Why the Law of Sines Holds

Draw the height h: a line from C straight down to AB, at a right angle. It splits the triangle into two right triangles. Both right triangles have the same height h. This is why the law works.

In the left triangle, h is opposite A and the hypotenuse is b. So $h = b\cdot \sin\left(A\right)$. In the right triangle, $h = a\cdot \sin\left(B\right)$. It is the same h, so $b\cdot \sin\left(A\right) = a\cdot \sin\left(B\right)$. Divide both sides by sin A and by sin B, and you get the law. If B is more than 90°, the same idea works, because B and $180^{\circ}-B$ have the same sine.