# How can one use the logarithm function to define angles?

In dealing with the complex logarithm function, I read that the imaginary part of $\log w$, is also called the argument of $w$, $\operatorname{arg }w$, and it is interpreted geometrically as the angle between the positive real axis and the half line from the origin through the point $w$.

Using the interpretation, is there a standard way to define angles in triangles in the plane? I tried to draw a crude little picture.

So something like $\Im(\log b)-\Im(\log a)$ would give the angle between the two dashed lines emanating out to $b$ and $a$, but how would one define the angle between the legs $ab$ and $ac$? I would like to avoid angles greater than $\pi$, so I'm hoping there's a definition that returns angles in $[0,\pi]$. Thanks.

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What if you look at $c - a$ and $b - a$? – Dylan Moreland Feb 1 '12 at 23:17
@DylanMoreland Ok, so a good formula is $\Im(\log(c-a))-\Im(\log(b-a))$? Is there a good way to mod out by $\pi$ to get values in $[0,\pi]$? I don't want to accidentally mod out and get $\pi=0$. – Dedede Feb 1 '12 at 23:23

Try $\text{Im} \log \left(\frac{c-a}{b-a}\right)$, using the principal branch of the logarithm (the one with the imaginary part in $(-\pi, \pi]$). The result is positive if leg $ac$ is counterclockwise from leg $ab$ and negative if it is clockwise. If you don't care whether it's clockwise or counterclockwise, take $\left|\text{Im} \log \left(\frac{c-a}{b-a}\right)\right|$.