Here’s a straightforward application of arctangent that, a generation ago, used to be taught in a typical Precalculus class (or, as it was called back then, analytical geometry).
Find the smallest angle between the lines and .
This problem is almost equivalent to finding the angle between the vectors and . I use the caveat almost because the angle between two vectors could be between and , while the smallest angle between two lines must lie between and .
This smallest angle can be found using the formula
where and are the slopes of the two lines. In the present case,
Not surprisingly, we obtain the same answer that we obtained a couple of posts ago using arccosine. The following picture makes clear why .