Please tell me why this statement is true for triangles with an obtuse angle, acute angle and the same two sides a and b:
Area_acute = a*b*sin(α)/2
Area_obtuse = a*b*sin(180-α)/2
I understand that the sine of an acute angle is the ratio of the opposing cathetus to the hypotenuse, but I cannot intuitively understand what the sine of an obtuse angle is. I also understand what it has to do with the trigonometric circle, but I don't see the connection to area in geometry here.
View $a$ as the base and $b\sin(\alpha)$ as the height.
Since the area of the triangle is $\frac12 ah$, we have the formula $\frac12 ab\sin(\alpha)$.
Notice that $b\sin(180^\circ - \alpha) = b\sin \alpha$.