b c \sin \theta = \frac{1}{2} b a \sin \beta \to \sin \frac{\theta}{a} = \sin \frac{\beta}{c}$