I've come across another question I'm having difficulty with.

The fixed points $\displaystyle A$ and $\displaystyle B$ represent the complex numbers $\displaystyle a$ and $\displaystyle b$ in an Argand diagram with origin $\displaystyle O$.

By writing $\displaystyle a = |a|e^{i\alpha}$ and $\displaystyle b = |b|e^{i\beta}$, show that $\displaystyle |Im(ab)| = 2\Delta$ where $\displaystyle \Delta$ is the area of triangle $\displaystyle OAB$.

My working takes me through these steps

$\displaystyle ab = |a||b|e^{i\alpha + i\beta} $

$\displaystyle ab = |a||b|(cos(\alpha + \beta) +isin(\alpha + \beta))$

Therefore $\displaystyle Im(ab) = |a||b|sin(\alpha + \beta)$

However from playing about with diagrams the area of the triangle $\displaystyle OAB$ looks like it should be $\displaystyle \Delta = \tfrac{1}{2}|a||b|sin(\alpha - \beta)$.

Does anyone have any thoughts on this?